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ADVANCED CALCULUS
A TEXT UPON SELECT PARTS OF DIFFERENTIAL CAL-
CULUS, DIFFERENTIAL EQUATIONS, INTEGRAL
CALCULUS, THEORY OF FUNCTIONS,
WITH NUMEROUS EXERCISES
BY
EDWIN BIDWELL WILSON, Ph.D.
PROFB880R OF MATHEMATICAL PHYSICS IN THE MASSACHUSETTS
INSTITUTE or TECHNOLOGY
GINN AND COMPANY
BOSTON • NKW YORK • CHICAOO • LONDON
ATLANTA • DALLAS • COLUMBUS • SAN FUANCISCO
COPTBIOHT, 1911, 1912, BT
EDWIN BIDWELL WILSON
ALL BIOHT8 RK8BBVKD
nUKTKO IX THB CyiTKD 8TATK8 OF AMERICA
187.7
303
CINN AND COMPANY . PRO.
rairroRS • boston . uajl
PREFACE
It 18 probable that almost every teacher of advanced calculus feels the
need of a text suited to present conditions and adaptable to his use. To
write such a book is extremely difficult, for the attainments of students
who enter a second course in calculus are different, their needs are not
uniform, and the viewpoint of their teachers is no less varied. Yet in
view of the cost of time and money involved in producing an Advanced
Calculus, in proportion to the small number of students who will use it,
it seems that few teachers can afford the luxury of having their own
text ; and that it consequently devolves upon an author to take as un-
selfish and unprejudiced a view of the subject as possible, and, so far as
in him lies, to produce a book which shall have the maximum flexibility
and adaptability. It was the recognition of this duty that has kept the
present work in a perpetual state of growth and modification during
five or six years of composition. Every attempt has been made to write
in such a manner that the individual teacher may feel the minimum
embarrassment in picking and choosing what seems to him best to meet
the needs of any particular class.
As the aim of the book is to be a working text or laboratory manual
for classroom use rather than an artistic treatise on analysis, especial
attention has been given to the preparation of numerous exercises which
should range all the way from those which require nothing but substi-
tution in certain formulas to those which embody important results
withheld from the text for the purpose of leaving the student some
vital bits of mathematics to develop. It has been fully recognized that
for the student of mathematics the work on advanced calculus falls in
a period of transition, — of adolescence, — in which he must grow from
close reliance upon his book to a large reliance upon himself. More-
over, as a course in advanced calculus is the ultima Thule of the
mathematical voyages of most students of physics and engineering, it
is appropriate that the text placed in the hands of those who seek that
goal should by its method cultivate in them the attitude of courageous
Ui
Ir PREFACE
exploren, and in its extent supply not only their immediate needs, but
much that may be useful for later reference and independent study.
With the hirge necessities of the physicist and the growing require-
of the engineer, it is inevitable that the great majority of our
of calculus should need to use their mathematics readily and
rigorously rather tlian with hesitation and rigor. Hence, although due
attention has been paid to modern questions of rigor, the chief desire
luis been to confirm and to extend the student's working knowledge of
those great algorisms of mathematics which are naturally associated
with the calculus. Tliat the compositor should have set " vigor " where
"rigor" was written, might appear more amusing were it not for the
suggested antithesis tliat there may be many who set rigor where vigor
thoold be.
As I have had practically no assistance with either the manuscript
or the proofs, I cannot expect that so large a work shall be free from
errors ; I can only have faith that such errors as occur may not prove
seriously troublesome. To spend upon this book so much time and
energy which could have been reserved with keener pleasure for vari-
ous fields of research would have been too great a sacrifice, had it not
been for the hope that I might accomplish something which should be
of material assistance in solving one of the most difficult problems of
mathematical instruction, — that of advanced calculus.
EDWIN BIDWELL WILSON
MAMACUUtSTTS ImSTITUTK OF TfiCUNOLOOY
CONTENTS
INTRODUCTOKY REVIEW
CHAPTER I
REVIEW OF FUNDAMENTAL RULES
SECTION PAOB
1. On differentiation 1
4. Logarithmic, exjwnential, and hyperbolic functions ... 4
6. Geometric i)roi)ertie8 of the derivative 7
8. Derivatives of higher order 11
10. The indefinite integral 15
13. Aids to integration 18
16. Definite integrals 24
CHAPTER II
REVIEW OF FUNDAMENTAL THEORY
18. Numbers and limits 3.3
21. Theorems on limits and on sets of points 37
23. Real functions of a real variable 40
26. The derivative 45
28. Summation and integration 50
PART I. DIFFERENTIAL CALCULUS
CHAPTER III
TAYLOR'S FORMULA AND ALLIED TOPICS
31. Taylor's Formula 55
33. IndeU'rminate forms, infinit<'siinals. iiifinitrs ..... 61
86. lnfinit(>simal analysis 68
40. Some differential geometry 78
CONTENTS
CHAPTER IV
lARTlAL DIFFERENTIATION; EXPLICIT FUNCTIONS
PAGE
87
4S. VmmMaaB of two or more variables
M. Plwt pwtial derivatives ^^
60. Derivatives of higher order
54. Taylor's Formula and applications '^^^
CHAPTER V
PARTIAL DIFFERENTIATION ; IMPLICIT FUNCTIONS
M. The simplest case; F(x,3f) = 0 1^7
M. More general cases of implicit functions 122
OS. Funetional determinants or Jacobians 129
65. EavclopM of curves and surfaces I^^
6S. More differential geometry ^^^
CHAPTER VI
COMPLEX NUMBERS AND VECTORS
70. Operators and operations I*^
71. Complex numbers 1^3
78. Functions of a complex variable 157
76. Vector sums and products 163
77. Vector differentiation 170
PART II. DIFFERENTIAL EQUATIONS
CHAPTER VII
OSNERAL INTRODUCTION TO DIFFERENTIAL EQUATIONS
81. Some geometric problems 170
8S. Problems in mechanics and physics 184
85. Lineal element and differential equation 191
87. The higher derivatives ; analytic approximations .... 197
CHAPTER VIII
TBI OOMMOHSR ORDINARY DIFFERENTIAL EQUATIONS
bj lepftratlng the variables 203
•I. iBlegrAUng factors 207
M. LliMAr eqaatkmn with constant coefficients 214
is. MattltMMOBf linear equations with constant coefficients . 228
CONTENTS vii
CHAPTER IX
ADDITIONAL TYPES OF ORDINARY EQUATIONS
•ECTIOX PAOB
100. Equations of the first order and higher degree .... 228
102. E(iuations of higher order ........ 234
104. Linear differential equations 240
107. The cylinder functions 247
CHAPTER X
DIFFERENTIAL EQUATIONS IN MORE THAN TWO VARIABLES
109. Total differential equations ........ 254
111. Systems of siinultaueou.s equations 260
113. Introduction to partial differential equations .... 267
116. Types of partial differential equations 273
PART III. INTEGRAL CALCULUS
CHAPTER XI
ON SIMPLE INTEGRALS
118. Integrals containing a parameter 281
121. Curvilinear or line integrals 288
124. Independency of the path 298
127. Some critical comments 308
CHAPTER XII
ON MULTIPLE INTEGRALS
129. Double sums and double integrals 315
133. Triple integrals and change of variable 326
13.'). Average values and higher integrals 332
137. Surfaces and surface integrals 338
CHAPTER XIII
ON INFINITE INTEGRALS
140. Convergence and divergence 852
142. The evaluation of infinite integrals 360
144. Functions defined by infinite integrals 368
^iii CONTENTS
CHAPTER XIV
SPECIAL FUNCTIONS DEFINED BY INTEGRALS
147. The Gamma and BeU fuuctioDS
160. Tbe error function
IAS. B««el f nnetloDf
PAGE
378
386
CHAPTER XV
THE CALCULUS OF VARIATIONS
155. The treatment of the simplest case 400
157. Variable limita and constrained minima 404
IM. SomegeiieralizaUons 409
PART IV. THEORY OF FUNCTIONS
CHAPTER XVI
INFINITE SERIES
162. CoDTergence or divergence of series 419
166. Series of functions 430
168. Manipulation of series 440
CHAPTER XVII
SPECIAL INFINITE DEVELOPMENTS
171. The trigonometric functions 453
176. Trigonometric or Fourier series 458
175. The TheU functions 467
CHAPTER XVIII
FUNCTIONS OF A COMPLEX VARIABLE
176. G«Bferal theorems 476
160. Chameierization of some functions 4S2
166. Conformal repn*M*ntaUon 490
165. Ittlefrala and tlieir iuvemion ....... 496
CONTENTS ix
CHAPTER XIX
ELLIITIC FUNCTIONS AND INTEGRALS
SECTION FAOE
187. Legendre's integral I and its inversion ftOS
190. Lej^eiidre'.s intcj^rals II and III . .511
192. Weierstrass's integral and its inversion 517
CHAPTER XX
FUNCTIONS OF REAL VARIABLES
194. Partial differential equations of physics 524
196. Harmonic functions; general theorems 530
198. Harmonic functions ; special theorems 537
201. The potential integrals 646
BOOK LIST 555
INDEX 557
ADVANCED CALCULUS
INTRODUCTORY REVIEW
CHAPTER I
REVIEW OF FUNDAMENTAL RULES
1. On differentiation. If the function f{x) is interpreted as the
curve y=f{j')* the (luotient of the increments Ay and Aa; of the
dependent and independent variables measured from (ar^, y^ is
y-y,^^y^ A/(a-) ^ /(x, + ^) -f{x,) ^ ^.
X — x^ ^x Ax Ax
and represents the slope of the secant through the points P(Xf^i y^ and
P\x^-\-A.x, y^j-f-Ay) on the curve. The limit approached by the quo-
tient Ay/ Ax when P remains fixed and Ax = 0 is the slope of the
tangent to the curve at the point P. This limit,
li,„ ^ = li.„ /(^. + ^-/(^o) =_^,(,^), (2)
is called the derivative of /(x) for the value x = x^^. As the derivative
may be computed for different points of the curve, it is customary to
speak of the derivative as itself a function of x and write
.. Ay ,. /(x -(- Ax) — /(x) , .«.
lim — ^ = hm '—^ — :^-^^=/'(x). (3)
A^-oAx Axio Ax
There are numerous notations for the derivative, for instance
/-(x) = ^ = '^ = />,/= />,y = y = i>/= i>y.
• Here and throughout the work, when' fitfiires are not given, the reader 8hou1d draw
graphs to ilhiHtrate the Ktatement.s. Training in making one's own illustrations, whether
graphical »)r analytic, is of great value.
1
S INTRODUCTORY REVIEW
The first five show distinctly that the independent variable is «, whereas
the bet thwe do not explicitly indicate the variable and should not be
need unless there is no chance of a misunderstanding.
t. The fundamental formulas of differential calculus are derived
dirooUy from the application of the definition (2) or (3) and from a
few fondamental propositions in limits. First may be mentioned
^ = ^ ^, where * = ^(y) and y =f(x), (4)
dx ay ax
dy dy dfi^ dy ^ ^
dx dx
D(u ±v) = Du±Dv, D(uv) — uDv + vDu. (6)
^(-) = ^ — ;^ » Di^)^^*^-^. (7)
It m»y be recalled that (4), which is the rule for differentiating a function of a
function, follows from the application of the theorem that the limit of a product is
Az A;? Aw
Um product of the limits to the fractional identity — = — -— ; whence
■^ Ax Ay Ax
,, A« ,, Az ,. Ay ,. *Az ,. Ay
lim — = lim lim -^ = limt lim -^,
AxAoAx AxAoAy AxsoAx iiy=o Ay az=oAx
which Is equivalent to (4). Similarly, if y =/(x) and if x, as the inverse function
of y, be written x=/-*(y) from analogy with y = 8inx and x = sin-iy, the
reUtion (6) follows from the fact that Ax/ Ay and Ay/ Ax are reciprocals. The next
three remit from the Immediate application of the theorems concerning limits of
■IBM, products, and quotients ({ 21). The rule for differentiating a power is derived
In eiM n is integral by the application of the binomial theorem.
^ = ^^^^ = .^- + ^^x.-. AX + . . . + (AX).-.,
and the limit when AzdbO Is clearly nx"-i. The result may be extended to rational
p
▼aloes of the index n by writing n = -. y = x«, y« = xp and by differentiating
bolh iidas of the equation and reducing. To prove that (7) still holds when n is
irrmtloBsl, It would be necessary to have a voorkahle definition of irrational numbers
sad to develop the properties of such numbers in greater detail than seems wise at
this point. The formula Is therefore assumed in accordance with the principle of
9f form (1 178), just as formulas like a^a'' = a*» + '« of the theory of
whieh may readily be proved for rational bases and exponents, are
without proof to hold also for Irrational bases and exponents. See, how-
ever, H l^Sft mmI the eserciaes thereunder.
• II b frs^ttMlly better to regard the quotient as the product u w-» and apply (6).
f For when a« *0, then Ay aq or Ay/Ax could not approach a limit.
FUNDAMENTAL RULES 8
3. Second may be mentioned the formulas for the derivatives of the
trigonometric and the inverse trigonometric functions.
D sin jr = cos a', Z) cos x = — sin ar, (8)
or D sin x = sin (x -f J tt), D cos x = cos (x -f- J it), (8*)
D tan X = sec*ir, Z> cot a; = — eschar, (9)
/) sec ar = sec x tan x, /) esc x = — esc x cot x, (10)
/) vers X = sin x, where vei-s x = 1 — cos x = 2 sin* ^ x, (11)
_. _, ±1 f — in quadrante I, II, ,^„.
^eos»x = -^==, 1^,,^ ,, jjj'j^; (13)
^tan->^=_l_, ^cot-x = -j^,, (14)
D sec-i X = ^^!^ — , / + »" quadrants I, III, . ^
xV?^ L- " " II, IV, ^^""^
:fc^ r — in quadrants I, III, .^ ^^
^V?3i' U" " n,iv, (i«)
It may be recalled that to differentiate sinx the definition is applied. Then
A sin X sin (x + Ax) — sin x sin Ax 1 — cos Ax .
= ^ = cos X sin X.
Ax Ax Ax Ax
It now is merely a question of evaluating the two limits which thus arise, namely,
,. sin Ax , ,. 1 — cos Ax „„,
lim and lim (18)
' Ax^o Ax ajt^o Ax
From the properties of the circle it follows that these are respectively 1 and 0.
Hence the derivative of sinx is cosx. The derivative of cosx may be found in
like manner or from the identity cosx = sin (J ir — x). The results for all the other
tilj^onometric functions are derived by expressing the functions in terms of sinx
and cosx. And to treat the inverse functions, it is suflHcient to recall the general
method in (5). Thus
if y = sin-i x, then sin y = x.
Differentiate both sides of the latter equation and note that cosy = i Vl — sin*y
= ± Vl — X* and the result for D sin-»x is immediate. To ascertain which sign to
use with the radical, it is suflBcient to note that ± Vl — x' is cosy, which is positive
when the angle y = 8in-»x is in quadrants I and IV, negative in II and III.
Sl>nilarlv for the other inverse functions.
4 INTEODUCTORY REVIEW
EXERCISES *
I. Carry through the derivation of (7) when n = p/^, and review the proofs of
tjpieal fonnulM eelectea from the list (6)-(17). Note that the formulae are often
glvan aa i>^" = »•«• "* I>«^^ I>* "i" u = coe u D^m, • • , and may be derived in this
form directly from the definition (8).
S. Derive the two limiu neceasary for the differentiation of sinx.
5. Draw grapha of the inverse trigonometric functions and label the portions
of the curvea which correspond to quadrante I, II, III, IV. Verify the sign in
(ItHlT) from the alope of the curves.
4. Find Dtanz and Dcotz by applying the definition (3) directly.
u 4- 0 u — X)
ft. Find D dnz by the identity sin u — sin c = 2 cos — -— sm — - —
6. Find Dtan-»x by the identity tan-^u - tan-ir = tan-i ~ and (3).
7. Differentiate the following expressions :
(a) cac2x-.cot2x, (/3) J tan»x - tanx + x, (7) x cos-i z - Vl - x^,
— i=. (.)8in-i_|=,
Vl-X« Vl + X»
(I) .ec-» ^ ^ , («) 8in-» ^^ , (f) X Va^ - x^ + a» sin-i ^ ,
2ax ^ . , X
(f ) a vers-* - - V2ax-x«, {$) cot-i - 2 tan
X* — a*-* a
What trigonometric identities are suggested by the answers for the following :
8. In B. O. Peirce's "Short Table of Integrals " (revised edition) differentiate the
right-hand members to confirm the formulas : Nos. 31, 45-47, 91-97, 125, 127-128,
181-185, 161-168, 214-216, 220, 260-269, 294-298, 300, 380-381, 386-394.
9. If X is measured in degrees, what is D sin x ?
4. The logarithmic, exponential, and hyperbolic functions. The
next set of formulas to be cited are
Dlog.« = i, /)log.x = l2|sf, (19)
DeT = «», Da* = a' log, a.t (20)
It oiay be recalled that the procedure for differentiating the logarithm is
A1QS.X )og.(x + Ax) ~ logaX 1 . X + Ax 1 , /, , Ax\Ai
Ax Ax
1 , x + Ax 1, A . Ax\:
= log, = _log„(l + _)
Ax X X \ X /
* The stodeot should keep on file his Dolutionii of at least the important exercises ;
•serdMS and couHiderable portionii of the text depend on previous
t A« le eoflloauiry, the rabeeript « will hereafter be omitted and the symbol log will
%km lofarlthm to the baae « ; any base other than e must be 8i>ecially designated
mwmk, TMa ubeei ralluu ia partloularly neoeMary with reference to the common base
10 Mtd la eompotatkNi.
FUNDAMENTAL RULES 5
If now Z/A2 be set equal to A, the problem becomet that of evaluating
Urn (l + iy= e = 2.71828. . ,• log^^e = 0.4M2M. • (21)
and hence if e be choeen as the base of the systenif D log x takes the simple f onn
1/x. The exponential functions e* and a' may be regarded as the inverse functions
of log X and lojjaX in deducing (21). Further it should be noted that it is frequently
useful to take the logarithm of an expression before differentiating. This is known
SLH logarithmic different ialion and is used for products and complicated powers and
roots. Thus
if V = x*y then logy = z logx,
and -/rsl + logx or / = x'(l + loga;).
1 L is I he expression j//y which Is called the logarithmic derivative of y. An especially
noteworthy property of the function y = Ce* is that the function and Its derivative
are equal, y' = y ; and more generally the function y = CV' is proportional to its
derivative, y* = ky.
5. The hyperbolic functions are the hyperbolic sine and cosine,
sinh X = — > cosh x = ; (22)
and the related functions tanha-, cotha-, sechar, cscha;, derived from
them by the same ratios as those by which the corresponding trigono-
metric functions are derived from sin a; and cosar. From these defini-
tions in terms of exponentials follow the formulas :
cosh''a;-sinh^a;=l, tanh^x -f- sech=*ic = 1, (23)
sinh (x ±y) = sinh x cosh y ± cosh x sinh y, (24)
cosh (x ±y) = cosh x cosh y ± sinh x sinh y, (25)
, X , (cosh ic 4-1 . ,x . Icosh x—1 ,-^^
cosh- = 4-^ , sinh- = ±^ , (26)
D sinh X — cosh x, D cosh x = sinh ar, (27)
D tanh x = sech^a-, D coth x = — csch^a;, (28)
D sech x = — sech x tanh Xj D csch ar = — cscha; coth x. (29)
The inverse functions are expressible in terms of logarithms. Thus
y = 8inh~*a;, x = sinhy =
Y^'
• The treatment of this limit is far from complete in the majority of texts. Reference
for a careful presentation may, however, be made to (Jranville's "Calculus," pp. 31-34,
ami Osgood's " Calculus," pp. 78-82. See also Ex. 1, (fi), in § ItiS below.
e INTRODUCTORY REVIEW
Here only the positive sign is available, for e' is never negative. Hence
sinh-» X = log{a; -|- Va^-fl), any x, (30)
oosh-»x = log(ar ± Vx«-l), x > 1, (31)
1, 1 + x
1, x-hl
lanh-»x=|log^, a.^<l, (32)
coth->x = hog^, ^^>1, (33)
Bech-» X = log 0 ± ^p--^J » ^ < 1» W
csch-»x = logf - 4- -J^ + 1 h any a;, (35)
Dsinh-^a;= /^ > Dcosh-^x = -7^=, (36)
D tanh->a; = q 5 = Z> coth-ia; = ^ » (37)
D sech-» X = — ^ , i) csch-i x = — " ' (38)
xVl~x*^ xVl + x^
EXERCISES
1. Show by Ic^rithmic difFerentiatiou that
D(ur«,...) = (| + ^ + ^ + ---)(umo...),
derive the rule : To differentiate a product differentiate each factor
add all the results thus obtained.
t. Sketch the graphs of the hyperbolic functions, interpret the graphs as those
ol the UiverBe functions, and verify the range of values assigned to x in (30)-(35).
3. Prove sundry of formulas (23)-(29) from the definitions (22).
4. Prove sundry of (30)-(38), checking the signs with care. In cases where
douUe signs remain, state when each applies. Note that in (31) and (34) the
4mAk tigti may be placed b^ore the log for the reason that the two expressions
9f9 fteiproealt.
$. Derive a fonnola for sinhu ± sinhv by applying (24) ; find a formula for
taoh I X analogous to the trigonometric formula tan \x = sinx/(l + cosx).
i. Tkt gMdermannian. The function 0 = gd x, defined by the relations
dnhx = Un0, 0 = gdx = tan-isinhx, — iir<0< + Jir,
ii called tba gudennannian of z. Prove the set of formulas :
eoihssaae^, tanhxssin^, C8chx=:cot0, etc.;
DgdzssMhx, « = gd-»0 = logUn(j0 + |ir), Dgd-i0 = 8ec0.
7. BobsUtoie the fonetions of ^ in Ex. 0 for their hyperbolic equivalents in
(tSK <Mi, (27 L ifciid riMlooe to simple known trigonometric formulas.
FUNDAMENTAL RULES
8. Differentiate the following expreasioni) :
(a) (X + 1)«(JC + 2)-»(x + 3)-«, (/J) x^', (y) log.(x + 1),
(8) x + logco8(x- Jw), (e) 2tan-ie', (f)z-Uuhx,
-(asinrnx— mcoimx)
9. Check sundry formulM of Peirce's "Table," pp. 1-61, 81-82.
in) X tanh-»x + \ log(l - x«), (0)
6. Geometric properties of the derivative. As the quotient (1) and
its limit (2) give the H\u\Hi of a secant and of the tangent, it appears
from graphical considerations that when the derivative is positive the
function is increasing with a*, but decreasing when the derivative is
negative.* Hence to determine the regioius in wh'vch a function is i»-
creasimj or decreasing y one may Jind the derivative inul dttennine the
regions in which it is positive or negative.
One must, however, be careful not to apply this rule Ujo blindly; for in so
simple a case as/(x) = logx it is seen that/'(x) = 1/x is positive when x > 0 and
negative when x < 0, and yet log x has no graph when x < 0 and is not considered
as decreasing. Thus the formal derivative may be real when the function is not
real, and it is therefora best to make a rough sketch of the function to corroborate
the evidence furnished by the examination of /'(x).
If x^ is a value of x such that immediately t upon one side of a; = ar^
the function f{x) is increasing whereas immediately upon tlie other
side it is decreasing, the ordinate y^=f(x^) will be a maximum or
minimum or f(x) will become positively or negatively infinite at x^.
If the case where /(x) becomes infinite be ruled out, one may say that
the /miction will have a minimum or tnaximum at x^ according as the
derivative changes from negative to positive or from positive to negative
when Xf moving in the positive directionj passes through the value x^.
Hence the usual rule for determining m^a^imu and minima is to find
the roots o//'(a-)=0.
This rule, again, must not be applied blindly. For first, /'(x) may vanish where
there is no maximum or minimum as in the case y = x' at x = 0 where the deriva-
tive does not change sign ; or second, /'(x) may change sign by becoming infinite
as in the case y = x^ at x = 0 where the curve has a vertical cusp, point down, and
a minimum ; or thirtl, the function /(x) may be restricted to a given range of values
a ^ X ^ 6 for X and then the values /(a) and/(6) of the function at the ends of the
interval will in general be maxima or minima without implying that the deriva-
tive vanish. Thus although the derivative is highly useful in determining maxima
and minima, it should not be trusted to the complete exclusion of the corroborative
evidence furnished by a rough sketch of the curve y =/(x).
* The construction of illustratiye figures is again left to the reader.
t The word " immediately " is necessary because the maxima or minima may be
merely relative ; in the case of several maxima and minima in an interval, some of
the maxima may actually be less ttiau some of the miuitaa.
g INTEODUCTOEY REVIEW
7. The derivative may be used to express the equations of the tangent
and normal, the values of the subtangent and subnormal, and so on.
Equation of tangent, y-y^^y[{^- ^o)» (^^)
BquaUon of normal, (y -y^y[-^{^- «o) = ^» (^^)
TM = Bubtangent = yjy'^, MN = subnormal = y^^ (41)
OT = x-intereept of tangent = x^ — yjy[y etc. (42)
The derivation of these results is sufficiently evi-
dent from the figure. It may be noted that the
subtangent, subnormal, etc., are numerical values
for a given point of the curve but may be regarded
as functions of x like the derivative.
In geometrical and physical problems it is frequently necessary to
apply the definition of the derivative to finding the derivative of an
unknown function. For instance if A denote the
area under a curve and measured from a fixed
ordinate to a variable ordinate, A is surely a func-
tion A{x) of the abscissa x of the variable ordinate.
If the curve is rising, as in the figure, then o mm'
MPQ'M* <AA< MQP'M', or y^x <^A<(y + Ay) Ax.
Divide by Aa; and take the limit when Ax = 0. There results
AA
lim y ^ lim — - ^ lim (y -f Ay).
Ax^O Ax^O Ax Ax = 0
Henoe
,. A A dA
lim -— = -j-
AxAo Ax ax
= y-
(43)
BoU^s Theorem and the Theorem of the Mean are two important
theorems on derivatives which will be treated in the next chapter but
may here be stated as evident from their geometric interpretation.
RoIIp's Theorem states that : If a function has a derivative at every
T
r
r
jK^
/ 0
{ ^ 0
y^ '\
Fio. 1
Fio. 2
Fig. 3
Tpoini of an interval and if the function vanishes at the ends of the in-
terval, then there is at least one point within the interval at which the
tUrhaiive vanishes. This is illustrated in Fig. 1, in which there are
two luch poinU. The Theorem of the Mean states that : If a function
FUNDAMENTAL RULES 9
h(Ui a derivative at each point of an interval, there i» at least on* point
in the interval such that the tangent to the curve y=z/(x) is parallel to
the chord of the interval. Tliis is illustrated in Fig. 2 in which there
is only one such jx)int.
Agahi care inuiit be exerclKed. In Fig. 8 the function vanUbes at A and B but
there is nu point at which the Hlope of the tangent is zero. Tbia ig not an excep-
tion or contrAtliction to Kolle's Theorem for the reaiion that tlie function does not
witisfy the conditionH of the theorem. In fact at the point P, although there i« a
tiuigent to the curve, there is no derivative ; tlie quotient (1) formed for the point P
becomes negatively infinite as Ax = 0 from one side, positively infinite aa Ax^O
from the other side, and therefore does not approach a definite limit as is required
in the definition of a derivative. Ttie hyixjthesis of the theorem is not satisfied and
there is no reason tliat tlie conclusion should hold.
EXERCISES
1. Determine the regions in which the following functions are increasing or
decreasing, sketch the graphs, and find the maxima and minima :
{a) ia^-x« + 2, 03) (x + l)t(x-5)», (7) log(x«~4),
(«) (X - 2)Vx - 1, (e) - (X + 2)Vl2-x2, (f) x» + ax + 6.
2. The ellipse is r = Vx* -f y* = e (d + x) referred to an origin at the focus.
Find the maxima and minima of the focal radius r, and state why Bjr = 0 does
not give the solutions while D^r = 0 does [the polar form of the ellipse being
r = i:(l — eco8 0)-i].
3. Take the ellipse as xV«l_+ yV^* = 1 ^"^ discuss the maxima and minima of
the central radius r = Vx'"* + y'^. Why does B^r = 0 give half the result when r is
expressed as a function of x, and why will D^r — 0 give the whole result when
X = acosX, y — 6sinX and the ellipse is thus expressed in terms of the eccentric
angle ?
4. If y = P(x) is a polynomial in x such that the equation P(x) = 0 hiis multiple
root,s, show that P'(x) = 0 for each multiple root. What more complete relationship
can be stated and "proved ?
5. Show that the triple relation 27 6^ -f 4 a* ^ 0 detennines completely the nature
of the roots of x* + ox + 6 = 0, and state what corresponds to each possibility.
6. Define the angle 6 between two intersecting curves. Show that
tantf = [/'(x,) - ^(Xo)] -^ [1 +/'(Xo)l7'W]
if y =/(x) and y = g{z) cut at the point (x^, i/p).
7. Find the subnormal and subtangent of the three curres
(a) y* = 4px, (/3) x« = 4py, (7) x« + v« = aV
8. The pedal curve. The locus of the foot of the perpendicular dropped from
a fixeil point to a vaiiable tangent of a given curve is called the pedal of the given
curve with respect to tlie given point. Show that if the fixed point is the origin,
the pedal of y =/(x) may be obtained by eliminating x^, y^ yo from the equations
10 INTRODUCTORY REVIEW
Find the pedal (a) of the hyperbola with respect to the center and (fi) of the
parabola with respect to the vertex and (7) the focus. Show (3) that the pedal of
the parabola with respect to any point is a cubic.
9. If the curve y =f(x) be revolved about the x-axis and if V{x) denote the
volume of revolution thus generated when measured from a fixed plane perpen-
dicular to the axis out to a variable plane perpendicular to the axis, show that
10. More generally if A (x) denote the area of the section cut from a solid by
a plane perpendicular to the x-axis, show that DxV=A (x).
11. If yt (^) denote the sectorial area of a plane curve r =f{<p) and be measured
from a fixed radius to a variable radius, show that D^A = I r".
12. If p, A, p are the density, height, pressure in a vertical column of air, show
that dp/dh = — p. 1( p = kp, show p = Ce-**.
13. Draw a graph to illustrate an apparent exception to the Theorem of the
Mean analogous to the apparent exception to Rollers Theorem, and discuss.
14. Show that the analytic statement of the Theorem of the Mean for/(x) is
that a value x = { intermediate to a and b may be found such that
m -f{a) =/'(f) (6 - a), a<i<b.
15. Show that the semiaxis of an ellipse is a mean proportional between the
x-intercept of the tangent and the abscissa of the point of contact.
16. Find the values of the length of the tangent (a) from the point of tangency
to the X-axis, (/3) to the y-axis, (7) the total length intercepted between the axes.
Consider the same problems for the normal (figure on page 8).
17. Find the angle of intersection of (a) y^ = 2 mx and x^ + y^ = a^,
,^ * A J 8a' , . X2 2/2 for 0<A<6
(/Dx« = 4ay and y = ^^-^-^, (y) -^--^ + ^^-^^ = 1 ^d,<,<„.
18. A constant length is laid off along the normal to a parabola. Find the locus.
19. The length of the tangent to x^ + y^ = a^ intercepted by the axes is constant.
20. The triangle formed by the asymptotes and any tangent to a hyperbola has
constant area.
21. Find the length PT of the tangent to x =Vc"-y'^ + c sech-i (y/c).
22. Find the greatest right cylinder inscribed in a given right cone.
23. Find the cylinder of greatest lateral surface inscribed in a sphere.
24. From a given circular sheet of metal cut out a sector that will form a cone
(without base) of maximum volume.
25. Join two points A^ B in the same side of a line to a point P of the line in
such a way that the distance PA + PB shall be least.
26. Obtain the formula for the distance from a point to a line as the minimum
diitiknce.
27. Te$l for maximum or minimum, {a) If /(x) vanishes at the ends of an inter-
val and is positive within the interval and if f'(x) = 0 has only one root in the
Interval, that root indicates a maximum. Prove this by Rollers Theorem. Apply
It In Ex«. 22-24. (/9) If /(x) becomes indefinitely great at the ends of an interval
*od /'(«) = 0 *>*■ ou\j one root in the interval, that root indicates a minimum.
FUNDAMENTAL RULES 11
Prove by Rollers Theorem, and apply In Exb. 26-20. Tbeae rules or yarioua modi-
fications of them generally suflHce in practical problem* to dlstinguiab between
maxima and minima without examining either the changes in sign of the first
derivative or the Bign of the second derivative ; for generally there is only one
root of /'(x) = 0 in the region considered.
28. Show that z-* sin x from z = Otoz = ^ir steadily decreases from 1 to 2/v.
29. If 0 < z < 1, show (or) 0 < z - log(l + z) < iz«, (/J) -l^ < z - log(l + z).
2 1 + z
30. If 0 > X > - 1, show that -x« < z - log(l + z) < -i-^.
m 1 "t" Z
8. Derivatives of higher order. The derivative of the derivative
(regjirded as itself a function of x) is the second derivative, and so on
to the nth derivative. Customary notations are :
/" w = ^ = 3 = ««V= uiy = >j" = i>y = /^'y,
/"'(.)./"(x),-,/-'(x); 3'g'-'g
The nth derivative of the sum or difference is the sum or difference of
the nth derivatives. For the nth derivative of the product there is a
special formula known as Leibniz^s Theorem. It is
D^(uv) = iru'U + nD^-^uDv + ^^^^^^^^ + urrv. (44)
^ I
This result may be written in symbolic form as
Leibniz's Theorem jr{uv) = {Du -f- Dvy^ (44')
where it is to be understood that in expanding (^Du -f Dvy the term
(Z>?<)* is to be replaced by Z>*m and (Dm)® by Ifiu = u. In other words
the powers refer to repeated differentiations.
A proof of (44) by induction will be found in § 27. The following proof is
interesting on account of its ingenuity. Note first that from
B (uc) = uDv + tjDu, D2 (uc) = D {uDv) + 7) {vJ>u),
and so on, it appears that D* (ur) consists of a sum of terms, in each of which there
are two differentiations, with numerical coeflScients independent of u and v. In like
manner it is clear that
D"(mj) = CoZ>u.t> + C^D^-^uDt^- •+ C,_iZ>uD"-io + C^uL^
is a sum of terms, in each of which there are n differentiations, with coefficients C
independent of u and v. To determine the C's any suitable functions u and r, say
u = e», r = e", uv = c<»+ «•)', L^ef^ = a*e«',
may be substituted. If the substitution be made and e(i+a>' be canceled,
e-(i+«)xi>.(„B) = (1 + a)- = Co + CjO + . . . + C.-iO— » + C,a-,
and hence the C's are the coefficients in the binomial expansion of (1 -} a)».
12 INTRODUCTORY REVIEW
Formula (4) for the derivative of a function of a function may be
extended to higher derivatives by repeated application. More genei-ally
any desired change of variable inay he made by the repeated use of (4)
tmd (5). For if x and y be expressed in terms of known functions
of new variables u and r, it is always possible to obtain the deriva-
tives D.y, Diy,' in terms of D.r, D^Vj • • -, and thus any expression
F(Xf y, y'f y", •••) may be changed into an equivalent expression
♦(«, p, v'f v"j •") in the new variables. In each case that arises the
transformations should be carried out by repeated application of (4)
and (5) rather than by substitution in any general formulas.
The following typical cases are illustrative of the method of change of variable.
Soppoee only the dependent variable y is to be changed to z defined as y =f{z) . Then
dx*~ dx \dx) ~dx\dx dzj dx* dz dx \dx dz)
~ dx^ dz dx\dz dz dx) ~ dx^ dz \dx) dz^
As the derivatives of y =f{z) are known, the derivative d^y/dx'^ has been expressed
in terms of z and derivatives of z with respect to x. The third derivative would be
found by repeating the process. If the problem were to change the independent
TtriaMe « to t, defined by x = /(«),
dv_dydz_dy /dz\ -i ■d^_d^rdy (dx\-^l
dx~ dzdx~ dz \dz) ' dx« ~ dxldz\dzj J *
d^_d^dz (dx\-^_ dy /dx\- ^dzd^_ fd^ ^ _ ^ ^1 ^ /^V
dx* " dz* dx\dz) dz \dz) dx dz^ ~ Idz^ dz dz^ dz] ' \dz) '
The change is thus made as far as derivatives of the second prder are concerned. If
the change of both dependent and independent variables was to be made, the work
would be similar. Particularly useful changes are to find the derivatives of y by x
when y and x are expressed parametrically as functions of t, or when both are ex-
prwind in terms of new variables r, ^ as x = r cos 0, y = r sin <f>. For these cases
■ee the exercises.
9. The concavity of a curve y =f(x) is given by the table :
if f"(r^) > 0, the curve is concave up at x = x^,
if f"(x^ < 0, the curve is concave down at a; = x^^
if f*\x^ = 0, an inflection point 2Xx = x^. (?)
Henoe the criterion for distinguishing between maxima and minima:
if /'(ar,^ = 0 and f"(x^ > 0, a minimum at a; = x^,
if f*(x^==0 and f"(x^ < 0, a maximum at a; = x^,
>^ /* W = <> and f"(x^ = 0, neither max. nor min. (?)
FUNDAMENTAL EULES 18
The question points are necessary in the third line because the state-
ments are not always true unless f'"(x^ =^ 0 (see Ex. 7 under § 39).
It may be recalled that the reason that the curve is concave up In CM6/"(a5^) > 0
ia because the derivative /'(x) Is then an increa«ing function in the neighborhood
of z = z„; whereas if /"(Zq) < 0, the derivative /'(z) is a decreasing function and
the curve is convex up. It should be noted that concave up is not the same as
concave toward the z-axls, except when the curve is below the axis. With regard
to the use of the second derivative as a criterion for distinguishing between maxiuia
and minima, it should be stated that in practical examples the criterion is of rela^-
tively small value. It is usually shorter to discuss the change of sign of /'(z) directly,
— and indeed in most cases either a rough graph of /(z) or the physical conditions
of the problem which calls for the determination of a maximum or minimum will
immediately serve to distinguish between them (see Ex. 27 above).
The second derivative is fundamental in dynamics. By definition the
average velocitt/ r of a particle is the ratio of the space traversed to the
time consumed, v = s/t. The actual velocity v at any time is the limit
of this ratio when the interval of time is diminished and approaches
zero as its limit. Thus
V = -— and V = lim -— = — • (4o)
A^ Ar = oA^ at
In like manner if a particle describes a straight line, say the ar-axis, the
average acceleration f is the ratio of the increment of velocity to the
increment of time, and the actual oAxeleration f at any time is the limit
of this ratio as A^ = 0. Thus
- Av , ^ ,. Av dv d}x ,.„
/=- and /=h,n_ = - = _. (46)
By NewtorCs Second Law of Motion, the force acting on the particle i»
equal to the rate of change of momentum with the time, momentum
being defined as the product of the mass and velocity. Thus
rf(mr) dv d}x ,._.
F = — H — ^ = m -r- = mf = m —r^^ > (47)
dt dt '' dt^ ^ '
where it has been assumed in differentiating that the mass is constant,
as is usually the cxise. Hence (47) ii])iK'ars as the fundamental equa-
tion for rectilinear motion (see also §§ 79, 84). It may be noted that
where 7"= 1 wv* denotes by definition the kinetic energy of the particle
For comments see Ex. 6 following.
14 INTRODUCTORY REVIEW
EXERCISES
I. Bute and prove the extenrion of Leibniz's Theorem to products of three or
more factors. Write out the square and cube of a trinomial.
t. Write, by Leibniz's Theorem, the second and third derivatives :
{a) ««8inx, OS) coBhxcosx, (7) xVlogx.
S. Write the nth derivatives of the following functions, of which the last three
■bould first be simplified by division or separation into partial fractions.
(a) \^rrT, (/3) log (ox + 6), (7) (X* + 1) (X + 1)- 8,
(I) COS ox, (<) csinx, (f) (1 - x)/(l + x),
4. If y and x are each functions of t, show that
dx d^y dy dH
d*y _ dt dt^ dt dt^ ^ xY'-y'xf'
dx«~ /dxV ~ x'*
a
d*v ^ x^jx'y''' - y^x^^O - 3 x''{xY' - y^x^Q
dx» x'« ' .
5. Find the inflection points of the curve x = 4 0 — 2 sin 0, y = 4 — 2 cos <f>.
6. Prove (47'). Hence infer that the force which is the time-derivative of the
momentum mo by (47) Is also the space-derivative of the kinetic energy.
7. If A denote the area under a curve, as in (43), find dA/dB for the curves
(a) y = o (1 — cos tf), z — a{B—^ sin ^), (/3) x = a cos ^, y = 6 sin B.
8. Make the indicated change of variable in the following equations :
dxi^
9. TroM^tmnaiionUi polar co^rdi-nalu. Suppose thatx = r cos 0,y=rsin0. Then
dz dr ^ , dy dr .
and «> on for higher derivatives. Find ^ and ^ = ^ -^ ^i^*r)^ - rl)}r
dx dx* (cos 0 D^r — r sin 0)«
10. Generalize formula (6) for the differentiation of an inverse function. Find
tf*s/<ly* and d«x/dy«. Note that these may also be found from Ex. 4.
11. A point deecribes a circle with constant speed. Find the velocity and
aeotleraiion of the projection of the point on any fixed diameter.
FUNDAMENTAL RULES 16
10. The indefinite integral. To integrate a function /(ar) ia to find
a function F{jr) the derituitire of whir h is f(x). The integral F(x) IB
not uniquely determined by the integrand f(x) ; for any two functions
which differ merely by an additive constant have the same derivative.
In giving formulas for integration the constant may be omitted and
understood; but in applications of integration to actual problems it
sliould always be inserted and must usually be determined to fit the
recjuirements of special conditions imposed upon the problem and
known as the inituil conditvuns.
It must not be thought that the constant of integration always appears added to the
function F(x). It may be combined with F(x) so a« to be some what disguised. Thus
logx, logx + C, logCx, log(x/C)
are all integrals of 1/x, and all except the first have the constant of integration C,
although only in the second does it appear as fonnally a<iditive. To illustrate the
determination of the constant by initial conditions, consider the problem of finding
the area under the curve y = cosx. By (43)
BxA = y = cosx and hence ^ = sin x + C.
If the area is to be measured from the ordinate x = 0, then ^ = 0 when x = 0, and
by direct substitution it is seen that C = 0. Hence A = sinx. But if the area be
measured from x = — J ir, then ^ = 0 when x = — ^ 7r and C = 1 . Hence ^ = 1 + sin x.
In fact the area under a curve is not definite until the ordinate from which it is
measured is specified, and the constant is needed to allow the integral to fit this
fnitial condition.
11. The fundamental formulas of integration are as follows :
Jl^loga-, Jx'' = ^x-if n=^-l, (48)
iff = e'y fa' = a' /log a, (49)
I sin ar = — cos x, I cos x = sin ar, (50)
I tan X — — log cos a*, I cot x = log sin x, (61)
I sector = tan a, I csc^a; = — cot a?, (62)
I tan X sec aj = sec a;, I cot a; esc x = — esc a^ (68)
with formulas similar to (60)-(53) for the hyperbolic functions. Also
I - — -^ = tan"* X or — cot~* a-, I - — -j = tanh"*x or coth"*x, (64)
16 INTRODUCTORY REVIEW
/=gin"'a or -cos-^x, / , • = ± sinh-^a, {hh)
VI -a^ J Vl + ar^*
— -= = 8ec-*a; or -C8C"*ar, / — . • = =F sech-^x, (56)
f ,^^ = ± co8h-»ar, r ^i= = qP csch-^a:, (57)
I —r== = vers-*ar, / secx = gd'^ic = logtanf j + g)* (58)
For the integrals expressed in terms of the inverse hyperbolic functions, the
logarithmic equivalents are sometimes preferable. This is not the case, however,
in the many instances in which the problem calls for immediate solution with
regard to x. Thus if y = Ml + x*)- i = sinh-i x + C, then x = sinh (y — C), and the
■oiution is effected and may be translated into exponentials. This is not so easily
accomplished from the form y = log (x + Vl + x^) + C. For this reason and
because the inverse hyi)erbolic functions are briefer and offer striking analogies
with the inverse trigonometric functions, it has been thought better to use them
in the text and allow the reader to make the necessary substitutions from the table
(30)- (35) in case the logarithmic form is desired.
12. In addition to these special integrals, which are consequences
of the corresponding formulas for differentiation, there are the general
rules of integration which arise from (4) and (6).
/dz dy Cdz
j (u-^v — w)= I u-\- I V- iw, (60)
uv= I uv'-\- I u'v, (61)
Of these rules the second needs no comment and the third will be treated later.
Especial attention should be given to the first. For instance suppose it were re-
qulr«d to integrate 2 logx/x. This does not fall under any of the given types ; but
? logx = ^^^^g^)" ^^Qg^ _dzdy
X dlogx dz dy dx
Here (log«)« ukes the place of z and logx takes the place of y. The integral is
therefore (loga)« ae may be verified by differentiation. In general, it may be
poMlMe to eee that a given integrand is separable into two factors, of which one
la Integrable when considered as a function of some function of x, while the other
la the derivative of tliat function. Then (69) applies. Other examples are :
fe>^' cfMx, J'un-»«/(l + a5«), Jx* sin (x«).
FUNDAMENTAL RULES 17
In the first, z = €» in integrable and aa y = sin z, y' = coax ; In the aeoond, x = ylM
integrable and a« y = tan-'x, ]/' = (! + ««)-» ; hi the third z = ainy la int^gmble
and a« y = x", / = 8x*. The reKulta are
e*'"', J(Un-»x)*, — Jco8(x»).
Thin method of integration at sight covers such a large percentage of ttie cajiea
tliat arise in geometry and physics tliat it must be thoroughly mastered.*
EXERCISES
1. Verify tin- fimdamentjil iiitci/nils M8)-(68) and give the hyperbolic analoguet
of (50)-(63).
2. Tabulate the inteKialH here expressed in terms of inverse hyperbolic func-
tions by means of the corresponding logarithmic equivalents.
3. Write the integrals of the following integrands at sight :
(rt) sin ax, (fi) cot(ax + />), (y) tanhSx,
(«) -A—.* (0 J— ^> (0 ^
a« + x'-» Vx« - a* V2ax-x*
(k) x^Vax^ 4- 6, (X) tan X sec' X, (fi) cot x log sin x,
(x-i - 1)» tanh-ix 2 + logx
, V , , , sinx , . 1
(p) ai + ^^'cosx, («r) « (t)
#
Vcosx Vl — x*8in-ix
4. Integrate after making appropriate changes such as sin'x = J — i co8 2x
or sec'x = 1 + tan*x, division of denominator into numerator, resolution of the
product of trigonometric functions into a sum, completing the square, and so on,
(a) cos2 2x, 03) 8in*x. (7) tan*x,
^'^ x» + 3x + 25' ^'^ ITTT' ^^' "^^ii;^'
(«c) sin 6x cos2x -f 1, (X) sinh mx sinh tix, (m) cos x cos 2 x cos 3 x,
^ /Mj J. /f jpw —1
(r) 8ec*xtanx— v2x, (o) — . (ir) -— •
^ ' ^ ' xs + ox + fc (OX"* + 6)p
• The use of differentials (§ liS) is perhaps more familiar than the use of derivatives.
Then J ' log r </x - (^2 log x rf log x - (log a-)«.
The use of this notation is left optional with the reader; it has some advantages and
some disadvantages. The essential thing is to keep clearly in mind the faot tliat the
problem is to be inspected with a view to detecting the function which will difTerentiate
into the given integrand.
18 INTRODUCTORY REVIEW
5. How are the following types integrated ?
(a) Bln*z coecx, morn odd, or m and n even,
(P) tan*x or cot"x when n is an integer,
(7) 8ec"x or c«c"x when n is even,
(«) tan"«x sec'x or cot"»x csc'x, n even.
6. Explain the alternative forms in (64)-(66) with all detail possible.
7. Find {a) the area under the parabola y* = 4px from x = 0 to x = a ; also
ip) the corresponding volume of revolution. Find (7) the total volume of an ellip-
soid of revolution (see Ex. 9, p. 10).
8. Show that the area under y = sin mx sin nx or y = cos mx cos nx from x = 0
to X = w is zero if m and n are unequal integers but ^ ir if they are equal.
9. Find the sectorial area of r = a tan 0 between the radii 0 = 0 and 0 = ^ir.
10. Find the area of the (or) lemniscate r^ = d^ cos20 and (P) cardioid r=l— cos0.
11. By Ex. 10, p. 10, find the volumes of these solids. Be careful to choose the
parallel planes so that A (x) may be found easily.
(a) The part cut off from a right circular cylinder by a plane through a diameter
of one base and tangent to the other. Ans. 2/3 tt of the whole volume.
(p) How much is cut off from a right circular cylinder by a plane tangent to its
lower base and inclined at an angle ff to the plane of the base ?
(7) A circle of radius 6 < a is revolved, about a line in its plane at a distance a
from its center, to generate a ring. The volume of the ring is 2^^(16^.
{i) The axes of two equal cylinders of revolution of radius r intersect at right
angles. The volume common to the cylinders is 16 r'/S.
12. If the cross section of a solid is A (x) = a(^' + a^x^ + a^ + a,, a cubic in x,
the volume of the solid between two parallel planes is \h{B -\- 4 M -\- B') where h
is the altitude and B and B' are the bases and M is the middle section.
18. Show that f — ^ = tan-i — t£.. ^
J 1 + x« 1 - ex
13. Aids to integration. The majority of cases of integration which
arise in simple applications of calculus may be treated by the method
of f 12. Of the remaining cases a large number cannot be integrated
at all in terms of the functions which have been treated up to this
point Thus it is impossible to express / , in terms
of elementary functions. One of the chief reasons for introducing a
variety of new functions in higher analysis is to have means for effect-
ing the integrations called for by important applications. The dis-
cussion of this matter cannot be taken up here. The problem of
integration from an elementary point of view caIIs for the tabula-
tion of some devices which will accomplish the integration for a
FUNDAMENTAL RULES 19
wide variety of integrands integ^ble in terms of elementary functions.
The devices which will be treated are :
Integration by parts, Resolution into partial fractions,
Various substitutions, Reference to tables of integrals.
Inteffration by parts Ih an application of (61) when written a«
Cuv' =uv— iu'v. (61')
That is, it may happen that the integrand can be written a« the product up' of two
factoFH, where «' i8 integrable and where u'v in also integrable. Then uv' is integrable.
For instance, logx is not integrated by the fundamental formulas ; but
f\ogx = flogx 'l = x logx — fx/x = X logx — X.
I lere log x is taken as u and 1 as v', so that v is x, u' is 1/x, and u'v = 1 is immedi-
ately integrable. This method applies to the inverse trigonometric and hyperbolic
functions. Another example is
Txsinx = — xcosx + Tcosx = sinx — xcosx.
Here if x = u and sinx = v\ both v' and u'v — — cosx are integrable. If the choice
sin x=u and x = r' had been made, i/ would have been integrable but u'tj= J x* coax
would have been less simple to integrate than the original integrand. Hence in
applying integration by parts it is necessary to look ahead far enough to see that
both v' and u'v are integrable, or at any rate that tj' is integrable and the integral
of m'o is simpler than the original integral.*
Frequently integration by parts has to be applied several times in succession. Thus
TxV = x«e^ - r2xe» if u = x«, v' = c»,
= x*c* — 2 xe* — Te* if u = x, o' = e*,
= xV — 2xc* + 2 6*.
Sometimes it may be applied in such a way as to lead back to the given integral
and thus afford an equation from which that integral can be obtained by solution.
For example,
jc* cosx = e» cosx + Je* sinx if u = coax, r'= e',
= e* cosx + e» sinx — Te* cosx I if u = sinx, r'= C
= e»(co8X + sinx) — Te* cosx.
Hence Te* cosx = J e» (cosx + sinx).
* The method of differentials may again be introduced if desired.
20 INTRODUCTORY REVIEW
14. For the inUgralion of a rational fraction f{x)/F{x) where /and F are poly-
nomiaU in x, the fraction i8 first resolved into partial fractions. This is accom-
plished as follows. First if / is not of lower degree than F, divide F into / until the
remainder is of lower degree than F. The fraction f/F is thus resolved into the
sum of a polynomial (the quotient) and a fraction (the remainder divided by F)
of which the numerator is of lower degree than the denominator. As the polyno-
mial is integrable, it is merely necessary to consider fractions f/F where / is of
lower degree than F. Next it is a fundamental theorem of algebra that a poly-
nomial F may be resolved into linear and quadratic factors
F(x) = k{x - a)'{x - b)P{x - c)y • - - {x^ -{■ mx -\- n)M(x2 +px 4- g)»'- • •,
where a, 6, c, • • • are the real roots of the equation F(x) = 0 and are of the respec-
tive multiplicities a, /9, 7, • • •, and where the quadratic factors when set equal to
zero give the pairs of conjugate imaginary roots of F = 0, the multiplicities of the
imaginary roots being m, », • • • • It is then a further theorem of algebra that the
fraction //F may be written as
m^ A, A, Aa B , ^^ ,
F(x) X - a (X - a)2 (x - a)« x - 6 (x - 6)^
^ M^x + N^ ^ M^ -\- N^ ^ ^ M^x -\- N^
X* ■\-mx-\- n (x* + OTX + n)2 (x^ + mx + n)M *
where there is for each irreducible factor of F a term corresponding to the highest
power to which that factor occurs in F and also a term corresponding to every
leaMr power. The coefficients A^ B, • • ., Jlf,,iV, • • • may be obtained by clearing
of fractions and equating coefficients of like powers of x, and solving the equations ;
or they may be obtained by clearing of fractions, substituting for x as many dif-
ferent values as the degree of F, and solving the resulting equations.
When f/F has thus been resolved into partial fractions, the problem has been
reduced to the integration of each fraction, and this does not present serious
difficulty. The following two examples will illustrate the method of resolution
into partial fractions and of integration. Let it be required to integrate
Jx(x-l)(x-2)(x24.x + l) ^" J (x-l)2(x-8)8*
The first fraction is expansible into partial fractions in the form
3^*4-1 _A B . C , Dx + E
x(x-l)(x-2)(x«4-x + l) ,x x-1 x-2 X24-X + I
Hence x« 4- 1 = ^(x - 1) (x - 2) (x« 4- x 4- 1) 4- Bx(x - 2) (x^ 4. x -1- 1)
4-Cx(x - 1) (x« 4- X 4- 1) 4- (Dx + F)x(x - 1) (X - 2).
Ratlier than multiply out and equate coefficients, let 0, 1, 2, - 1, - 2 be substi-
tuted. Then
l = 2i4, J=-8B, 6 = 14C, D-^ = l/21, F-2D = l/7,
r ^-»-> = f— - r ^ 4. f— L_ _ c ^^-»-5
J»(x-l)(x-2)(x«4.x4-l) J 2x J8(x-l)'^J 14(x-2) J 21(xa4x + l)
*a*'**"I*'«<'-^)+n^°«(*-2)-Aiog(x«4-x4-l) ?-tan-i?^.
FUNDAMENTAL RULES 21
la the aeooild OMe the form to be aMumed for the expansion Ui
2x» + 6 A , B , C , D E
(x-l)«(x-8)« x-1 (x-l)« (a;-8) (x-8)« (z-8)«
2z« + 6 = -4(x - 1) (X - 8)» + B(x - 8)« + C{x - l)«(x - 8)«
+ D(x-l)«(x-8) + JS:(z-l)«.
The 8ubetitution of 1, 3, 0, 2, 4 gives the equations
8=-8B, eO = 4E, 9^ + 3C-D + 12 = 0,
^_C + 2> + 6 = 0, ^ + 3C + 3Z) = 0.
'I'he solutions are — 9/4, — 1, + 9/4, — 8/2, 16, and the integral becomes
8 16
2(x-8) 2(x-8)«*
The importance of the fact that the method of partial fractions shows that any
rational fraction may be integrated and, moreover, that the integral may at most con-
sist of a rational part plus the logarithm of a rational fraction plus the inverse
tangent of a rational fraction should not be overlooked. Taken with the method
of substitution it establishes very wide categories of integrands which are inte-
grate in terms of elementary functions, and effects their integration even though
by a somewhat laborious method.
15. The method of substitution depends on the identity
f/^''^"f/^'^^^^^% '^ a; = 0(y), (690
which is allied to (59). To show that the integral on the right with respect to y
is the integral of /(x) with respect to x it is merely necessary to show that its
derivative with respect to x is /(x). By definition of integration,
and if/^*(y)^%=-^^*(y)^f/i=nMm
by (4). The identity is therefore proved. The method of integration by substitu-
tion is in fact seen to be merely such a systematization of the method based on
(AJ)) and set forth in § 12 as will make it practicable for more complicated problems.
Again, differentials may be used if preferred.
Let R denote a rational function. To effect the integration of
fsin X R (sin*x, cosx), let cos x = y, then j—R{l — y*, y) ;
I coax R (cos*x, sinx), let sin x = y, then Th (1 — y*, y) ;
/R(,inx,co,x), let un| = ,, then j;«(ji^. L_g)j^.
The last substitution renders any rational function of sin x and cos x rational in
the variable y ; it should not be used, however, if the previous ones are applicable
— it is almo.»^t certain to give a more difficult final rational fraction to integTAte.
00
INTRODUCTORY REVIEW
A large number of geometric problems give integrands which are rational in x
and In some one of the radicals Va» + x*, Va* - x*, Vx^ - a^. These may be con-
▼6rt«d Into trigonometric or hyperbolic integrands by the following substitutions :
Cr{x^ Va*-!*) x = a8iny, ^^(asiny, acosy)acosy;
x = ata.ny, ri2(atany, asecy)asec2y
•^ x = a8inhy, l 12 (a sinh y, a cosh y) a cosh y ;
fR{x, Vx« - a«)
X = o sec y
JS(asecy, a tan y) a sec y tan y
y
a cosh y, /B (a cosh y, a sinh y) a sinh y.
It frequently turns out that the integrals on the right are easily obtained by
methods already given ; otherwise they can be treated by the substitutions above.
In addition to these substitutions there are a large number of others which are
applied under si)ecific conditions. Many of them will be found among the exer-
ciiies. Moreover, it frequently happens that an integrand, which does not come
under any of the standard types for which substitutions are indicated, is none the
leas integrable by some substitution which the form of the integrand will suggest.
Tables of integrals, giving the integrals of a large number of integrands, have
been constructed by using various methods of integration. B. O. Peirce's " Short
Table of Integrals " may be cited. If the particular integrand which is desired does
not occur in the Table, it may be possible to devise some substitution which will
reduce it to a tabulated form. In the Table are also given a large number of
reduction formulas (for the most part deduced by means of integration by parts)
which accomplish the successive simplification of integrands which could perhaps
be treated by other methods, but only with an excessive amount of labor. Several
of these reduction formulas are cited among the exercises. Although the Table is
useful in performing integrations and indeed makes it to a large extent unneces-
■ary to learn the various methods of integration, the exercises immediately below,
which are constructed for the purpose of illustrating methods of integration, should
be done without the aid of a Table.
EXERCISES
1. Integrate the following by parts :
' {a) j'x coeh x, » (/S) Ttan-i x,
t. If P(x) la a polynomial and P'(x), P"(x), • • • its derivatives, show
(a) /P(x)e- = 1 e«rP(x) - 1 P'(x) + -1 P"(x) 1
•^ o> I. CL or J
(7) J'x'-logx,
ifi) fP(x)coia« = l8lnax
•^ a
P(x)-ip"(x) + lp.'(x)-...]
+ lco.ax[ll-(x)-ip"'(x) + JjiH(,)_...],
■ad (>) dsrin • ilmlUr ranlt for the Integrand P(x) sin ax.
FUNDAMENTAL RULES 23
3. By guccessive integration by part8 and Bubsequent solution, show
c«*8lnte =
e^{afiinbx — 6cosbr)
e^(b8into + aco8te)
(7) jxt^'cmx = j*5e*'[6x(8inx + 2 coax) — 4 8lnx — 8 coax].
4. Trove by integration by parts the reduction formulas
/ V r • - ^ 8in"'+ixco8"-ix . n — 1 /• . _ ^ „
(a) / Kin'»xco8"x = + I Bin"* x cos" -^x,
,^. r^ tan'^-ixsec"* m — 1 r^ „
(a) / tan-'xsec'x = / tan^-^xsec'x,
^ 'J m + n-1 m+n-W
^^ ' J (x2 + a*)" ~ 2 (n - 1) a* L (x* + a')"- 1 "^ ^ ^~ ^ J (x« + a«)«-ij'
^ ^ J (logx)» ~ (n - 1) (log X)" -1 n-lJ (log x)- -1 '
5. Integrate by decomposition into partial fractions :
<">/(x-i)(x-2)' <^>/^^3^' ^^^/rr^'
^ ^J (x + 2)2(x + l)' ^''J 2x« + x» ' ^*'Jx(l + x«)**
6. Integrate by trigonometric or hyperbolic substitution :
(rt) rVa2-x2, (/3) r Vx2 - a2, (7) TVoM^,
•' (a-x«)
7. Find the areas of these curves and their volumes of revolution :
{a) xl + yJ = ai, (/S) a^* = a^x* - x«, (7) (-Y + (-^ = 1.
8. Integrate by .converting to a rational algebraic fraction :
?in3x ,_ r cosSx . . r 8in2x
/8in3x r COS3X / \ f sinzx
a- ws'-^'x + ft^sin^x * J a^ cos^ x + 6^ sin^ x * J a^ cos« x + 6* sin*
Jo + 6co8X* »/ a + 6cosx + csinx* •'1 +
X
1 — cosx
sinx
9. Show that jR(x^ Vo + 6x4- ex*) may be treated by trigonometric substitu-
tion ; distinguish between 6* — 4 ac ^ 0.
10. Show that CrIx, \I ) is made rational by ir = Hence Infer
J \ \cx + d/ cx + d
that fR(x, V{x — a){x — /3)) is rationalized by j/« = ^~^. This accomplishes
•/ X — a
the integration of R (x, Va + fix + ex*) when the roots of a + te + c«* = 0 are
real, that is, when If^ — Aac> 0.
24 INTRODUCTORY REVIEW
U. Show that /«[x, (^p (^)"' • • •]' ^^«^ ^^« exponents m, n,
... are raUonal, U raUonalized by y* = ^^^^ if A: is so chosen that km, Ten, - • • are
ex + 1*
Integen.
la. Show that C{a-\- hy)^ may be rationalized if p or gf orj) + 9 is an integer.
By MtUng X" = y show that fx* (a + te«)p may be reduced to the above type and
hence U Integrable when ^^-^ orp or ^^-^ + P is integral.
n n
13. If the roote of a + te + cx« = 0 are imaginary, Je (x, Va + 6x + cx«) may
be rationalized by y = Va + 6x + cx^ ::f x Vc.
14. Integrate the following .
/«*' /* X* / y\ r ^
V?Ti* ' J \/(l-x2)8' "^ (X - d) Va + to + cx^'
''x(l + x«)i J x^ •^ Vl-x» *
16. In view of Ex. 12 discuss the integrability of :
let X = ay^,
//- r x^ (]
Bln"»xcoff»x, let sinx=Vy, (/3) \ ——== <
•^ Vox — x* U
Vax-x2 l^^ Vox - x2 = xy.
16. Apply the reduction formulas, Table, p. 66, to show that the final integral for
f-S= is f 1_ or f 1_ or r-4=
•^ Vl-X« •^ Vl-X2 •^ Vl-X3 •^xVl-X*
aooording m m is even or odd and positive or odd and negative.
17. Prove sundry of the formulas of Peirce's Table.
18. Show that if H (x, Va* — x'-') contains x only to odd powers, the substitu-
tion t = Va* — x» will rationalize the expression. Use Exs. 1 (f) and 6 (e) to
compare the labor of this algebraic substitution with that of the trigonometric or
hyperbolic.
16. Definite integrals. If an interval from x = a\.ox = bhQ divided
into i» tuooessive intervals Aa^i, Aa-,, • • -, Aa;, and the value /(^,) of a
function f(x) be computed from some point ^,- in each interval Aa;,- and
be multiplied by Ajr<, then the limit of the sum
Urn r/X^i) CLt, 4-/(^0 Ax, + . . . + f{Q Ax J = fj(x) dx, (62)
FUNDAMENTAL RULES
25
when each interval becomes infinitely short and their number n be-
L'oraes infinite, is known as the definite integral ot f(x) from a to 6, and
is designated as indicated. If y=f{x) be graphed, the sum will \w
represented by the area under
a broken line, and it is clear
that the limit of the sum, tliat
is, the integral, will be repre-
sented by the area under the
curve y=f(x) and between
the ordinates x = a and x — b.
Thus the definite integral, de-
fined arithmetically by (62),
may be connected with a geo-
metric concept which can serve to suggest properties of the integral
much as the interpretation of the derivative as the slope of the tan-
gent served as a useful geometric representation of the arithmetical
definition (2).
For instance, if a, i, c are successive values of a;, then
«^i ^.
«i
in b X
£f{x)dx+rfix)dx=rf{x)d^
(63)
is the equivalent of the fact that the area from a to c is equal to the
sum of the areas from a to b and b to c. Again, if Aa; be considered
positive when x moves from a to i, it must be considered negative
when X moves from b to a and hence from (62)
///(^)rfx = -X/(x)<fa.
(64)
Finally, if M be the maximum of f(x) in the interval, the area under
the curve will be less than that under the line y = M through the
highest point of the curve ; and if m be the minimum of /(x), the
area under the curve is greater than that under y = vi. Hence
■I (b - a) < fy(x) dx < M(b - a).
(66)
There is, then, some intermediate value m< fA< M such that the inte-
gral is equal to fi(b — a); and if the line y=fi cuts the curve in a
point whose abscissa is $ intermediate between a and b, then
/:
j^f(x)dx = ^(f>-o^
This is the fundamental Theorm) nf tin
^ ('■-«)/{()■ (66')
Mr,fn for detinue integrals.
26 INTRODUCTORY REVIEW
The definition (62) may be applied directly to the evaluation of the definite in-
tegrals of the simplest functions. Consider first 1/x and let a, b be positive with a
\em than 6. Let the interval from o to 6 be divided into n intervals Ax, which are
in geometrical progreasion in the ratio r so that Xi = a, Xg = ar, • • •, x„+i = ar«
and Axi = a(r-1), Ar, = ar(r-1), Ax, = ar^ (r - 1), • • -, Ax, = ar--i(r- 1) ;
whence 6- o = Axi + Ax, + ••• + Ax, = a(r»- 1) and r* = b/a.
Choose the points (i in the intervals Ax^ as the initial points of the intervals. Then
I, e, f» a ar ar^-^
But r = Vb/a or n = log (b/a) -i- logr.
„ Axi . Ax, , . Ax, , ,. , 6 r — 1 , 6 h
Hence — » + =2 + . . . 4. :=:2 = n (r — 1) = log - • = log
ii (« ^H a logr a log(l + A)
Now if n becomes infinite, r approaches 1, and h approaches 0. But the limit of
log (1 + h)/h as A = 0 is by definition the derivative of log (1 + x) when x = 0 and
is 1. Hence
^a X ii = »Lti ta in J a
As another illustration let it be required to evaluate the integral of cos^ x from
0 to I X. Here let the intervals Ax,- be equal and their number odd. Choose the f s
aa the initial points of their intervals. The sum of which the limit is desired is
w = coa^ 0 • Ax + cos2 Ax • Ax + cos^ 2 Ax • Ax + • • •
+ cos2 (n — 2) Ax • Ax + cos^ (n — 1) Ax • Ax.
But nAx = J T, and (n - 1) Ax = ^ tt - Ax, (n - 2) Ax = i tt - 2 Ax, • . .,
mad cos {\ IT — y) = sin y and sin^y + cos^y = l.
Henoe r = Ax [cos* 0 + cos^ Ax + cos* 2 Ax + ... + sin* 2 Ax + sin^ Ax]
V
Hence f*c(Mflxdx= lim [J nAx + i Ax] = lim (J 7r + i Ax) = i ?r.
Indications for finding the integrals of other functions are given in the exercises.
It should be noticed that the variable x which appears in the expression of the
definite integral really has nothing to do with the value of the integral but merely
•enrei as a symlwl useful in forming the sum in (62). What is of importance is
tlM function /and the limite a, b of the interval over which the integral is taken.
^ fix) dx = f^ /(<) di=f^ /(y) dy=f /(•) d«.
TteTtrtebto in the integrand disappears in the integration and leaves the value of
tb§ latiglll M a function of thu limits a and b alone.
FUNDAMENTAL RULES 27
17. If the lower limit of the integral be fixed, the value
X
of the integral is a function of the upjx?? limit regarded as variable
To find the derivative <l>'(^), form the quotient (2),
' f(x)dx- I f(x)dx
By applying (63) and (65% this takes the simpler form
' f(x)dx
where ^ is intermediate between b and h -f Ai. Let A6 = 0. Then ^
approaches h and /(^) approaches /(^). Hence
*'(i) = ^j["/W'^ =/(*)• (66)
If preferred, the variable h may be written as x, and
♦(x)=jr/(x)<fo, *'(x)=£jr/(x)<fe=/(x). (66-)
This equation will establisli the relation between the definite integral
and the indefinite integral. For by definition, the indefinite integral
F{x) of f(x) is any function such that F'(x) equals /(a;). As *'(x) =/(x)
it follows that px
j f(x)dx = F(x)^C. (67)
Hence except for an additive constant, the indefinite integral of / is
the definite integral of / from a fixed lower limit to a variable upper
limit. As the definite integral vanishes when the upper limit coincides
with the lower, the constant C is — F{a) a"
X
fix) dx = F{b) - FCa). (67^)
Hence, the dejinlte integral of f(x) from a to b is the difference between
the values of ant/ Indejinite integral F(x) taken for the upper and lower
limits of the definite integral; and if the indefinite integral of / is
known, the definite integral may be obtained without applying the
definition (62) to/
28 INTRODUCTORY REVIEW
The great importance of definite integrals to geometry and physics
lies in that fact that mamj quantities connected with geometric figures
or physical bodies maij be expressed simply for small portions of the
figures or bodies and may then be obtained as the sum of those quanti-
ties taken over all the small portions, or rather, as the limit of that sum
when the portions become sm^ler and smaller. Thus the area under a
curve cannot in the first instance be evaluated ; but if only that portion
of the curve which lies over a small interval Aa; be considered and the
rectangle corresponding to the ordinate /(^) be drawn, it is clear that
the area of the rectangle is /(^) Aa;, that the area of all the rectangles is
the sum %f{i)\x taken from a to b, that when the intervals Aa; approach
zero the limit of their sum is the area under the curve ; and hence that
area may be written as the definite integral oif(x) from a to b*
In like manner consider th£ mass of a rod of variable density and suppose the
rod to lie along the x-axis so that the density may be taken as a function of x.
In any small length Ax of the rod the density is nearly constant and the mass of
that part is approximately equal to the product pAx of the density p{x) at the
initial point of that part times the length Ax of the part. In fact it is clear that
the mass will be intermediate between the products wiAx and 3fAx, where m and
Jf are the minimum and maximum densities in the interval Ax. In other words
the mass of the section Ax will be exactly equal to p (^) Ax where ^ is some value of
X In the interval Ax. The ma.ss of the whole rod is therefore the sum 2p(|)Ax
taken from one end of the rod to the other, and if the intervals be allowed to
approach zero, the mass may be written as the integral of p{x) from one end of
the rod to the other, t
Another problem that may be treated by these methods is that of finding the
total pressure on a vertical area submerged in a liquid, say, in water. Let w be the
weight of a column of water of cross section 1 sq. unit and
of height 1 unit. (If the unit is a foot, lo = 62.5 lb.) At a
point h units below the surface of the water the pressure is
wh and upon a small area near that depth the pressure is
approximately whA if A be the area. The pressure on the
area A is exactly equal to w^A if f is some depth interme-
diate between that of the top and that of the bottom of
the area. Now let the finite area be ruled into strips of height A^. Consider the
product wkb{h) AA where b(h) =f{h) is the breadth of the area at the depth h. This
• The {•§ may evidently be so chosen that the finite sum 2i/*(^)Ax is exactly equal to
Um area under the curve ; but still it is necessary to let the intervals approach zero and
thos replace the eom by an integral because the values of f which make the sum equal
to the aiea are unknown.
t Thb and nimilar problems, here treated by using the Theorem of the Mean for
bitegrals, may be treated from the point of view of differentiation as in § 7 or from that
of Dohanel'a or Osgood's Theorem as in §§ :U, .•«. It should be needless to state that in
aay particular problem some one of the three methods is likely to be somewhat preferable
to either of the others. The reason for layiug such emphasis upon the Theorem of the
Mean hers and in the ezerclites below is that the theorem is in itself very important and
■eede to be thoroughly mastered.
FUNDAMENTAL RULES 29
iM approximately the preieure on the strip as it is the pressure at the top of the strip
iimltiplied by the approximate area of the strip. Then u>(6({) AA, where | is some
value bi'twi'eii A and h + AA, is the actual pressure on the Mtrip. (It is sufficient to
write the pressure as approximately toA6(A)AA and not trouble with the (.) The
total preHsure is then Zw^b{^) A/t or better the limit of that 8um. Then
P = lim Vtc{6(e)dA= r w>A6(A)dA,
where a is the depth of the top of the area and b that of the bottom. To evaluate
the pressure it is merely necessary to find the breadth 6 as a function of A and
integrate.
EXERCISES
1. If ilc i8 a constant, show I k/{x)dx — k i f{x)dx.
2. Show that f {u±v)dx=^ f udx ± f vdx.
3. If, from a to 6, V(x)</(x) < 0(z), show f ^(z)(tc < T f{^x)dx < f 0(x)dz.
Ja Ja Ja
4. Suppose that the minimum and maximum of the quotient Q(x) =/(x)/0(z)
of two functions in the interval from a to 6 are m and M^ and let 0(z) be positive
so that
m<q{x) = ^<M and m0(a;) </(x) < Jf0(x)
0(X)
are true relations. Show by Exs. 3 and 1 that
Cnx)dx fW)dx
m<^^ <^f and ^^ = ^=Q{^) = 1^,
fj{x)dx fj^"")^ ^^^^
where ( is some value of x between a and h.
5. If m and M are the minimum and maximum of f{z) between a and 6 and if
^ (x) is always positive in the interval, show that
m f 0(x)dx < f /{x)<f>{x)dx <M f 0(x)dx
Ja Ja Ja
and f /(x)^(x)dx = n C it>{x)dx =/(f) C 0(x)dx.
Ja Ja Ja
Note that the integrals of [3f -/(x)]0(x) and [/(x)-m]0(x) are positive and
apply Ex. 2.
6. Evaluate the following by the direct application of (62) :
{a) / xdx = — - — , (/9) / e*dx = €* - «-.
Ja 2 J a
Take equal intervals and use the rules for arithmetic and geometric progressions.
7. Evaluate (a) C x"dx = (6«« +» _ a* +»), (/J) C c*dz = -^ (c* - c^.
Ja m + 1 *'« logc
In the first the intervals should be taken in geometric progression with f^ = b/a.
30 INTRODUCTORY REVIEW
8. Show direcUy that (a) f'sin^xdac = i ir, (/3) f cog»xdx = 0, if n is odd.
«/0 •'0
9. With the aid of the trigonometric formulas
coflX + co82a; + ••• + co8(n — l)x = i [sinnxcot^x — 1 — cosnx],
ainx + 8in2x + • • • + 8in(n— 1) x = i[(l — co8nx)cotix — sinnx],
abow (a) f coaxdx = sinft- sina, (/3) J* sinxdx = cosa- cos6.
10. A function is said to be even if /(- x) =/(x) and odd if /(- x) = -/(x).
Show (a) f ^ /(x) dx = 2 j^ /(*) d«, / even, O^) f ^ /(x) dx = 0, / odd.
11. Show that if an integral is regarded as a function of the lower limit, the
upper limit being fixed, then
♦'(a) = ^ f'f{x)dx = -/(a), if *(a) = f /(x)dx.
12. Use the relation between definite and indefinite integrals to compare
X
/(x)(ix = (6-a)/(f) and F{h)- F{a) = {h - a)F\i),
the Theorem of the Mean for derivatives and for definite integrals.
IS. From consideration of Exs. 12 and 4 establish Cauchy''s Formula
A* *(6)--*(a) *'(f)
which states that the quotient of the increments AF and A* of two functions, in
any interval in which the derivative *'(x) does not vanish, is equal to the quotient
of the derivatives of the functions for some interior point of the interval. What
would the application of the Theorem of the Mean for derivatives to numerator
and denominator of the left-hand fraction give, and wherein does it differ from
Cauchy*8 Formula ?
14. Discuss the volume of revolution of y =/(x) as the limit of the sum of thin
cylinders and compare the results with those found in Ex. 9, p. 10.
15. Show that the mass of a rod running from a to b along the x-axis is
J *(6* — o*) if the density varies as the distance from the origin {k is a factor of
proportionality).
16. Show (a) that the mass in a rod running from a to & is the same as the area
under the curve y = p{x) between the ordinates x = a and x = 6, and explain why
thifl ibould be seen intuitively to be so. Show (/3) that if the density in a plane slab
boanded by the x-axia, the curve y =/(x), and the ordinates x = a and x = 6 is a
Jftb
yp (x)dx ; also (7) that the mass
a
pb
of the oorrafponding volume of revolution is / iry^p (x) dx.
17. An iaoecelet triangle has the altitude a and the base 26. Find (a) the mass
on the Mnimptlon that the density varies as the distance from the vertex (meas-
ured elong the altitude). Find (/J) the mass of the cone of revolution formed by
revoking the triangle about iu altitude if the law of density is the same.
FUNDAMENTAL RULES 81
18. In a plane, the moment of inertia I ef a particle qf mass m with retpeei to a
point in (leflncd sm tlie prrxluct mr* of the mtit» by the square of its distance from the
|M>iiit. ExtJMui this tlfllnition from particlf« to Ixxlies.
(a) Show that tlit* inoiiientH of inertia of a rod running from a to b and of a
circular slab of radius a are respectively
l=Cx^p{x)dx and I = f 2in*p{r)dr, p the density,
if the point of reference for the rod is the origin and for the slab is the center.
(/3) Show that for a rod of length 2 1 and of uniform density, / = ^ ^fI* wltli
respect to the center and / = J MP with respect to the end, M being the total mass
of the rod.
(7) ¥(>T a unifonn circular slab with respect to the center / = | Ma*.
(8) Ft)r a unifonn nxl of length 2/ with respect to a jMiint at a dist<ince d from
its center is / = M (\ /* -f (f^). Take the rod along the axis and let the point be
(a, p) with d« = a» + /32.
19. A rectangular gate holds in check the water in a reservoir. If the gate is
submerged over a vertical distance // and has a breadth B and the top of the
gate is a units below the surface of the water, find the pressure on the gate. At
what depth in the water is the point where the pressure is the mean pressure
over the gate ?
20. A dam is in the form of an isosceles trapezoid 100 ft. along the top (which
is at tlie water level) and 60 ft. along the bottom and 30 ft. high. Find the pres-
sure in tons.
21. Find the pressure on a circular gate in a water main if the radius of the
circle is r and the depth of the center of the circle below the water level is d^r.
22. In space, moments of inertia are defined relative to an axis and in tlie for-
nnila I = mr^, for a single particle, r is the perpendicular di.stance from the
particle to the axis.
{a) Show that if the density in a solid of revolution generated by y =f{x) varies
only with the distance along the axis, the moment of inertia about the axis of
I rry*p (x) dx. Apply Ex. 18 after dividing the solid into disks.
„
(/3) Find the moment of inertia of a sphere about a diameter in case the density
is constant ; / = ^ 3/a- = y"; trpa^.
(7) Apply the result to find the moment of inertia of a spherical shell with
external and internal radii a and 6 ; / = | M{a^ — 6^)/(a* — 6'). Let 6 = a and
thus find / = J Ma- as the moment of inertia of a spherical surface (shell of negli-
gible thickness).
(5) For a cone of revolution / = 1*5 Ma* where a is the radius of the base.
23. If the force of attraction exerted by a mass m upon a point is krnf{r) where
r is the distance from the ma.s8 to the point, show that the attraction exerted at
the origin by a ro<l of density p (x) running from a to 6 along the x-axis is
i4 = r kf{x) p (x) dx, and that A - kM/ab, 3f = p (6 - a),
is the attraction of a uniform roil if the law is the Law of Nature, that \a^
fir) = l/r».
^2 INTRODUCTORY REVIEW
84. SuppoM that the density p in the slab of Ex. 16 were a function p (x, y) of
both z and y. Show that the mass of a small slice over the interval AXf would be
of the form
Azj p(x,y)dy = *(e)Ax andthat J *(x)Ax=j J p{x,y)dy\dx
would be the expreasfon for the total mass and would require an integration with
respect to y In which x was held constant, a substitution of the limits f{x) and 0
for y, and then an integration with respect to x from a to 6.
85. Apply the considerations of Ex. 24 to finding moments of inertia of
(a) a uniform triangle y = »nx, y = 0, a; = a with respect to the origin,
(p) a uniform rectangle with respect to the center,
(7) a uniform ellipse with respect to the center.
86. Compare Exs. 24 and 16 to treat the volume under the surface z = p(x, y)
and over the area bounded by y =/(x), y = 0, x = a, a; = 6. Find the volume
(a) under z = xy and over y^ = 4px, y = 0, x = 0, x = 6,
{$) under « = x* + y' and over x* + y^ = a^, y = 0. x = 0, x = Q,
(y) under _ + ^ + _ = 1 and over _ + ^ = l, y = 0, x = 0, x = a.
87. Discuss sectorial area j jr^d<f> in polar coordinates as the limit of the sum
of small sectors running out from the pole.
88. Show that the moment of inertia of a uniform circular sector of angle a
r*d0 in polar coordinates.
89. Find the moment of inertia of a uniform (a) lemniscate r^ = a^ cos^ 2 0
and (/}) cardioid r = a(l — cos0) with respect to the pole. Also of (7) the circle
r = 2 a 006^ and (a) the rose r = a sin 2 0 and (e) the rose r = a sin 3 0.
CHAPTER II
REVIEW OF FUNDAMENTAL THEORY*
18. Numbers and limits. The concept and theory of real number^
integral, mtiunal, and irrational, will not be set forth in detail here.
Some matters, however, which are necessary to the proper understand-
ing of rigorous methods in analysis must l3e mentioned ; and numerous
points of view which are adopted in the study of irrational number
will be suggested in the text or exercises.
It is taken for granted that by his earlier work the reader has become familiar
with the use of real numbers. In particular it is assumed that he is accustomed
to represent numbers as a scale, that is, by points on a straight line, and that he
knows tliat when a line is given and an origin chosen upon it and a unit of measure
and a positive direction have been chosen, then to each point of the line corre-
sponds one and only one real number, and conversely. Owing to this correspond-
ence, that is, owing to the conception of a scale, it is possible to interchange
statements about numbers with statements about points and hence to obtain a
more vivid and graphic or a more abstract and arithmetic phraseology as may be
desired. Thus instead of saying that the numbers Zi, Xj, • • • are increasing algebra-
ically, one may say that the points (whose coSrdinates are) Xi, Xa, • • • are moving
in the positive direction or to the right ; with a similar correlation of a decreasing
suite of numbers with points moving in the negative direction or to the left. It
should be remembered, however, that whether a statement is couched in geometric
or al^'ebraic terms, it is always a statement concerning numbers when one has in
mind the point of view of pure analysis.t
It may be recalled that arithmetic begins with the integers, including 0, and
with addition and multiplication. That second, the rational numbers of the
form p/q are introduced with the operation of division and the negative rational
numbers with the operation of subtraction. Finally, the irrational numbers are
introduced by various processes. Thus V2 occurs in geometry through the
necessity of expressing the length of the diagonal of a square, and Vs for the
diagonal of a cube. Again, ir is needed for the ratio of circumference to diameter
in a circle. In aljjebra any equation of odd degree has at least one real root and
hence may be regarded as defining a number. But there is an essential difference
iK'tween rational and irrational numbers in that any rational number is of the
• The object of this chapter is to set forth systematically, with attention to precision
of HtHtcineiit and accuracy of proof, those fundamental detinitions and theorems which
lie at the basis of calculus and whicli have been given in the previous chapter from an
iutuitive rather than a critical iH»int »»f view.
t Some illustrative graphs will be given ; the student should make many others.
S3
84 INTRODUCTORY REVIEW
form ± p/g with q ^0 and can therefore be written down explicitly ; whereas
the irrational numbers arise by a variety of processes and, although they may be
represented to any desired accuracy by a decimal, they cannot all be written
down explicitly. It is therefore necessary to have some definite axioms regulating
the e«ential properties of irrational numbers. The particular axiom upon which
stress will here be laid is the axiom of continuity, the use of which is essential
to the proof of elementary theorems on limits.
19. Axiom of Continuity. If all the points of a line are divided into
two classes such that every point of the first class precedes every point of
the second class^ there must he a point C such that any point preceding
C is in the first class and any point succeeding C is in the second class.
This principle may be stated in terms of numbers, as : If all real num-
bers be assorted into two classes such that every number of the first class
is algebraically less than every number of the second class, there must he
a number N such that any number less than N is in the first class and
any number greater than N is in the second. The number N (or point C)
is called the frontier number (or point), or simply the frontier of the
two classes, and in particular it is the upper frontier for the first class
and the lower frontier for the second.
To consider a particular case, let all the negative numbers and zero constitute
the first class and all the positive numbers the second, or let the negative numbers
alone be the first class and the positive numbers with zero the second. In either
case it is clear that the classes satisfy the conditions of the axiom and that zero is
the frontier number such that any lesser number is in the first class and any
greater in the second. If, however, one were to consider the system of all positive
and negative numbers but without zero, it is clear that there would be no number
N which would satisfy the conditions demanded by the axiom when the two
classes were the negative and positive numbers ; for no matter how small a posi-
tive number were taken as JV, there would be smaller numbers which would also
be positive and would not belong to the first class ; and similarly in case it were
attempted to find a negative N. Thus the axiom insures the presence of zero in
the system, and in like manner insures the presence of every other number — a
matter which is of importance because there is no way of writing all (irrational)
numbers in explicit form.
Further to appreciate the continuity of the number scale, consider the four
significations attributable to the phrase "(Ae interval from a to 6." They are
CL^x^h^ a<x^6, a^x<h^ a<x <b.
That is to say, both end points or either or neither may belong to the interval. In
the case a is absent, the interval has no first point ; and if b is absent, there is no
la«t point. Thus if zero is not counted as a positive number, there is no least
positive number ; for if any least number were named, half of it would surely be
lea, and hene^ the absurdity. The axiom of continuity shows that if all numbers
be divided Into two classes as required, there must be either a greatest in the first
class or a least In the second — the frontier — but not both unless the frontier is
counted twice, once In each class.
FUNDAMENTAL THEORY 35
20. Definition of a Limit. If x it a variahle which takes on succes-
sive values a-,, r^, • • • , ar<, Xjy • • • , the variable x is said to approach the cotv-
stant I as a limit if the numerical difference between x and I ultimately
becomes^ and for all succeeding values of x remains,
less than any preassiyned nutnber rm matter how n 1 ii a< ' ) ' x*
small. The numerical difference Ix'tween x and /
is denoted by j^z^ — /| or |/ — :z;| and is called the absolute value of the
difference. The fact of the approach to a limit may be stated aa
|a; — /| < < for all x's subsequent to some x
or x = l-\-rii \yf\< t for all ar's subsequent to some ar,
where c is a positive number which may be assigned at pleasure and
must be assigned Ijefore the attempt be made to find an x such that
for all subsequent x's the relation |a; — /| < c holds.
So long as the conditions required in the definition of a limit are satisfied there
is no need of bothering about how the variable approaches its limit, whether from
one side or alternately from one side and the other, whether discontinuously as in
the case of the area of the polygons used for computing the area of a circle or
continuously as in the case of a train brought to rest by its brakes. To speak
geometrically, a point x which changes its position upon a line approaches the
point / as a limit if the point x ultimately comes into and remains in an assigned
interval, no matter how small, surrounding I.
A variable is said to become infinite if the numerical value of the
variable ultimately becomes and remains greater than any preassigned
nunil)er A', no matter how large.* The notation is a; = oo, but had best
Ix* read " x becomes infinite," not " x equals infinity."
Theorem 1. If a variable is always increasing, it either becomes
infinite or approaches a limit.
That the variable may increase indefinitely is apparent. But if it does not
become infinite, there must be numbers K which are greater than any value of
the variable. Then any number must satisfy one of two conditions : either there
are values of the variable which are greater than it or there are no values of the
variable greater than it. Moreover all numbers that satisfy the first condition are
less than any number which satisfies the second. All numbers are therefore
divided into two cla-sses fulfilling the requirements of the axiom of continuity, and
there must be a number N such that there are values of the variable greater than
any number N — e which is less than N. Hence if e be assigned, there is a value of
the variable which lies in the interval ^ — e < x ^ iV, and as the variable is always
increiusing, all subsequent values must lie in this interval. Therefore the variable
approaches ^ as a limit.
•This definition means what it says, and no more. Later, additional or (lifT»'rent
meanings may be assigned to infinity, but not now. Loose and extniiuous t ouctiits in
this connection are ahuost certain to introduce errors and confusion.
86 INTRODUCTORY REVIEW
EXERCISES
1. If Xi, X,, . . •, X,, • • •, X, + p, • • • is a suite approaching a limit, apply the defi-
nition of a limit to show that when e is given it must be possible to find a value of
» ao great that |x, +p - x,| < e for all values of p.
2. If Xi, x«, • • • is a suite approaching a limit and if yi, ye, • • • is any suite such
that I y, — z, I approaches zero when n becomes infinite, show that the y's approach
a limit which is identical with the limit of the x's.
8. As the definition of a limit is phrased in terms of inequalities and absolute
▼aluea, note the following rules of operation :
c b . a a
(a) If a > 0 and c> 6, then - > - and - < r »
^ ' a a c b
(^ ja + 6 + c+ ...|^|a| + |6| + |c|+ •••, (7) \abc- • .\ = \a\>\b\.\c\- . >,
where the equality sign in (p) holds only if the numbers a, 6, c, • • • have the same
sign. By these relations and the definition of a limit prove the fundamental
theorems:
If lim X = X and lim y = T, then Urn {x ±y) = X ± Y and lim xy = XY.
4. Prove Theorem 1 when restated in the slightly changed form : If a variable
X never decreases and never exceeds K, then x approaches a limit N and N ^ K.
Illustrate fully. State and prove the corresponding theorem for the case of a
variable never increasing.
6. If Xi, Xg, • • • and yi, y2, • • • are two suites of which the first never decreases
and the second never increases, all the y's being greater than any of the x's, and if
when e is assigned an n can be found such that Vn — ^n< «, show that the limits
of the suites are identical.
6. If Xi, Xj, • • • and yi, y2, • • • are two suites which never decrease, show by Ex. 4
(not by Ex. 3) that the suites Xi + yi, X2 + y2, • • • and x^yi, X2y2, • • • approach
limits. Note that two infinite decimals are precisely two suites which never de-
crease a8 more and more figures are taken. They do not always increase, for some
of the figures may be 0.
7. If the word " all " in the hypothesis of the axiom of continuity be assumed to
refer only to rational numbers so that the statement becomes : If all rational
numbers be divided into two classes • • • , there shall be a number N (not neces-
sarily rational) such that • • • ; then the conclusion may be taken as defining a
number as the frontier of a sequence of rational numbers. Show that if two num-
bers X, y be defined by two such sequences, and if the sum of the numbers be
diffined as the number defined by the sequence of the sums of corresponding terms
M In Er, 6, and if the product of the numbers be d^ned as the number defined by
the sequence of the products as in Ex. 6, then the fundamental rules
X+r=r+X, XY=YX, {X-\-Y)Z = XZ+YZ
of arithmetic hold for the numbers X, T, Z defined by sequences. In this way a
complete theory of irrationals may be built up from the properties of rationals
oombine<l with the principle of continuity, namely, 1° by defining irrationals as
frontiers of sequences of rationals, 2° by defining the operations of addltibn, multi-
plication, ... as operations upon the rational numbers in the sequences, 8° by
showing that the fundamenUl rules of arithmetic still hold for the irrationals.
FUNDAMENTAL THEORY 87
8. Apply the principle of continuity to show that there i« a podtiTe namber x
such that X* = 2. To do this it should be shown that the ratioiutls are divioiUe
into two classes, those whose square is leas than 2 and thoae whoae square la not
less than 2 ; and that these classes satisfy the requirements of the axiom of conti-
nuity. In like manner if a is any positive number and n Ls any positive integer,
show that there is an x such that x* = a.
21. Theorems on limits and on sets of points. The theorem on
limits which is of fundaiiiental algebraic importance is
Thkorkm 2. If R (x, y, «,•••) be any rational function of the variables
^> y> *> • •> ^^d ^^ these variables are approaching limits X, K, Z, •••,
then the value of R approaches a limit and the limit is R(Xj K, Z, •••),
provided there is no division by zero.
As any rational expression is made up from its elements by combinations of
addition, subtraction, multiplication, and division, it \s sufficient to prove the
theorem for these four operations. All except the last have been indicated in the
above Ex. 3. As multiplication has been cared for, division need be considered
only in the simple case of a reciprocal 1/x. It must be proved that if limz = X,
then lim (1/x) = 1/X. Now
X X\ \x\\X\
This quantity must be shown to be less than any assigned c. As the quantity is
complicated it will be replaced by a simpler one which is greater, owing to an
increase in the denominator. Since x .^ X, x — X may be made numerically as
small as desired, say less than e', for all x's subsequent to some particular x. Hence
if f' be taken at least as small as \\X\, it appears that |x| must be greater than
\\X\. Then
'^""^1 < l^~^l = _il_ , by Ex. 3 la) above,
and if t' be restricted to being less than ^(A'j^e, the difference is less than e and
-the theorem that lim (1/x) = l/X is proved, and also Theorem 2. The necessity
for the restriction X ^i 0 and the corresponding restriction in the statement of
the theorem is obvious.
Theorem 3. If when c is given, no matter how small, it is possible
to find a value of n so great that the difference |ic,^.p — a*,! between ar,
and every subsequent term x^^^ in the suite Xj, a*,, •••, a;,, ••• is less
than €, the suite approaches a limit, and conversely.
The converse part has alreatly been given as Ex. 1 above. The theorem itself Is
a consequence of the axiom of continuity. First note that as |x,+p — x,| < « for
all x's subsequent to x„, the x's cannot become infinite. Suppose 1° that there
is some number / such that no matter how remote x, is in the suite, there are
always subsequent values of x which are greater than I and others which are Ie«
than /. As all the x's after x, lie in the inter>'al 2r and a« / is less than some x's
and greater than others, I must lie in that interval. Hence |< ~ x.^.,! < S« for all
38 INTRODUCTORY REVIEW
«'» subsequent to x.. But now 2 e can be made as small as desired because e can be
taken as small as desired. Hence the definition of a limit applies and the x's
approach / as a limit.
Suppose 2® that there is no such number /. Then every number k is such that
either it is possible to go so far in the suite that all subsequent numbers x are
as great as fc or it is possible to go so far that all subsequent x's are less than k.
Hence all numbers k are divided into two classes which satisfy the requirements of
the axiom of continuity, and there must be a number N such that the x's ultimately
come to lie between iV — e' and N + e', no matter how small c' is. Hence the x's
approach -Y as a limit. Thus under either supposition the suite approaches a limit
and the theorem is proved. It may be noted that under the second supposition the
x's ultimately lie entirely upon one side of the point N and that the condition
jx, + p — x,|<ei8not used except to show that the x's remain finite.
22. Consider next a set of points (or their correlative numbers)
without any implication that they form a suite, that is, that one may
be said to be subsequent to another. If there is only a finite number
of points in the set, there is a point farthest to the right and one
farthest to the left. If there is an infinity of points in the set, two
possibilities arise. Either V it is not possible to assign a point K so
far to the right that no point of the set is farther to the right — in
which case the set is said to be unlimited above — or 2° there is a
point K such that no point of the set is beyond K — and the set is
said to be limited above. Similarly, a set may be limited below or un^
limited below. If a set is limited above and below so that it is entirely
contained in a finite interval, it is said merely to be limited. If there
is a point C such that in any interval, no matter how small, surround-
ing C there are points of the set, then C is called a j)oint of condensa-
tion of the set (C itself may or may not belong to the set).
Theorem 4. Any infinite set of points which is limited has an
upper frontier (maximum?), a lower frontier (minimum?), and at
least one point of condensation.
Before proving this theorem, consider three infinite sets as illustrations :
(a) 1, 1.9, 1.99, 1.999, • • ., (/j) _ 2, • . ., - 1.99, - 1.9, - 1,
(7) -l,-i,-i,.--,i,i,l.
In (a) the element 1 is the minimum and serves also as the lower frontier ; it is
clearly not a point of condensation, but is isolated. There is no maximum ; but 2
Is the upper frontier and also a point of condensation. In (/3) there is a niaxinmm
— 1 and a minimum — 2 (for — 2 has been incorporated with the set). In (7) there
Is a maximum and minimum ; the point of condensation is 0. If one could be sure
that an infinite set had a maximum and minimum, as is the case with finite
■etc, there would be no need of considering upper and lower frontiers. It is clear
that If the upper or lower frontier belongs to the set, there is a maximum or
miuimum and the frontier is not necessarily a point of condensation ; whereas
FUNDAMENTAL THEOKY 89
if the frontier does not belong to the aet^ it i» nectuarUy a point of eondenaaUon amd
the correttponding extreme point w muurtngr.
To prove that there in an upper frontier, divide the poinU of the line into two
classes, one consisting of points whicli are to the left of some point of the let, the
other of pf>ints which are not to the left of any point of the set — then apply the
axiom. Similarly for the lower frontier. To show the existence of & point of con-
densation, note that as there is an infinity of elements in the set, any point p is such
that either there is an infinity of points of the set to the right of it or there is not.
Hence the two classes into which all points are to be assorted are suggested, and
the application of the axiom offers no difficulty.
EXERCISES
1. Ill a manner analogous to the proof of Theorem 2, show that
la) lim = -, Ifi) lim 1- = -» (7) Hm . =— 1.
2. Given an infinite series S = Ui + Wi + u$ + • • • . Construct the suite
Si = Ui, Sj = Ml + Uj, Sf = Ui + Ma + U,, . . ., Si = Ui +M, + . . . + Ui, . . .,
where Si is the sum of the first t terms. Show that Theorem 3 gives : The neces-
sary and sufficient condition that the series 6' converge is that it is possible to find
an n so large that |Sn + p — S^\ shall be less than an assigned e for all values of p.
It is to be understood that a series converges when the suite of 6"s approaches a limit,
and conversely.
3. If in a series wi — it^ + Ug — M4 + • • • the terms approach the limit 0, are
alternately positive and negative, and each term is less than the preceding, the
series converges. Consider the suites i>i, St, -So, • • • and Sj, S^, S^, • • • .
4. Given three infinite suites of numbers
2i,a^, •••,x», •••; yi, ys, •••, y»., •••; zi, za, • • •, z., • •
of which the first never decreases, the second never increases, and the terms of the
tliird lie between corresponding terms of the first two, x^ ^ Zn = Vm- Show that
the suite of z's has a point of condensation at or between the limits approached by
the x's and by the y's ; and that if lim x = lim y = /, then the z's approach / as a
limit.
5. Restate the definitions and theorems on sets of points in arithmetic terms.
6. Give the details of the proof of Theorem 4. Show that the proof as outlined
gives the least point of condensation. How would the proof be worded so as to give
the greatest jKjint of condensation ? Show that if a set is limited abovei^it has an
upper frontier but need not have a lower frontier.
7. If a set of points is such that between any two there is a third, the set is said
to be dense. Show that tlie rationals form a dense set ; also the irrationals. Show
that any point of a dense set is a jKnnt of condensation for the set.
8. Show that the rationals p/q where g < /T do not form a dense set — in fact
are a finite set in any limited interval. Hence in regarding any irrational as the
limit of a set of rationals it is necessary that the denominators and also the numer-
ators should become infinite.
40 INTRODUCTORY REVIEW
9. Show that if an infinite set of points lies in a limited region of the plane,
Bay in the rectangle a^x^b, c^y^d, there must be at least one point of
condensation of the set. Give the necessary definitions and apply the axiom
of continuity successively to the abscissas and ordinates.
23. Real functions of a real variable. If x be a variable which
takes on a certain set of values of which the totality may be denoted
by [x] and if y is a second variable the value of which is uniquely
determined for each x of the set [x], then y is said to be a function of
X dejined over the set [x']. The terms " limited," " unlimited," " limited
above," " unlimited below," • • • are applied to a function if they are
applicable to the set [y] of values of the function. Hence Theorem 4
has the corollary :
Theorem 5. If a function is limited over the set [a?], it has an
upper frontier M and a lower frontier m for that set.
If the function takes on its upper frontier M, that is, if there is a
value x^ in the set [x'\ such that /(aj^) = M, the function has the abso-
lute maximum M at ic^; and similarly with respect to the lower
frontier. In any case, the difference M — m between the upper and
lower frontiers is called the oscillation of the function for the set [x\
The set [a;] is generally an interval.
Consider some illustrations of functions and sets over which they are defined.
The reciprocal 1/x is defined for all values of x save 0. In the neighborhood of 0
the function is unlimited above for positive x's and unlimited below for negative x's.
It should be noted that the function is not limited in the interval 0 < x ^ a but is
limited in the_ interval e ^ x ^ a where e is any assigned positive number. The
function + Vx is defined for all positive x's including 0 and is limited below. It
is not limited above for the totality of all positive numbers ; but if K is assigned,
the function is limited in the interval O^x^K. The factorial function x ! is de-
fined only for positive integers, is limited below by the value 1, but is not limited
above unless the set [x] is limited above. The function ^(x) denoting the integer
not greater than x or " the integral part of x " is defined for all positive numbers
— for instance ^(3) = ^(tt) = 3. This function is not expressed, like the elemen-
tary functions of calculus, as a " formula " ; it is defined by a definite law, however,
and is just as much of a function as x* + 3x + 2 or J sin2 2x 4- logx. Indeed it
should be noted that the elementary functions themselves are in the first instance
defined by definite laws and that it is not until after they have been made the
subject of considerable study and have been largely developed along analytic lines
that they appear as fonnulas. The ideas of function and formula are essentially
distinct and the latter is essentially secondary to the former.
The definition of function as given above excludes the so-called multiple-valued
functions such as vx and sin-* x where to a given value of x correspond more than
one value of the function. It is usual, however, in treating multiple-valued func-
tions to resolve the functions into different parts or branches so that each branch
Im a slnL^b'-valued function. Thus + Vx is one branch and - Vx the other branch
FUNDAMENTAL THEORY 41
of Vx ; iu fact when x Is positive the symbol y/x is usually restricted to mean
merely + Vx and thus becomes a Kitigle-valuud symbol. One branch of sin"* x cofr-
KiHtM of the values between — \ir and + ^ ir, other branches give values between
\ IT and j IT or — J ir and — f ir, and so on. Hence the term " function " will be
riKtricted in tliin cliapter to the single-valued functions allowed by the definition.
24. If x — a i» any point of an interval over which f(x) is defined^
the function f(x) is said to be continuous at the point x^a if
lim/(aj) =/(a), no matter how x =xa.
The function is said to be continuums in the interval if it is contintunu
tit ererif point of the Interval. If the function is not continuous at the
point a, it is said to be discontinuous at a ; and if it fails to be con-
tinuous at any one point of an interval, it is said to be discontinuous
in the interval.
Thkokem G. If any finite number of functions are continuous (at a
point or over an interval), any rational expression formed of those
functtions is continuous (at the point or over the interval) provided no
division by zero is called for.
Theorem 7. If y=f{x) is continuous at x^ and takes the value
i/q z=f{jr^ and if z = <j>{if) is a continuous function of y at y = y^^ then
X = <^[/(^)] will l)e a continuous function of x at x^.
In regard to the definition of continuity note that a function cannot be con-
tinuous at a point unless it is defined at tliat point. Thus e-^f^ is not continuous
at X = 0 because division by 0 is impossible and the function is undefined. If, how-
ever, the function be defined at 0 a8/(0) = 0, the function becomes continuous at
X = 0. In like manner the function 1/x is not continuous at the origin, and in this
case it is impossible to assign to/(0) any value which will render the function
continuous ; the function becomes infinite at the origin and the very idea of be-
coming infinite precludes the possibility of approach to a definite limit. Again, the
function E (x) is in general continuoius, but is discontinuous for integral values
of X. When a function is discontinuous at x = a, the amount of the discantiwuitif is
tlie limit of the oscillation 3f — m of the function in the interval a — d<x<a-|-4
surrounding the point a when S approaches zero as its limit. The discontinuity
of E{x) at each integral value of x is clearly 1 ; that of 1/x at the origin is infi-
nite no matter what value is assigned to/(0).
In case the interval over which /(x) is defined has end points, say a Sx ^6,
the question of continuity at x = a must of course be decided by sllowing x to
approach a from the right-hand side only ; and similarly it is a question of left-
handed approach to 6. In general, if for any reason it is desired to restrict the
approach of a variable U) its limit to being one-sided, the notations x = o+ and
X = 6- respectively are used to denote approach through greater values (right-
handed) and through lesser values (left-handed). It is not necessary to make this
spetnfication in the case of the ends of an inter\'al ; for It is understood that x
shall take on only values in the interval in question. It should be noted that
7
42 INTRODUCTORY REVIEW
lim fix) =/(xo) when « = «©+ in no wise implies the continuity of /(x) at xo ; a
timple example is that of E{x) at the positive integral points.
The proof of Theorem 6 is an immediate corollary application of Theorem 2. For
lim R [/(z), 0 (X) . . •] = « [lim /(x), lim 0 (x), ...] = « [/(lim x), 0 (lim x), • • •],
and the proof of Theorem 7 is equally simple.
Theorem 8. If f(x) is continuous at x = a, then for any positive
c which has been assigned, no matter how small, there may be found a
number S such that \f{x)—f(a)\<€ in the interval |a;— a|<8, and
hence in this interval the oscillation of f(x) is less than 2 c. And
conversely, if these conditions hold, the function is continuous.
This theorem is in reality nothing but a restatement of the definition of conti-
nuity combined with the definition of a limit. For "lim/(x) =f{a) when x = a,
no matter how " means that the difference between /(x) and /(a) can be made as
small as desired by taking x sufficiently near to a ; and conversely. The reason
for this restatement is that the present form is more amenable to analytic opera-
tions. It also suggests the geometric picture which corre-
sponds to the usual idea of continuity in graphs. For the
theorem states that if the two lines y =/(a) ± € be drawn,
the graph of the function remains between them for at least
the short distance 8 on each side of x = a ; and as e may be
assigned a value as small as desired, the graph cannot exhibit
breaks. On the other hand it should be noted that the actual
physical graph is not a curve but a band, a two-dimensional region of greater or
less breadth, and that a function could be discontinuous at every point of an
interval and yet lie entirely within the limits of any given physical graph.
It is clear that 5, which has to be determined subsequently to e, is in general
more and more restricted as c is taken smaller and that for different points it is
more restricted as the graph rises more rapidly. Thus if /(x) = 1/x and e = 1/1000,
a can be nearly 1/10 if Xq = 100, but must be slightly less than 1/1000 if Xo = 1, and
something less than 10- « if x is 10- «. Indeed, if x be allowed to approach zero, the
value i for any assigned e also approaches zero ; and although the function
/(x) = 1/x is continuous in the interval 0 < x ^ 1 and for any given xo and « a
number i may be found such that |/(x) — /(xo) | < c when |x — Xo] < 5, yet it is not
poMible to assign a number J which shall serve uniformly for all values of Xq.
25. Theorem 9. If a function f{x) is continuous in an interval
a^x^b with end points, it is possible to find a S such that
\f(x) — /(xo)| < « when |a; — Xo | < 8 for all points Xq ; and the function
is said to be uniformly continuous.
The proof is conducted by the method of reductio ad absurdum. Suppose e
I* BMigned. Consider the suite of values J, J, J, . . . , or any other suite which
approaches zero m a limit. Suppose that no one of these values will serve as a d
for all points of the Interval. Then there must be at least one point for which \
will not serve, at least one for which \ will not serve, at least one for which \ will
not serve, and so on indefinitely. This infinite set of points must have at least one
FUNDAMENTAL THEORY 48
point of condensation C such that in any interval surrounding C there are points for
which 2-* will not nerve vm B, no matter how large k. But now by hypothesis /(z)
iH continuous at C and hence a iminl>er J can be found such that |/(x) — /(C)|< \ <
when |x — Xo| < 2 a. The utu^illation of /(x) in the whole interval 41 is leas than «.
Now if xo he any i)oint in the middle half of this interval, |xo— C\<8; and if x
satiHiien tlie relation |x — Zo| < '« it miiMt Htill lie in the interval ii and the differ-
ence |/(x) — /(xo) I < «, being surely not greater than the oscillation of /in the whole
interval. Hence it is possible to surround C with an interval so small that the
sanie d will serve for any point of the interval. This contradicts the former con-
clusion, and hence the hypothesis upon which that conclusion was based must have
been false and it must have been possible to find a 8 which would serve for all
IH)ints of the interval. The reason why the proof would not apply to a function
like 1/x defined in the interval 0 < x ^ 1 lacking an end jx^int is precisely that
the point of condensation (J would be 0, and at 0 the function is not continuous
and |/(x) -/(C) |<i«, |x-C|<2« could not be satisfied.
Thkorem 10. If a function is continuous in a region which includes
its end points, the fun(;tion is limited.
TiiKoRKM 11. If a function is continuous in an interval which includes
its end points, the function takes on its upper frontier and has a maxi-
mum M'f similarly it has a minimum 7ti.
These are successive corollaries of Theorem 9. For let e be assigned and let 8
be determined so as to serve uniformly for all points of the interval. Divide the
interval b — a into n successive intervals of length 8 or less. Then in each such
interval /cannot increase by more than e nor decrease by more than e. Hence/
will be contained between the values /(a) + ne and /(a) — n<, and is limited. And
/(x) has an upper and a lower frontier in the interval. Next consider the rational
function l/(3f — /) of/. By Theorem 6 this is continuous in the inter>'al unless
the denominator vanishes, and if continuous it is limited. This, however, is impos-
sible for the reason that, as 3f is a frontier of values of /, the difference M — J
may be made as small as desired. Hence l/(3f — /) is not continuous and there
must be some value of x for which / = M.
Theorem 12. If f(x) is continuous in the interval a^x^b with end
points and if f(f>) and f(b) have opposite signs, there is at least one
point $, a < $ <bj in the interval for which the function vanishes.
And whether /(o) and/(/>) have opposite signs or not, there is a point
^, a <$ < by sucli that /(^) = /x, where /i is any value intermediate be-
tween the maximum and minimum of /in the interval
For convenience suppose that /(a) < 0. Then in the neighborhood of x = a the
function will remain negative on account of its continuity ; and in the neighbor-
hood of 6 it will remain positive. Let $ be the lower frontier of values of x which
make/(x) positive. Suppose that/(^) were either positive or negative. Then as
/ is continuous, an interval could be chosen surrounding { and so small that/ re-
mained positive or negative in that interval. In neither case could ( be the lower
frontier of positive values. Hence the contradiction, and /(f) must be xero. To
44 INTRODUCTORY REVIEW
prove the second part of the theorem, let c and d be the values of x which make
/ a minimum and maximum. Then the function /-m has opposite signs at c and
d, and mu«t vanish at some point of the interval between c and d ; and hence a
fortiori at some point of the interval from a to b.
EXERCISES
1. Note that z is a continuous function of x, and that consequently it follows
from Theorem 6 that any rational fraction P{x)/Q{x), where P and Q are poly-
nomials in z, must be continuous for all x's except roots of Q (x) = 0.
2. Graph the function x — ^ (x) f or x ^ 0 and show that it is continuous except
for integral values of x. Show that It is limited, has a minimum 0, an upper fron-
tier 1, but no maximum.
3. Suppose that/(x) is defined for an infinite set [xj of which x = a is a point
of condensation (not necessarily itself a point of the set). Suppose
lim [/(xO-/(^")] = 0 or \f{x')-f{x'')\<€,\x'-a\<8,\x''-a\<8,
when x' and x" regarded as independent variables approach a as a limit (passing
only over values of the set [x], of course). Show that/(x) approaches a limit as
x^a. By considering the set of values of /(x), the method of Theorem 3 applies
almost verbatim. Show that there is no essential change in the proof if it be
assumed that x' and x" become infinite, the set [x] being unlimited instead of
having a point of condensation a.
4. From the formula sin x < x and the formulas for sin u — sin tj and cos m — cos r
show that A sin x and A cosx are numerically less than 2 1 Ax | ; hence infer that sin x
and cosx are continuous functions of x for all values of x.
5. What are the intervals of continuity for tanx and esc x ? If c = 10-*, what
are approximately the largest available values of 5 that will make |/(x) — /(x©) l<*
when Xq = 1°, 30°, 60°, 89° for each ? Use a four-place table.
6. Let /(x) be defined in the interval from 0 to 1 as equal to 0 when x is irra-
tional and equal to \/q when x is rational and expressed as a fraction jp/q in lowest
terms. Show that/ is continuous for irrational values and discontinuous for
rational values. Ex. 8, p. 39, will be of assistance in treating the irrational values.
7. Note that in the definition of continuity a generalization may be introduced
by allowing the set [x] over which / is defined to be any set each point of which
is a point of condensation of the set, and that hence continuity over a dense set
(Ex. 7 above), say the rationals or irrationals, may be defined. This is important
because many functions are in the first instance defined only for rationals and are
subsequently defined for irrationals by interpolation. Note that if a function is
continuous over a dense set (say, the rationals), it does not follow that it is uni-
formly continuous over the set. For the point of condensation C which was used
in the prcKif of Theorem 9 may not be a point of the set (may be irrational), and
the proof would fall through for the same reason that it would in the case of 1/x
In the Interval 0 <x ^ 1, namely, because it could not be affirmed that the function
WM continuous at C. Show that if a function is defined and is uniformly continu-
otu over a dense set, the value /(x) will approach a limit when x approaches any
value a (not necessarily of tiie set, but situated between the upper and lower
FUNDAMENTAL THEORY 46
frontiers of the set), and that if this limit be defined a« the value of /(a), the
function will remain continuous. Ex. 8 may be used to advantage.
8. By factoring (x + Ax)" — z", show for integral values of n that when
0 ^ X ^ A', tluMi A (x") < n A'" -^ Ax for small Ar's and consequently z" is uniformly
continuouH in the interval O^x^ K. If it be assumed that z" has been defined
only for rational x's, it follows from Kx. 7 that the definition may be extended
to all x's and that the resulting x» will be continuous.
9. Suppose (a) that/(x) +f{v) =/(x + y) for any numbers x and y. Show that
/(n) = n/"(l) anil ri/{l/n) =/(l), and hence infer that f{x) = a/(l) = C% where
C =/(l), for all rational x'h. From Ex. 7 it follows that if /(x) is continuous^
/(x) = Cx for all x's. Consider (fi) the function /(x) such that/(x)/(i/) =/(« + y).
Show that it is Ce* = a'.
10. Show by Theorem 12 that if y =/{x) is a continuous constantly increasing
function In the interval a ^ x ^ 6, then to each value of y corresponds a single value
of X 8o that the function x =/-» {y) exists an<l is single-valued ; show also that
it Is continuous and constantly Increasing. State the corresponding theorem if
/(x) is constantly decreasing. The function f-^{y) is called the inverne function
to/(x).
11. Apply Ex. 10 to discuss y = Vx, where n Is integral, x is positive, and only
lM)sitlve roots are taken into consideration.
12. In arithmetic it may readily be shown that the equations
a'^* = a* + ••, (a"«)* = a***, a^b" = (ah)*,
are true when a and b are rational and positive and when m and n are any positive
and negative integers or zero, (a) Can It be inferred that they hold when a
and b are positive irrationals ? (/S) How about the extension of the fundamental
inequalities
x">l, when x>l, x»<l, when 0^x<l
to all rational values of n and the proof of the inequalities
x«">x» if m>n and x>l, x'»<x» if m>n and 0<x<l.
(7) Next consider x as held constant and the exponent n as variable. Discuss the
exponential function a' from this relation, and Exs. 10, 11, and other theorems that
may seem necessary. Treat the logarithm as the inverse of the exponential.
26. The derivative. If x = a is a jjoint of an interval over which
fi/) is defined and if tJte quotient
ax a
approaches a limit when h approaches zero, iw matter lioWy the function
f(.r) is said to be differentiable at x = a and the value of the limit ef
the quotient is the derivative f\a) off at x = a. In the case of differ-
entiability, the definition of a limit gives
ft+JihzIial =/'(„) + ,, or /(a + A) -/(a) = */•(«) + ^ (1)
wliere lini rj = 0 when lini h = 0, no matter how.
46 INTRODUCTORY REVIEW
In other words if « is given, a « can be found so that|i7|<e when |A[<«. This
shows that a function differentiable at o as in (1) is continuous at a. For
\/{a + A) -/(a) 1 ^ \r{a) 1 3 + e«, \h\<8.
If the limit of the quotient exists when h = 0 through positive values only, the
function ha« a right-hand derivative which may be denoted by/' (a+) and similarly
for the left-hand derivative/' (a-). At the end points of an interval the derivative
is always considered as one-handed ; but for interior points the right-hand and left-
band derivatives must be equal if the function is to have a derivative (unqualified).
The function is said to have an ivfinUe derivative at a if the quotient becomes infi-
nite as A = 0 ; but if a is an interior point, the quotient must become positively
Infinite or negatively infinite for all manners of approach and not positively infinite
for some and negatively infinite for others. Geometrically this allows a vertical
tangent with an inflection point, but not with a cusp as in Fig. 3, p. 8. If infinite
derivatives are allowed, the function may have a derivative and yet be discontin-
uous, as is suggested by any figure where /(a) is any value between lim /(x) when
X = a+ and lim/(x) when « = a-.
Theorem 13. If a function takes on its maximum (or minimum) at
an interior point of the interval of definition and if it is differentiable
at that point, the derivative is zero.
Theorem 14. Rollers Theorem. If a function f(x) is continuous over
an interval a^x^h with end points and vanishes at the ends and has
a derivative at each interior point a <x < b, there is some point i,
a<$<by such that /' (^) = 0.
Theorem 16. Theorem of the Mean. If a function is continuous over
an interval a ^ x ^ b and has a derivative at each interior point, there
is some point $ such that
where h ^ b ^ a* and ^ is a proper fraction, 0 < d < 1.
To prove the first theorem, note that iff {a) = 3f, the difference /(a + h) -f{a)
cannot be positive for any value of h and the quotient Af/A cannot be positive
when A > 0 and cannot be negative when A < 0. Hence the right-hand derivativf?
cannot be positive and the left-hand derivative cannot be negative. As these two
must be equal if the function has a derivative, it follows that they must be zero,
and the derivative is zero. The second theorem is an immediate corollary. For as
the function is continuous it must have a maximum and a minimum (Theorem 11)
both of which cannot be zero unless the function is always zero in the interval.
Now If the function is identically zero, the derivative is identically zero and the
theorem Is true ; whereas if the function is not identically zero, either the maximum
or minimum must be at an Interior point, and at that point the derivative will vanisli.
• Thai the theorem is true for any part of the interval from a to 6 if it is true for the
whole Interval foUowB from the fact that the conditions, namely, that / be continuous
•Ad that/' exist, hold for any part of tlie Interval if they hold for the whole.
FUNDAMENTAL THEORY 47
Po prove the last theorem construct the auxiliary function
As ^ (a) = f (6) = Of Rollers Theorem shows that there is some point for which
^'({) = 0, and if thig value be subKtituted in the expression for ^'(2) the solution
f<>r/'(0 K»ve8 the result denianded by the theorem. The proof, however, requires
the UHC of tlie function ^ (/) and iU* derivative and Ih not complete until it is shown
tliat ^ (/) really sjitiHfles the conditions of Rolle's Theorem, namely, is continuous
in the interval a^x^b and ha« a derivative for every point a <x < b. The con-
tinuity is a consequence of Theorem 6 ; that the derivative exists follows from the
tlirect application of the definition combined with the assumption that the deriva-
tive of /exists.
27. TiiEOKKM 16. If a function has a derivative which is identically
zero in the interval a ^ x ^ b, the function is constant ; and if two
functions have derivatives equal throughout the interval, the functions
differ by a constant.
Thkokkm 17. If f(x) is differentiable and becomes infinite when
X == a, the derivative cannot remain finite as a: = a.
Theorkm 18. If the derivative f'(x) of a function exists and is a
continuous function of x in the interval a ^ x ^ b^ the quotient A//A
converges uniformly toward its limit /'(a*).
These theorems are consequences of the Theorem of the Mean. For the first,
f{a-{-h)-f{a) = hf{a-\-eh) = 0, if h^b-a, or f{a + h)=f{a).
Hence /(x) is constant. And in case of two functions/and 0 with equal derivatives,
the difference \f/ (x) =/(x) — <f>{x) will have a derivative that is zero and the differ-
ence will be constant. For the second, let x,, be a fixed value near a and suppose that
in the interval from Xq to a the derivative remained finite, say less than K. Then
\f{xo + h) -/(xo)| = \hr{xo + eh)\^\h\K.
Now let Xq + A approach a and note that the left-hand term becomes infinite and
the supposition that/' remained finite is contradicted. For the third, note that/',
being continuous, must be uniformly continuous (Theorem 9), and hence that if « is
piven, a i may be found such that
/(x + A>-/(x)
•/'(x)^|/'(x-K9/i)-/'(x)|<«
when I A|< 3 and for all x's in the interval ; and the theorem ia prove<l.
Concerning derivatives of higher order no special remarks are necessary. Each
is the derivative of a definite function — the previous derivative. If the deriva-
tives of the first n onlers exist and are continuous, the derivative of onler n + 1
may or may not exist. In practical applications, however, the functions are gen-
erally indefinitely differentiable except at certain isolated points. The proof of
I^ibniz's Theorem (§ 8) may be reviseii so as to depend on elementary proccMOS.
Let the formula b« assumed for a given value of n. The only terms which can
48 INTRODUCTORY REVIEW
eontiibate to the term D^I>* + ^-*v in the formula for the (n + l)st derivative of
iM are the terms
ii(ii-l)...(i»-<.H2)^,,..^^,., n(n-l)...(n-i + l)^.^^
1.2...(<-1) 12. ..i
in which the flret factor la to be differentiated in the first and the second in the
■eoond. The sum of the coeflBcienta obtained by differentiating is
m(m-l)...(n-<-K2) n(n- 1). ■ • (n- i + 1) _ (n + l)n. ■ .(n- t + 2)
1.2. ..(<-!) 1.2...i 1.2. ..I
which i« precisely the proper coeflScient for the term I)^uJ> + 1 - % in the expansion
of the (n + l)8t derivative of uv by Leibniz's Theorem.
With regard to this rule and the other elementary rules of operation (4)-(7) of
the previous chapter it should be remarked that a theorem as well as a rule is in-
Tolved — thus: If two functions u and t> are differentiable at x^, then the product
MV Is differentiable at aj^, and the value of the derivative is u {x^ v' (Xq) + u' (x^) d (x^).
And similar theorems arise in connection with the other rules. As a matter of fact
the ordinary proof needs only to be gone over with care in order to convert it into
a rigorous demonstration. But care does need to be exercised both in stating the
theorem and in looking to the proof. For instance, the above theorem concerning
ft product Is not true if infinite derivatives are allowed. For let u be — 1, 0, or + 1
according as z is negative, 0, or positive, and let v = x. Now v has always a deriva-
tive which is 1 and u has always a derivative which is 0, + oo, or 0 according as x
is negative, 0, or positive. The product uc is |x|, of which the derivative is — 1 for
negative x's, + 1 for positive x's, and nonexistent for 0. Here the product has no
derivative at 0, although each factor has a derivative, and it would be useless to have
a formula for attempting to evaluate something that did not exist.
EXERCISES
1. 8how that if at a point the derivative of a function exists and is positive, the
function must be increasing at that point.
2. Suppose that the derivatives /'(a) and f'{b) exist and are not zero. Show
that /(a) and f{b) are relative maxima or minima of / in the interval a^x^b, and
determine the precise criteria in terms of the signs of the derivatives /'(a) and/'(6).
8. Show that if a continuous function has a positive right-hand derivative at
every point of the interval a ^ x ^ 6, then f{b) is the maximum value off. Simi-
larly, If the right-hand derivative is negative, show that/(6) is the minimum off.
4. Apply the Theorem of the Mean to show that if /'(x) is continuous at a, then
«',»"Ao X —X ^ '
«* and ^ being regarded as Independent.
5. Porm the Increments of a function /for equicreacent values of the variable :
A,/=/(a + A) -/(o), ^J = f{a + 2h)-f{a + A),
V = /(« + 8*)-/(a + 2A),....
FUNDAMENTAL THEORY 49
Thf-rto are called first differences ; the differences of these differences are
A}/=:f(a + 2h)- 2/(a + A) +/(a),
A.;/=/{a -^Sh)- 2/(a + 2/*) +/(a + A), • • •
wiiirli iire called the second differences; in like manner there are third differences
Af/ = f{a + 8 A) - 8/(a + 2 A) + 8/(o + h) -/(a), • • •
and 80 on. Apply the Law of the Mean to all the differences and show that
A?/ = W'(a + e^h + e^h ^if = h*r\a + eyh + B^h + ^,A), ...
Hence show that if the first n derivatives of / are continuous at o, then
/"(a) = lim ^ , r\a) = Hm ^ , • • • , /<-)(a) = Hm ^ •
6. Cauchy's Theorem. If /(j) and 0(jr) are continuous over a^x^b^ have
derivatives at each interior point, and if 0'(x) does not vanish in the interval,
f{b)-f{a) ^/-(f) ^^ /(a + A)^/(a) ^ /(a -t- (9A)
0 (6) - 0 (a) 0'(f ) 0 (a + A) - 0 (a) 0'(a + ^A)
Prove tliat this follows from the application of Rolle's Theorem to the function
^(x) =:/(x)-/(a)- [0(x)- 0(a)]Mz:Zj^).
0(6) -0(a)
7. One application of Ex. 6 is to the theory of indeterminate fonns. Show that
if /(a) = 0(a) = 0 and if /'(z)/0'(x) approaches a limit when x = a, then /(x)/0 (x)
will approach the same limit.
8. Taylor's Theorem. Note that the form f{b) =f{a) + (6 — «)/'(!) is '^'"e way
of writing the Theorem of the Mean. By the application of Rolle's Theorem to
^(x)=/(6)-/(x)-(5-x)Ax)-(5-x)^M-^ig5|^^
show f(b) = f{a) + (6 - a)r{a) + l^I^/"(e),
and to ^(x)=:/(6)-./(x)-(6-x)/'(x)-i^:^/-(x) <^^^/(.-i)(x)
2 ^ ' (n-1)! J
show /(6) = f{a) + (6 - a)/'(a) + <^_^/-(a) + • • •
ib-a)^ <^Zl^V->(f).
(n - 1) I "^ ^ ' n\ ^ '
What are the restrictions that must be imposed on the function and its derivatives ?
9. If a continuous function over a^x^b has a right-hand derivative at each
l>niiit of the interval which is zem, show that the function is constant. Apply Ex. 2
to the f unctions /(x) + e (x — a) and/(x) — e(x — a) to show that the maximum
difference between the functions is 2 e (6 — a) and that / must therefore be constant.
50
INTRODUCTORY REVIEW
10. State and prove the theorems implied in the formulas (4)-(6), p. 2.
11. Consider the extension of Ex. 7, p. 44, to derivatives of functions defined
over a dense set. If the derivative exists and is uniformly continuous over the dense
set, what of the existence and continuity of the derivative of the function when its
definition is extended as there indicated ?
12. If /(x) has a finite derivative at each point of the interval a^x^b, the
derivative f'{x) must take on every value intermediate between any two of its values.
To show this, take first the case where /'(a) and /'(6) have opposite signs and show,
by the continuity of / and by Theorem 13 and Ex. 2, that /'(^) = 0. Next if
f'(a)</i<f\b) without any restrictions on /'(a) and/'(6), consider the function
/(x) — /*x and its derivative f'{x) — /jl. Finally, prove the complete theorem. It
should be noted that the continuity of /'(x) is not assumed, nor is it proved ; for
there are functions which take every value intermediate between two given values
and yet are not continuous.
28. Summation and integration. Let/(x) be defined and limited
over the interval a ^ x ^ b and let ilf, w, and 0 = M — m be the
upper frontier, lower fron-
tier, and oscillation of f(x)
in the interval. Let n — 1
points of division be intro-
duced in the interval divid-
ing it into n consecutive
intervals S^ 82
L of
yl
Mi
m:
/
A
mi \
^
/
m
0
c
I
ii
I
) X
which the largest has the
length A and let Af„ m,-, 0,-,
and /(^,) be the upper and lower frontiers, the oscillation, and any
value of the function in the interval 8,-. Then the inequalities
mhi ^ rriihi ^ /(^,) 8.- ^ M,.8,. ^ il/8..
will hold, and if these terms be summed up for all n intervals,
will also hold. Let « = 2wi,8,, o- = 2/(^.0 ^o and 5 = 5^,8.. From (^)
it is clear that the difference S — s does not exceed
(M - m){h - a) = 0(b - a),
the product of the length of the interval by the oscillation in it. The
values of the sums 5, «, a will evidently depend on the number of parts
into which the interval is divided and on the way in which it is divided
into that nunilxjr of parts.
TiiKORKM 19. If n' additional jmints of division be introduced into
the interval, the sum S* constructed for the n -h n' — 1 points of division
FUNDAMENTAL THEORY 51
canuot be greater than S and cannot be less than S by more than
n'OA. Similarly, s' cannot be less tlian » and cannot exceed s by more
than 7t'OA.
rHKOKEM 20. There exists a lower frontier L for all possible methods
ot constructing the sum S and an upjMjr frontier / for s.
TiiKoKKM 21. l)arbuux-8 Theorem. When c is assigned it is possible
to find a A so small tliat for all methods of division for which i< ^ A,
the sums S and s shall dififer from their frontier values L and / by less
tlian any preassigned i.
To prove the first theorem note that although {A) is written for the whole inter-
val from a to 6 and for the sums constructed on it, yet It applies equally to any
I)art of the interval and to the sums constructed on that part. Hence if Si = Mtii be
the part of S due to the interval 3, and if .S',' be the part of S' due to this interval
after the introduction of some of the additional points into it, m,5,- ^ Sj ^ Si = Mtii.
Hence S/ is not greater than <S,- (and as this is true for each interval «,, S' is not
greater than <S) and, moreover, -S,- — 5^ is not greater than 0,-3,- and a fortiori not
greater than OA. As there are only n' new points, not more than n' of the intervals
d( can be affected, and hence the total decrease S — S' in S cannot be more than
n'OA. The treatment of s is analogous.
Inasmuch as (A) shows that the sums .S and s are limited, it follows from Theo-
rem 4 that they possess the frontiers required in Theorem 20. To prove Theorem 21
note first that as L is a frontier for all the sums S, there is some particular sum 8
which differs from I, by as little as desired, say J e. For this S let n be the number
of divisions. Now consider S' as any sum for which each 5, is less than A = J */nO.
U the sum S" be constructed by adding the n points of division for S to the points
of division for S\ S" cannot be greater than S and hence cannot differ from L by
so much as j e. Also S" cannot be greater than S' and cannot be less than S' by
more than nOA, which is \ e. As iS" differs from L by less than \e and S' differs
from 6" by less than \ e, .S' cannot differ from L by more than «, which was to be
proved. The treatment of s and / is analogous.
29. If indices are introduced to indicate the interval for which the
frontiers L and I are calculated and if fi lies in the interval from a to A,
then Zf and /^ will l)e functions of fi.
Theorkm 22. The equations L^ ^ L^ -^ L^y a<c<b; Li = -L^-y
Li =z fi(b — a), ni^fi^ My hold for L, and similar equations for L As
functions of )8, L^ and l^ are continuous, and if f(x) is continuous,
they are differentiable and have the common derivative /()8).
To prove that L^ = L^ -f- L*, consider c as one of the joints of division of the
interval from a to b. Then the sums S will satisfy S^ = S^ ■{- 5*, and as the limit
of a sum is the sum of the limits, the corresponding relation must hold for the
frontier L. To show that L^ = — L^ it is merely necessary to note that S^ = — <8j^
because in passing from 6 to a the inten'als 3,- must be taken with the sign oppotita
to that which they have when the direction is from a to 6. From (A) It appears
that m (6 - a) ^ S^ ^ AT {b - a) and hence in the limit m (6 - n) s L^ ^ -V (6 — a).
52 INTRODUCTOKY KEVIEW
Henoe there is a value fi^m^n^M, such that L^ = fi{b — a). To show that L^
U a continuous function of /S, take A' >|3f | and |m|, and consider the relations
x,a+* _ L$ = LS + L| + * - Li = L^+* = M, k|< a:,
Li-'^^Li = Li-^-Li-^-Ll_^=-Ll_,=-^% |/|<JS:.
Hence if « is assigned, a a may be found, namely 5 < f/A", so that |L| ** — X^| < c
when A < a and L^ is therefore continuous. Finally consider the quotients
= II and = II ,
A —h
where M is some number between the maximum and minimum of f{x) in the inter-
val /S ^ a; ^ /3 + A and, if / is continuous, is some value /(f) of / in that interval
and where /t' =/({') is some value of / in the interval /3— A^x^/3. Now let
A = 0. As the function / is continuous, lim /(f) = /(/3) and lim /(f ') = f{p) . Hence
the right-hand and left-hand derivatives exist and are equal and the function L^
has the derivative /(/3). The treatment of I is analogous.
Theorem 23. For a given interval and function /, the quantities I
and L satisfy the relation I ^ L] and the necessary and sufficient con-
dition that L = l is that there shall be some division of the interval
which shall make S (3/,. — m,) S, = 20,8, < c.
If L^ == /^, the function / is said to be integrable over the interval
from atob and the integral i f(x) dx is defined as the common value
Li = l^. Thus the definite integral is defined.
Theorem 24. If a function is integrable over an interval, it is inte-
grable over any part of the interval and the equations
rf(x) dx-i- f f(x) dx = Cfix) dx,
f f(x)dx = - r f(x)dx, rf(x)dx = fi(b-a)
hold ; moreover, / f{x)dx = F(I3) is a continuous function of p ; and
if /(x) is continuous, the derivative F'(p) will exist and hef(fi).
By (A) the sums S and « constructed for the same division of the interval satisfy
the relation S - « ^ 0. By Darboux's Theorem the sums S and s will approach the
values L and I when the divisions are indefinitely decreased. Hence L — l^O.
Now it L = l and a A be found so that when 5,- < A the inequalities S — i < ^ e and
/ - * < J « hold, then S - « = 2 (3f,- - m.) 5, = 20,5, < •; and hence the condition
ZOtBi < • U seen to be necessary. Conversely if there is any method of division such
that ZOA < «, then .S _ « < « and the lesser quantity L — I must also be less than e.
But If the difference between two consUint quantities can be made less than e,
where c ie arbitrarily assigned, the constant quantities are equal ; and hence the
FUNDAMENTAL THEORY 58
condittun )h Been to be alao sufficient. To Hhow that if a f uuctlou U Intagnble over
an interval, it iH integrable over any part of the interval, it la merely neoeMary to
ghow that if Li = /„, then L^ = l^ where a and p are two point« of the interval.
Here the condition ZO<d{<« applies; for if ZO<a< can be made lew than f for the
whole Interval, iUi value for any part of the interval, being leas than for the whole,
uiuBt be leas than «. The rest of Theorem 24 is a corollary of Theorem 22.
30. Thkokkm 25. A function is integrable over the interval a^x^b
if it is continuous in that interval.
Thkokkm 26. If the interval a^x^b over which f(x) is defined
and limited contains only a finite number of points at which / is dis-
continuous or if it contains an infinite number of points at which / is
discontinuous but these points have only a finite number of points of
condensation, the function is integrable.
Thkokkm 27. lif(j') is integrable over the interval a^x^b, the
sum o'=2/(^,)8, will approach the limit I f(x)dx when the indi-
vidual intervals S,- approach the limit zero, it being immaterial how
they approach that limit or how the points ^< are selected in their
respective intervals S,.
Thkokkm 28. If /(j*) is continuous in an interval a^x^by then
f{x) has an indefinite integral, namely I f(x)dxj in the interval.
Theorem 25 may be reduced to Theorem 23. For as the function is continuous,
it is possible to find a A so small that the oscillation of the function in any interval
of length A shall be as small as desired (Theorem 9). Suppose A be chosen so that
the oscillation is less than e/{b — a). Then 20,3,- < e when 5, < A ; and the function
is integrable. To prove Theorem 26, take first the case of a finite number of discon-
tinuities. Cut out the discontinuities surrounding each value of x at which/ is dis-
continuous by an interval of length 8. As the oscillation in each of these intervals
is not greater than 0, the contribution of these intervals to the sum SO.-a, is not
greater than Ond, where n is the number of the discontinuities. By taking 8 small
enough this may be made as small as desired, say less than \ e. Now in each of the
remaining parts of the interval a ^ x ^ 6, the function / is continuous and hence
integrable, and consequently the value of 20,3,- for these portions may be made ap
small as desired, say \ e. Thus the sum 20,5,- for the whole interval can be made
as small as desired and/(x) is integrable. When there are points of condensation
they may be treated just as the isolated points of discontinuity were treated. After
they have been surrounded by intervals, there will remain over only a finite num-
ber of discontinuities. Further details will be left to the reader.
For the proof of Theorem 27, appeal may be taken to the fundamental relation
{A) which shows that a^c^S. Now let the number of divisions increase indefi-
nitely and each division become indefinitely small. As the function is int^rable,
S and s approach the same limit j f(x)dx, and consequently c which is included
between them must approach that limit. Theorem 28 is a corollary of Theorem 24
54 INTRODUCTORV REVIEW
which bcates that a«/(x) is continuous, the derivative of f f{x) dx is /(a;). By defi-
nition, the indefinite integral is any function whose derivative is the integrand.
Hence f /(x)dx is an indefinite integral of /(x), and any other may be obtained
by adding to this an arbitrary constant (Theorem 16). Thus it is seen that the
proof of the existence of the indefinite integral for any given continuous function
U made to depend on the theory of definite integrals.
1. Rework some of the proofs in the text with I replacing L.
2. Show that the L obtained from Cf{x), where C is a constant, is C times the L
obtained from/. Also if m, r, w are all limited in the interval a^x^b, the L JEor
the combination u -\- v — w will be L{u) ■}- L (v) — L (ly), where L {u) dienotes the L
for u, etc. State and prove the corresponding theorems for definite integrals and
hence the corresponding theorems for indefinite integrals.
3. Show that 20,5,- can be made less than an assigned e in the case of the func-
tion of Ex. 6, p. 44. Note that t = 0, and hence infer that the function is integrable
and the integral is zero. The proof may be made to depend on the fact that there
are only a finite number of values of the function greater than any assigned value.
4. State with care and prove the results of Exs. 3 and 5, p. 29. What restric-
tion is to be placed on f{x) if /(f) may replace /* ?
5. State with care and prove the results of Ex. 4, p. 29, and Ex. 13, p. 30.
6. If a function is limited in the interval a^x^b and never decreases, show
that the function is integrable. This follows from the fact that SO,- ^ O is finite.
7. More generally, let/(x) be such a function that SO,- remains less than some
number K, no matter how the interval be divided. Show that/ is integrable. Such
a function is called & function of limited variation (§ 127).
8. Change of variable. Let f{x) be continuous over a^x^b. Change the
variable to x = 0(«), where it is supposed that a = <f,{t^) and b = <t>{t^), and that
^(0» ^'(0» and/[0 (t)] are continuous in t over t^^t^ t^. Show that
J fix) dx= f[4> {t)] <t>\t) dt or f fix) dx= f f[<p it)] <l>\t) dt.
Do this by showing that the derivatives of the two sides of the last equation with
respect to t exist and are equal over t^^t^ t^, that the two sides vanish when
t = fj and are equal, and hence that they must be equal throughout the interval.
9. Osgood' 8 Theorem. Let a,- be a set of quantities which differ uniformly from
/iii) «< by an amount i-.Ji, that is, suppose
o^i = /(f .) Si + r.«i, where | f,- 1 < e and a ^ f ^ 6.
Prove that if /is integrable, the sum Sor,- approaches a limit when a, = 0 and that
tlie limit of the aum is f /(x)dx.
10. Apply Ex. 0 to the case Y = /'Ax + fAx where/' is continuous to show
directly that/(6) -/(a) = J* f'ix)dx. Also by regarding Ax = 0'(O At + fAf, apply
to Ex. 8 to prove the rule for change of variable.
PART 1. DIFFERENTIAL CALCULUS
CHAPTER III
TAYLOR'S FORMULA AND ALLIED TOPICS
31. Taylor's Formula. The object of Taylor's Formula is to express
the value of a function f(x) in terms of the values of the function and
its derivatives at some one point x=^a. Thus
fix) =fia) + (x- a)fXa) + ^~^f"ia) + • • •
Such an expansion is necessarily true because the remainder R may be
considered as defined by the equation; the real significance of the
formula must therefore lie in the possibility of finding a simple ex-
pression for R, and there are seveml.
Theorem. On the hypothesis that f{x) and its first n derivatives
exist and are continuous over the interval a^x^b^ the function may
be expanded in that interval into a polynomial in ar — a,
f(x) =f(a) + (X - a)f(a) + ^^^V'(«) + ' ' •
with the remainder R expressible in any one of the forms
«=^>(^)=^^J^/-(0
= (;rri)7jr*'"-'/'"H« + A-orf<,
(2)
where h = x — a and a <,$< x or $ = a -{-6h where 0 < ^ < 1.
A first proof may be made to depend on RoUe's Theorem as indicate in Ex. 8,
l». 41». Let X be reganled for the moment as constant, say equal to b. Construct
65
56 DIFFERENTIAL CALCULUS
the function ^ (x) there indicated. Note that ^ (a) = ^ (6) = 0 and that the deriva-
tive ^'(dt) is merely
r{x) = - l^Z:^/(») (X) + n ^4F^ r/(^) - /(«) - (^ - «)/'(«)
(n — 1)! (0— a)* L
- .-^^V-.>.)].
By Rolle's Theorem ^'(() = 0. Hence if f be substituted above, the result is
f{b) =/(a) + (6 - a)r{a) + • • • + ^^^3^'-^^" "''(«> + ^^^/^''MD,
after striking out the factor — (6 — ^)'»-i, multiplying by (6 — a)''/n, and transposing
f{b). The theorem is therefore proved with the first form of the remainder. This
proof does not require the corUinuity of the nth derivative nor its existence at a and at b.
The second form of the remainder may be found by applying Rolle's Theorem to
^(x)=/(6)-/(x)-(6-x)/'(x) i^^Z^V'-^Mx)-(&-a;)P,
where P is determined so that R = {b — a) P. Note that ^ (6) = 0 and that by
Taylor's Formula ^ (a) = 0. Now
nx)=-^]~'^!^~'f^''\^)-\-P or P=/W(^)^^~^^""' since ^'(^) = 0.
Hence if {be written f=a+^A where h=b—a, then b—^ = b—a—Oh={b—a){l — 0).
And B = (6- a) P = (6- a) C-a)-^!-^)-^,.,^^) ^ (6 - g). (1 - ^). -.
The second form of R is thus found. In this work as before, the result is proved
for X = 6, the end point of the interval a^x^b. But as the interval could be
considered as terminating at any of its points, the proof clearly applies to any x
in the interval.
A second proof of Taylor's Formula, and the easiest to remember, consists in
integrating the nth derivative n times from a to x. The successive results are
fy(''y{x)dx =/— i(x)T=/(«-i)(x) _/(n-l)(a).
f f'f<''Hx)dx^z= r7('-i)(x)dx- r/('»-i)(a)dx
•fa va va */a
= /<''-2)(x) -fin-2^a) _ (x _ a)/(«-i)(a).
J* '^ 'j[/<»> (X) dx« ==/<*-«) (X) -/<»-») (a) - (X - a)/(«-2) (a) - ^^^I^
L'"f, '^^"^ ^''^ ^ = -^^^^ - •^(«> - (^ - «)-^' («)
21*^^' (n - 1) !
The formula is therefore proved with R in the form f "- f /<''>(x) dx". To trans-
form thi« to the ordinary form, the Law of the Mean may be applied ((66), § 16). For
m(»-o)<r>-)(x)dx<Jif(x-a), mi^LlL^< r... r>)(x)dx'.<3f<^- "^
TAYLOR'S FORMULA; ALLIED TOPICS 6T
wtiBre m is the leant and ^f the greatetit value of /(">(/) from a to x. There !• than
some {nt(>niu>diate value /<")(() = ft 8uch tlutt
»/ u "^ <J H I
ThiH {troof requireH that the nth derivative be continuous and U leet genend.
The third proof in obuiiiied by applying successive integrations by parts to the
obvious identity /(a + h) -/{a) = f /'(a + A - «) (tt to make the integrand contain
Jo
higher derivativeH.
/(a + h)-fia) = r */'(« ■\-h-t)dt = ^f'{a + A - ^l* + f V"(a + A - 0*
Ju J 0 Jo
Jo Jo
=*/'(«)+^/"(»)+--+^^/'"-»(«)+X'(£fl7/<"'<''+*-')<tt
This, however, is precisely Taylor's Formula with the third form of remainder.
If the point a about which the function is expanded is ar = 0, the
expansion will take the form known as Maclaurin's Formula :
/(x) =/(0) + x/'(0) + f5/"(0) +. . . + (£^/<-" (0) + R, (3)
R = ?;/(")(te) = ^-^j (l-fl)— /<•>(*.)= ^-A_.J'r-'f<->(^^-t)dt.
32. Both Taylor's Formula and its special case, Maclaurin's, express
a function as a polynomial in h = x — a, of which all the coefiBcients
except the last are constants while the last is not constant but depends
on h both explicitly and through the unknown fraction B which itself is
a function of h. If, however, the nth derivative is continuous, the coeffi-
cient/^"^(a -H dA)/;i I must remain finite, and if the form of the deriva-
tive is known, it may be possible actually to assign limits between
which /^"^(a + Bh)/n ! lies. This is of great imiK)rtance in making
approximate calculations as in Exs. 8 ff. below; for it sets a limit to
the value of R for any value of n.
Theorem. There is only one possible expansion of a function into
a polynomial in h = x — a of which all the coefficients except the last
are constant and the last finite; and hence if such an expansion is
found in any manner, it must be Taylor's (or Maclaurin's).
To prove this theorem consider two polynomials of the nth order
Co + c^h + c,A« + . . + c«-iA--i + c,A»'= Co + CjA + C^« + . . . + C7,_iA"-» + C^\
which represent the same function and hence are equal for all values of A from 6
to & — a. It follows that the coefficients must be equal. For let A approach Oi
58 DIFFERENTIAL CALCULUS
The terms containing h will approach 0 and hence Cq and Cq may be made as
nearly equal as desired ; and as they are constants, they must be equal. Strike
them out from the equation and divide by h. The new equation must hold for all
values of h from 0 to 6 — a with the possible exception of 0. Again let h = 0 and
now It follows that Cj = C,. And so on, with all the coefficients. The two devel-
opments are seen to be identical, and hence identical with Taylor's.
To illustrate the application of the theorem, let it be required to find the expan-
sion of tan 2 about 0 when the expansions of sinx and cosx about 0 are given.
sinx = X - Jx« + xh«« + P«^ cosx = 1 - ^x^ + ^\x* + Qx«,
where P and Q remain finite in the neighborhood of x = 0. In the first place note
that tan x clearly has an expansion ; for the function and its derivatives (which
are combinations of tan x and sec x) are finite and continuous until x approaches i ir.
By division,
x + ^x3+ W x^
l-\x* + iiX*+ Qx«)x- Jx3 + T^^x6: + Px7
x-|x«+ ^\x^\+ Qx7
^^'\
Hence tan x = x + ix* + Ax5H x'. where S is the remainder in the division
cosx
and is an expression containing P, Q, and powers of x ; it must remain finite if P
and Q remain finite. The quotient S/cos x which is the coefficient of x' therefore
remains finite near x = 0, and the expression for tan x is the Maclaurin expansion
up to terms of the sixth order, plus a remainder.
In the case of functions compounded from simple functions of which the expan-
sion is known, this method of obtaining the expansion by algebraic processes upon
the known expansions treated as polynomials is generally shorter than to obtain
the result by differentiation. The computation may be abridged by omitting the
last terms and work such as follows the dotted line in the example above ; but if
this is done, care must be exercised against carrying the algebraic operations too
far or not far enough. In Ex. 6 below, the last terms should be put in and carried
far enough to insure that the desired expansion has neither more nor fewer terms
than the circumstances warrant.
EXERCISES
1. Assume R = (& - a)* P ; show B = ^"(^ - <^)"~ Vn) m,
^ ' ' (n-l)\k -^ ^^'
2. Apply Ex. 6, p. 29, to compare the third form of remainder with the first.
3. Obtain, by differentiation and substitution in (1), three nonvanishing terms :
(a) sin-^x, a = 0, (/3) tanh x, a = 0, (7) tan x, a = ^ ir,
(«) cscx, a = Jir, (<) e**^^:^ a = 0, (f) logsinx, a = ^ t.
4. Find the nth derivatives in the following cases and write the expansion :
(a) sin X, o = 0, (/3) sin x, a = | r, (7) c*, a = 0,
(«) c, a = 1, (*) logx, a = 1, (f) (1 + X)*, a = 0.
TAYLOR'S FORMULA; ALLIED TOPICS 59
5. By algebraic proceases find the Maclaurin expansion to the tenn in 2* :
(or) sec 05, 03) tanh «, (7) — Vl-x*,
(a) e=^8inx, (f) [Iog(l-a5)]«, (f) + Vcosh z,
(1;) c**"', (^) logcosx, (t) log Vl + X*.
The expaiiKlons needed in this work may be found by differentiation or taken
from B. (). Peirce'8 "Tables." In (7) and (f) apply the binomial theorem of Ex.
4 (J). In (ri) let y = sin z, expand e*, and substitute for y the expansion of sin z.
In {$) let cosz = 1 — ]/. In all cases show that the coefficient of the term in a^
really remains finite when z = 0.
6. If /(a + A) = Cq 4- c^h + c^A* + • • • + c„_iA»-i + c^h", show that In
f V(a + A)dA = c^ + ^;i« + ^/i* 4- • • + ^=^ A« + r*r,A«dA
Jo 2 3 u Jo
the last term may really be put in the form PA» +* with P finite. Apply Ex. 6, p. 29.
Jf** dx
^ . etc., to find developments of
0 Vl-z*
(a) sin -J z, (/3) tan-i z, (7) sinh-i z,
(,),ogl±f. (.)/%-<!., (n/'^<to.
1 — z Jo Jo z
In all these cases the results may be found if desired to n terms.
8. Show that the remainder in the Maclaurin development of f is less than
x'*€F/n ! ; and hence that the error introduced by disregarding the remainder in com-
puting (^ is less than xy^e^/n !. How many terms will suffice to compute e to four
decimals ? How many for ^ and for e^-^ ?
9. Show that the error introduced by disregarding the remainder in comput-
ing log (1 + z) is not greater than z"/^ if z > 0. How many terms are required for
the computation of log 1| to four places ? of log 1.2 ? Compute the latter.
10. The hypotenuse of a triangle is 20 and one angle is 31°. Find the sides by
expanding sinz and cosz about a = J tt as linear functions of z — ^ ir. Examine
the term in (z — ^ tt)* to find a maximum value to the error introduced by
neglecting it.
11. Compute to 6 places: (a) e^, (/3) log 1.1, (7) sin 30', (5) cos 30'. During
the computation one place more than the desired number should be carried along
in the arithmetic work for safety.
12. Show that the remainder for log(l + z) is less than z»/n(l + z)" if z<0.
Compute (a) log 0.9 to 5 places, (/3) log 0.8 to 4 places.
13. Show that the remainder for tan-*z is less than x'*/n where n may always
be taken as odd. Compute to 4 places tan-* \.
14. The relation J tt = tan-i 1 = 4 tan-i J — Un-i , Jg enables | v to be found
easily from the series for tan-* z. Find | ir to 7 places (intermediate work carried
to 8 places).
15. Computalion qf logarithms, (a) If a = log V» * = ^^K H» c = J^K ll» ^«"
log2 = 7a- 26 + 3c, log3 = 11 a - 36 + 5c, log 6 = 16a- 46 + 7c.
^ Jt.(t-l)...(fc-n-fl>^,
■ 1 . 2... n
or Rn<
60 DIFFERENTIAL CALCULUS
Now a =- log(l - ^), 6 = - log(l - y^^), c = log(l + ^\) are readily computed
and hence log 2, log 8, log 5 may be found. Carry the calculations of a, &, c to
10 places and deduce the logarithms of 2, 3, 5, 10, retaining only 8 places. Com-
pare Peirce'8 " Tables," p. 109.
(/J) Show that the error in the series for log ^j— ^ is less than ^ • Com-
pute log 2 corresponding to x = :J to 4 places, log If to 5 places, log 1^ to 6 places.
give an estimate of R^n+u »"<! compute to 10 figures log 3 and log 7 from log 2
and log 6 of Peirce's *' Tables " and from
81 7*
41og8-41og2-log6 = log — , 41og7 - 51og2 - log3 - 21og6 = log^j— ^.
16. Compute Ex. 7 (e) to 4 places f or z = 1 and to 6 places for x = \.
17. Compute sin-i 0.1 to seconds and sin-i ^ to minutes.
18. Show that in the expansion of (1 + Jc)* the remainder, as x is > or < 0, is
fc.(fc-l)...(fc-n + l) xn I ^^^
1.2...n (l+x)«- *i'
Hence compute to 5 figures Vl03, V98, V28, ^250, VlOOO.
19. Sometimes the remainder cannot be readily found but the terms of the
expansion appear to be diminishing so rapidly that all after a certain point appear
negligible. Thus use Peirce's "Tables," Nos. 774-789, to compute to four places
(estimated) the values of tan 6°, log cos 10°, esc 3°, sec 2°.
20. Find to within 1% the area under cos (x^) and sin (x^) from 0 to J tt.
21. A unit magnetic pole is placed at a distance L from the center of a magnet
of pole strength M and length 2 1, where l/L is small. Find the force on the pole
if (a) the pole is in the line of the magnet and if (/3) it is in the perpendicular
bisector.
Ana. {a) i^ (1 + e) with < about 2 ^^V , (/S) ?^ (1 - c) with e about ? (^ •
22. The formula for the distance of the horizon is D =vTa where D is the
distance in miles and h is the altitude of the observer in feet. Prove the formula
and show that the error is about \% for heights up to a few miles. Take the radius
of the earth as 8060 miles.
28. Find an approximate formula for the dip of the horizon in minutes below
the horizontal if h in feet is the height of the observer.
24. If S is a circular arc and C its chord and c the chord of half the arc, prove
S = J (8 c - C) (1 + e) where e is about SV7680 R* if R is the radius.
25. If two quantities differ from each other by a small fraction e of their value,
show that their geometric mean will differ from their arithmetic mean by about
1 1* of iu value.
26. The algebraic method may be applied to finding expansions of some func-
tions which become infinite. (Thus if the series for cosx and sinx be divided to
find cotx, the initial term is 1/z and becomes infinite at x = 0 just as cotx does.
TAVLOR'S FORMULA; ALLIED TOPICS 61
Such expansions are not Maclaurin developments but are analogouji to tbem.
The function zcotz would, however, have a Maclaurin development and tlie
expansion fuund for cot z is this development divided by x.) Find the develop-
ments about X = 0 to terms in x* for
(a) cotz, 05) cot^z, (7) C8C«, («) ok^x,
(<) cotzcscz, (f) l/(tan-*z)*, (if) (sin z — tan z)-»
27. Obtain the expansions :
(a) log8inz = logz-iz«-Tj5Z< + /«, (/9) log tanx = logz + Jz«+ ^z* + • -.,
(>) likewise for log versz.
33. Indeterminate forms, infinitesimals, infinites. If two functions
f(x) and <^(^) are detined for x = a and if <^('') =^ 0, the quotient//^ is
defined for x = a. But if <f> (a) = 0, the quotient f/<f> is not defined for a.
If in this case/ and <f> are defined and continuous in the neighborhood
of a and f(a) ^ 0, the quotient will l)ecome infinite as a- == « ; whereas
if /(«) = 0, the l)ehavior of the quotient //<^ is not immediately appar-
ent but gives rise to the indeterminate form 0/0. In like manner if/
and <f> become infinite at a, the quotient f/<f> is not defined, as neither
its numerator nor its denominator is defined ; thus arises the indeter-
minate form 00 /oo. The question of determining or evaluating an
indeterminate form is merely the question of finding out whether the
quotient f/<f> ai)i)roaches a limit (and if so, what limit) or becomes
}X)sitively or negatively infinite when x approaches a.
Theorem. UHospitaVs Rule. If the functions /(j-) and ^(jr), which
give rise to the indeterminate form 0/0 or oo/oo when x == a, are con-
tinuous and differentiable in the interval a < x ^ b and if b can be
taken so near to a that <t>'(x) does not vanish in the interval and if the
quotient /'/<^' of the derivatives approaches a limit or becomes posi-
tively or negatively infinite as a; = a, then the quotient f/<ft will ap-
proach that liiuit or become positively or negatively infinite as the case
may be. Hence an indetermimite form 0/0 or oo/ao vmi/ be replaced by
the quotient of the derivatives of numerator and denominator.
Case I. /(a) = 0 (a) = 0. The proof follows from Cauchy's Formula, Ex. 6, p. 40.
0(z) 0(z)-0(a) 0'(f)
Now If z = a, so must {, which lies between x and a. Hence if the quotient on the
right approaches a limit or becomes positively or negatively inflnite, the same is
tnie of that on the left. The necessity of inserting the restrictions that / and 0
shall be continuous and differentiable and that 0' shall not have a root indefinitely
near to a Is apparent inm\ the fact that Cauchy's Formula is proved only for funo>
ti(»nK that satisfy these conditions. If the derived form/'/^' should also be Inde-
terminate, the rule could again be applied and the quotient/"/^" would replace
j'/<tt' with the understanding that proper restrictions were satisfied by/*, ^\ and 4r,
62 DIFFERENTIAL CALCULUS
Ca8» II. /(a) = ^ (a) = 00. Apply Cauchy's Formula as follows :
m-m ^/(x) i-/(&)//(x) _r(g) a<x<6,
0(x) - 0(6) 0(x) 1 - 0(6)/0(x) 0'(f) ' X < $ <6,
where the middle expression is merely a different way of writing the first. Now
suppose that/'(x)/0'(x) approaches a limit when x = a. It must then be possible to
take 6 so near to a that/'({)/0'(f) differs from that limit by as little as desired, no
matter what value { may have between a and 6. Now as / and 0 become infinite
when X = a, it is possible to take x so near to a that /(6)//(x) and 0 (6)/0 (x) are
as near zero as desired. The second equation above then shows that/(x)/0(x),
multiplied by a quantity which differs from 1 by as little as desired, is equal to
a quantity /'(f)/0'(i) which differs from the limit of /'(x)/0'(x) as x = a by as little
as desired. Hence //0 must approach the same limit as/V0'. Similar reasoning
would apply to the supposition that/V0' became positively or negatively infinite,
and the theorem is proved. It may be noted that, by Theorem 16 of § 27, the form
/70' is sure to be indeterminate. The advantage of being able to differentiate
therefore lies wholly in the possibility that the new form be more amenable to
algebraic transformation than the old.
The other indeterminate forms 0- oo, 0", 1", 00*^, oo — 00 may be reduced to thie
foregoing by various devices which may be indicated as follows :
0-oo = - = -, 00 = eiogoo^goiogo = eo-oo .. 00 — 00 = loge"-« = log — .
00 0
The cajse where the variable becomes infinite instead of approaching a finite value
a is covered in Ex. 1 below. The theory is therefore completed.
Two methods which frequently may be used to shorten the work of evaluating
an indeterminate form are the method of E -functions and the application of Taylor's
Formula. By definition an E -function for the point x = a is any continuous function
which approaches a finite limit other than 0 when x = a. Suppose then that/(x) or
0(x) or both may be written as the products E^f^ and ^5^20^ Then the method of
treating indeterminate forms need be applied only to/j/0j and the result multiplied
by lim EJE^. For example,
lim .^ ~" = lim (x2 + ax + a^) lim ^~^ = Sa^ lim ^~^ = Sa^.
xi«sin(x — a) x=a x = asin(x — a) x = asin(x — a)
Again, suppose that in the form 0/0 both numerator and denominator may be de-
veloped about X = a by Taylor's Formula. The evaluation is immediate. Thus
tanx - sinx _ (x + |x8 + Px^') - (x - ^ x» + Qx^) _ | -{-(P - Q)a;2
a;Mog(l + X) ~ x2(x - ^x^ + B,x^) ~ \-\x-k-Rx^ '
and now if x = 0, the limit is at once shown to be simply \.
When the functions become infinite at x = a, the conditions requisite for Taylor's
Formula are not present and there is no Taylor expansion. Nevertheless an expan-
sion may sometimes be obtained by the algebraic method (§ 32) and may frequently
be used to advantage. To illustrate, let it be required to evaluate cot x — 1 /x which
is of the form » — « when x = 0. Here
«lni z-}i»+(Jt» a!l-Jx«+<Jx« xV 3 ^ j
TAYLOR'S FORMULA; ALLIED TOPICS 68
where S remaiiiB finite when x^O. If this value be BubsUtut«d for ootz, then
lira ('cot* - 1\ = lim(i - ix + Sx«- i\ = llm(- 1* + fix»\ = 0.
34. An infinitesimal is a varinhle which U ultimately to approach the
limit zt'vo ; an infinite is a variable which U to become either jjos it iveli/
or negatively infinite. Thus the increments Ay and Ax axe finite quan-
tities, but when tliey are to serve in the definition of a derivative they
must ultimately approach zero and hence may be called infinitesimals.
The form 0/0 represents the quotient of two infinitesimals ; ♦ the form
«/oo, the quotient of two infinites ; and 0- oc, the product of an infin-
itesimal by an infinite. If any infinitesimal a is chosen as the primary
infinitesimal, a second infinitesimal ^ is said to be of the aams order as
a if the limit of the quotient p/a exists and is not zero when a = 0 ;
whereiis if the quotient p/a becomes zero, p is said to be an infinites-
imal of higher order than or, but of lower order if the quotient becomes
infinite. If in i)articular the limit /S/a* exists and is not zero when
a = 0, then p is said to be of the nth order relative to a. The deter-
mination of the order of one infinitesimal relative to another is there-
fore essentially a problem in indeterminate forms. Similar definitions
may be given in regard to infinites.
Theorem. If the quotient ^/a of two infinitesimals approaches a
limit or becomes infinite when a = 0, the quotient p /a' of two infin-
itesimals which differ respectively from ^ and a by infinitesimals of
higher order will approach the same limit or become infinite.
Theorem. DuhamePs Theorem. If the sum 2a, = a, -f- ^j -I ha.
of n positive infinitesimals approacrhes a limit when their number n
l)ecomes infinite, the sum 2^,- = P^-\- P^A h A, where each ft differs
uniformly frouL the corresponding or, by an infinitesimal of higher
order, will approach the same limit.
As nr' — rtr is of higher order than a and p' — ^ oi higher order than /?,
\\m^^^^~^ = Q, Wm^-^^O or - = 1 + i;, ^ = 1 + f ,
a /S a /S
where t; and f are infinitesimals. Now or' = a(l + ij) and /S' = /5(1 -|- f). Hence
?L = ?.\±1 a..d lm,^ = Um£.
a' a 1 + 1» ^ a* a
provided /3/a approaclies a limit; whereas If fi/a becomes mnnne, so wiii fi/a .
In a more complex fraction such as (/3 — 7)/<r it is not permissible to replace fi
* It cannot be emphasized too strongly that in the symbol 0/0 the 0*» are merely sym-
ImiHp for a mode of variation just as ac is; they are not actual 0*!» and some other nota-
tion would hv far preferable, likewise forO* », 0*, etc
64 DIFFEKE.NTIAL CALCUXUS
and y individually by infiaitasimals of higher order ; f or /3 — 7 may itself be of
higher order than /3 or 7. Thus tan x — sin x is an infinitesimal of the third order
relative to x although tan x and sin x are only of the first order. To replace tan x
and sin x by infinitesimals which differ from them by those of the second order or
even of the third order would generally alter the limit of the ratio of tan x — sin x
to X* when x = 0.
To prove Duhamel's Theorem the /S's may be written in the form
ft = a,(l + rii), t = 1, 2, • • ., n, |»;.| <e,
where the ij's are infinitesimals and where all the t;'s simultaneously may be made
leas than the assigned e owing to the uniformity required in the theorem. Then
lO^i + /5, + • • • + /3*) - (o-i + 0-2 + • • • + «^") I = Ni^'i + '^2^2 + • • • + •rinC(n\<^^a.
Hence the sum of the /3's may be made to differ from the sum of the ar's by less
than «2rt, a quantity as small as desired, and as Str approaches a limit by hypoth-
esis, so 2/3 must approach the same limit. The theorem may clearly be extended
to the caae where the a's are not all positive provided the sum S | af| of the abso-
lute values of the a's approaches a limit.
35. If y =f(x), the differential of y is defined as
dy =f\x) Aa:, and hence dx = l- Ao;. (4)
From this definition of dy and dx it appears that dy/dx =f'(x), where
the quotient dy/dx is the quotient of two finite quantities of which dx
may be assigned at pleasure. This is true if x is the independent
variable. If x and y are both expressed in terms of #,
x = x(t), y=ry(t), dx = Dp^dt, dy^D^ydt;
and 2 == A^ = ^-^' ^^ ^^'**^^ ^^ W' § 2-
From this appears the important theorem : The quotient dy/dx is the
derivative of y with respect to x no matter what the indejjendent variable
may be. It is this theorem which really justifies writing the derivative
as a fraction and treating the component differentials according to the
rules of ordinary fractions. For higher derivatives this is not so, as
may be seen by reference to Ex. 10.
As Ay and Aa; are regarded as infinitesimals in defining the derivar
tive, it is natural to regard dy and dx as infinitesimals. The difference
Ay — dy may be put in the form
Ay-.y=[/(-^^)-/(-)-^>(.)]^,
(5)
wherein it appears that, when Aa; = 0, the bracket approaches zero.
Hence arises tlie theorem: Ifxis the independent variable and if Ay
and dy are regarded as injinitesimals, the difference Ay — dy is an infin-
itenimal of hiijUer order than Aa-. This has an application to the
TAYLOR'S FORMULA; ALLIED TOPICS 65
subject of change of variable in a definite integral. For if jcas^^Q,
then dx = ^\t) dt, and api)arently
f f(x)dx= rf[<i>(t)]i»'(t)dt,
where ^ {t^ = a and ^ {t^ = ft, so that t ranges from t^ to t^ when x
ranges from a to h.
\\\\i this substitution is too hasty ; for the dx written in the integrand
is really Ax, which differs from dx by an infinitesimal of higher order
when X is not tlie independent variable. The true condition may be
seen by comparing the two sums
2)/(^,) A:r„ ^f{^ (^.)] ^\t^ M^ M = dt,
the limits of which are the two integrals above. Now as Aa; differs
from dw = <fi'(f)dt by an infinitesimal of higher order, 8of(x)Ax will
differ from fl<t>(t)^<t>'(t)dt by an infinitesimal of higher order, and
with the proper assumptions as to continuity the difference will be uni-
form. Hence if the infinitesimals f(x)Ax be all positive, Duhamel's
Theorem may be applied to justify the formula for change of variable.
To avoid the restriction to positive infinitesimals it is well to replace
Duhamel's Theorem by the new
TiiKOREM. Osgood's Theorem. Let a^, a^, • • •, a^ be n infinitesimals
and let at,, differ uniformly by infinitesimals of higher order than Aar
from the elements /(a*,)Aa',. of the integrand of a definite integral
I f{x) dx, where / is continuous ; then the sum 2a = a^ -f or^ H \-a^
approaches the value of the definite integral as a limit when the num-
ber n becomes infinite.
Let ai =f{Xi)£uCi-\-iiLxi, where |f,| <e owing to the uniformity demanded.
Then j^^^r,- ]^/(x,)A2.-[ = |2jr.Ax.|<e2^Ax. = «(6-a).
But as/ is continuous, the definite integral exists and one can make
Y^f{Xi)^i- j f{z)di\<ty andhence I Va, - f /(x)dx <t(6- a + 1).
It tluTefore appears tliat Sa, may be made to differ from the integral by as little
as desired, and Sa,- must then approach the integral as a limit. Now if this theo-
rem be applied to the ca.se of the change of variable and if it be assumed that
/[0(O] and 4>'{t) are continuous, the infinitesimals Ajt, and dx,- = 0'((<) dt^ will
differ unifonnly (compare Theorem 18 of § 27 and the above theorem on Ay — dy)
by an infinitesimal of higher order, and so will the infinitesimals /(jr,) Axv and
/[^ ('•)] 0'(^) ''^• Hence the change of variable suggested by the hasty substitution
is justified.
66 DIFFERENTIAL CALCULUS
EXERCISES
1. Show that rHoepitars Rule applies to evaluating the indeterminate form
/(x)/^(x) when x becomes infinite and both /and 0 either become zero or infinite.
2. Evaluate the following forms by differentiation. Examine the quotients
for left-hand and for right-hand approach ; sketch the graphs in the neighborhood
of the points.
a'—b' / V 1. tanx — 1 / v i- i
(a) lim , OS) lim ; — » (7) limxlogx,
(«> limxc-*, (e) lim(cota;)«'i°% (f) limxi"^
3. Evaluate the following forms by the method of expansions :
(a) lim (\ - cota x) . 03) lim ^ "/*""" , (7) Hm ^ ,
^ x = 0\X* / ^'a; = OX— tanx a;=ll — X
. , ,. , ^ V / X .. X sin (sin x) — sin2 X ,^, ,. ef^—e-''—2x
(«) hm (cschx— cscx), (c) 11m ^^ ^ , (f) lim :
^ ' j:=o ■ x=o x» x=o X — sinx
4. Evaluate by any method: ^
, , ,. c^— 6-^^+ 2sinx — 4x ,^, ,. /tanx\^
(a) hm , (/3) lim ,
a: = 0 X^ a; = 0\ X /
^^xcos»x-log(l + x)-sin-4x^^ ^.^ log (x - ^jQ ^
kj.0 x^ x=^n tanx
<'>iL"iK^+5r-'"'"'«(^+i)]
5. Give definitions for order as applied to infinites, noting that higher order
would mean becoming infinite to a greater degree just as it means becoming zero
to a greater degree for infinitesimals. State and prove the theorem relative to quo-
tients of infinites analogous to that given in the text for infinitesimals. State and
prove an analogous theorem for the product of an infinitesimal and infinite.
6. Note that if the quotient of two infinites has the limit 1, the difference of
the infinites is an infinite of lower order. Apply this to the proof of the resolution
in partial fractions of the quotient /(x)/F(x) of two polynomials in case the roots
of the denominator are all real. For if F{x) = (x — a)*Fi(x), the quotient is an
infinite of order k in the neighborhood of x = a ; but the difference of the quotient
and /{a)/{x — a)*Fj (a) will be of lower integral order — and so on.
7. Show that when x = +qo, the function e=" is an infinite of higher order
than X" no matter how large n. Hence show that if P(x) is any polynomial,
Urn P(x)e-* = 0 when X = + 00.
8. Show that (log «)"• when x is infinite is a weaker infinite than x* no matter
how large m or how small n, supposed positive, may be. What is the graphical
Interpretation ?
9. If P ia a polynomial, show that lim /*(- |e~^ = 0. Hence show that the
xAO \x/
Madaiirin development of c"^ i8/(x) = e"^ = —f(n)(0x) if /(O) is defined as 0.
TAYLOR'S FORMULA; ALLIED TOPICS 67
10. T!ie higher (lifferentiala are defined a« d»y = /<")(2) ((Lr)* where x i« Uken
•A» tae iiutependent variable. Show that d^'x = 0 for I: > 1 if x is the independent
variable. Show tliat the higher derivatives D^, D/y, • • • are not the quotients
(Py/dx*, d^y/dx*, • • • if x and y are expressed in terms of a third variable, but that
the relatioiiH are
J _ d^ydx — d^zdy _, _dx(dxd^ — dyd*x) — Sd^(dxd*y — dyd»x)
^'^- d? ' '*'" (te*
The fact that the quotient d»y/dx», n > 1, Is not the derivative when x and y are
expressed parametricjiliy inilitat<*H agaiuKt the usefulness of the higher differentials
and emphasizes the advanUige of working with derivatives. The notation c^y/dx"
is, however, used for the derivative. Nevertheless, as indicated in £xs. 16-19,
higher differentials may be used if proper care is exercised.
11. Compare the conception of higher differentials with the worlc of £x. 5, p. 48.
12. Show that in a circle the difference between an infinitesimal arc and its
chord is of the third order relative to either arc or chord.
13. Show that if /9 is of the nth order with respect to a, and y is of the first
order with respect to a, then /S is of the nth order with respect to 7.
14. Show that the order of a product of infinitesimals is equal to the sum of the
orders of the infinitesimals when all are referred to the same primary infinitesimal
a. Infer that in a product each infinitesimal may be replaced by one which differs
from it by an infinitesimal of higher order than it without affecting the order of the
product.
15. Let A and B be two points of a unit circle and let the angle -4 OB subtended
at tlie center be the primary infinitesimal. Let the tangents at A and B meet at
7', and OT cut the chord AB in M and the arc AB in C. Find the trigonometric
expression for the infinitesimal difference TC — CM and determine its order.
16. Compute d* (x sin x) = (2 cos x — x sin x) dx^ + (sin x + x cos x) d^x by taking
the differential of the differential. Thus find the second derivative of x sinx if x is
the independent variable and the second derivative with respect to t if x = 1 + ^•
17. Compute the' first, second, and third differentials, d*x ^ 0.
(a)x*cosx, (/3) Vl — X log (1 — x), (7) x€**sinx.
18. In Ex. 10 take y as the independent variable and hence express D^, D/y
in terms of l>yX, D'jx. Cf. Ex. 10, p. 14.
19. Make the changes of variable in Exs. 8, 9, 12, p. 14, by the method of
differentials, that is, by replacing the derivatives by the corresponding differential
expressions where x is not assumed as independent variable and by replacing these
diffen-ntirtls by their values in terms of the new variables where the higher differ-
entials of the new independent variable are set equal to 0.
20. Reconsider some of the exercises at the end of Chap. I, say, 17-19, 22, 28,
27, from the point of view of Osgood's Theorem instead of the Theorem of the Mean.
21. Find tlie areas of the bounding suifaces of the solids of Ex. 11, p. 18.
ea DIFFERENTIAL CALCULUS
22. AMume the law F = kmm'/i^ of attraction between particles. Find tho
attraction of :
(a) a circular wire of radius a and of mass 3f on a particle m at a distance r from
the center of the wire along a perpendicular to its plane ; Ans. kMmr {a^ + r^)~^.
{fi) a circular disk, etc., as in (a) ; Ans. 2 kM7na-^(l - r/Vr^ + or).
(7) a semicircular wire on a particle at its center ; Ans. 2kMm/Tra^.
(a) a finite rod upon a particle not in the line of the rod. The answer should
be expressed in terms of the angle the rod subtends at the particle.
(«) two parallel equal rods, forming the opposite sides of a rectangle, on each
other.
23. Compare the method of derivatives (§ 7), the method of the Theorem of the
Mean (§ 17), and the method of infinitesimals above as applied to obtaining the for-
mulas for (a) area in polar coordinates, (/3) mass of a rod of variable density, (7) pres-
sure on a vertical submerged bulkhead, (5) attraction of a rod on a particle. Obtain
the results by each method and state which method seems preferable for each case.
24. Is the substitution dx, = <t>'{t)dt in the indefinite integral jf{x)dxto obtain
the indefinite integral ff[(p («)] 0'(O ^^ justifiable immediately ?
36. Infinitesimal analysis. To work rapidly in the applications of
calculus to problems in geometry and physics and to follow readily the
books written on those subjects, it is necessary to have some familiarity
with working directly with infinitesimals. It is possible by making use
of the Theorem of the Mean and allied theorems to retain in every ex-
pression its complete exact value ; but if that expression is an infini-
tesimal which is ultimately to enter into a quotient or a limit of a sum,
any infinitesimal which is of higher order than that which is ultimately
kept will not influence the result and may be discarded at any stage of
the work if the work may thereby be simplified. A few theorems
worked through by the infinitesimal method will serve partly to show
how the method is used and partly to establish results which may be
of use in further work. The theorems which will be chosen are :
1. The increment Ax and the differential dx of a variable differ by
an infinitesimal of higher order than either.
2. If a tangent is drawn to a curve, the perpendicular from the curve
to the tangent is of higher order than the distance from the foot of the
perpendicular to the point of tangency.
3. An infinitesimal arc differs from its chord by an infinitesimal of
higher order relative to the arc.
4. If one angle of a triangle, none of whose angles are infinitesimal,,
differs infinitesimally from a right angle and if h is the side opposite
and if ^ is another angle of the triangle, then the side opposite <f> is
h sin ^ except for an infinitesimal of the second order and the adjacent
Bide is A cos ^ except for an infinitesimal of the first order.
TAYLOR'S FORMULA; ALLIED TOPICS
69
The first of these theorems has been proved in | 86. The aecpnd follows froon
it and f rum the idea of tani^ncy. Fur talce the z-azis coincident with the tang«nt
or parallel to it. Then the perpendicular is Ay and the distance from ito foot to the
point of tangency Ih A/. The quotient Ay /Ax approaches 0 as itu limit becauae the
tangent is horizontal ; and the tlieorem is proved. The theorem would remaba true
if the perpendicular xocre replaced by a line making a constant angle with the tangmU
and the distanre from the point of tangency to the foot qf the perpendicular were r«-
placed by Uic distance to the foot of the oblique line. For if Z PMN = 6,
PM
TM
PiV csc^
FN
C8C^
TN~~PN cote
TN , PN ^ .
TN
^
and therefore wl»en P approaches T with B constant, P3f/ rjf approaches zero and
PM is of higher order than TU.
The third theorem follows without difficulty from the assumption or theorem
that the arc has a length intennediate between that of the chord and that of the
sum of the two tangents at the ends of the chord. Let B^ and B^ be the an^e«
between the chord and the tangents. Then
s-AB AT+ TB-AB _ AM {sec 6>^ - 1) + MB (sec B^ - 1)
AM-\-MB^ AM-\-MB ~ AM ^ MB ' ^^
Now as ^J3 approaches 0, both sec^i — 1 and aecB^—l approach 0 and their
coefficients remain necessarily finite. Hence the difference between the arc and
the chord is an infinitesimal of higher order than the chord. As
the arc and chord are therefore of the same order, the difference
is of higher order than the arc. This result enables one to replace
the arc by its chord and vice versa in discussing infinitesimals of
the first order, and for such purposes to consider an infinitesimal
arc as straight. In discussing infinitesimals of the second order, this substitution
would not be permissible except in view of the further theorem given below in
§ 37, and even then the substitution will hold only as far as the lengths of arcs are
concerned and not in regard to directions.
For the fourth theorem let B be the angle by which C departs from 90° and with
the perpendicular BM as radius strike an arc cutting BC. Then by trigonometry
= AM+ MC = hco»<t> + BMtSLnB,
- *"'"- + JJJlf (sec tf-1).
AC
BC = A sin 0
Now tan B is an infinitesimal of the first order with respect to B ;
for its Maclaurin development begins with B. And sec B — 1
is an infinitesimal of the second order; for its development
begins with a tenn in B^. The theorem is therefore proved.
This theorem is freciuently applied to infinitesimal triangles,
that is, triangles in which h is to approach 0.
37* As a further discussion of the thini theorem it may be recalled that by defi-
nition the length of the arc of a curve is the limit of the length of an inacribed
polygon, namely,
a= lim (Vax? + Ay? + Vajt* + Ay.f + • • • + Va/,« + Ay « »•
70 DIFFERENTIAL CALCULUS
Now VAx* + Av* - Vdx^+ dy^
VAx2 + a7^ + Vdx2 + dy2
_ (Ax - dz) (Ax -i- dx) + (Ay - dy) (Ay + dy)
~ VAx2 + Ay2 + Vdx2 + dy2
VAx* + Ay2 — vdxM- dy2 (Ax — dx) Ax + dx
and — ~"
VAx2 + Ay* V Ax2 + Ay2 VAx^ + Ay^ + Vdx^ + dy^
{Ay - dy) Ay + dy
VAx2 + Ay2 Vax2 + Ay2 + Vdx^ + dy^
But Az — dx and Ay — dy are infinitesimals of higher order than Ax and Ay.
Hence the right-hand side must approach zero as its limit and hence VAx^ 4- Ay^
differs from Vdx* + dy'^ by an infinitesimal of higher order and may replace it in
the sum
a = lim V Vax ? + Ay ? = lim V -Vdx^ + dy2 = f'Wl + y'^dx.
The length of the arc measured from a fixed point to a variable point is a func-
tioc of the upper limit and the differential of arc is
d8 = d rVl + y'2dx = VlTy^dx =Vdx2 + dy2.
To find the order of the difference between the arc and its chord let the origin
be taken at the initial point and the x-axis tangent to the curve at that point.
The expansion of the arc by Maclaurin's Formula gives
8(x) = 8 (0) + xs'(O) + i xV'(O) + i x3s-'((9x),
yV"
where «(0) = 0, 5^(0) = VT+y^|o = 1, s"(0)
Vi + r'
Owing to the choice of axes, the expansion of the curve reduces to
y =f{x) = y (0) + xy'(O) + i x^'iOx) = I x^y'\ex),
and hence the chord of the curve is
= 0.
0
c(x) = Vx* + y2 = X Vl + Jx2[y''(6'x)]2 = X (1 + «2P),
where P is a complicated expression arising in the expansion of the radical by
Maclaurin's Formula. The difference
»(X)- C(X) = [X + ixV"(^x)] - [X (1 + X2P)] = X3(is'"((9x) - P).
This is an infinitesimal of at least the third order relative to x. Now as both s(x)
and c (x) are of the first order relative to x, it follows that the difference s{x) — c (x)
must also be of the third order relative to either s (x) or c (x). Note that the proof
MRimes that y" is finite at the point considered. This result, which has been
found analytically, follows more simply though perhaps less rigorously from the
fact that sec ^j - 1 and sec ^j - 1 in (6) are infinitesimals of the second order with
$^ and B^
38. The theory of contact of plane curves may ^be treated by means
of Taylor's Formula and stated in terms of inlinitesimals. Let two
curves y = /(x) and y = g{x)hQ tangent at a given point and let the
TAYLOR'S FORMULA; ALLIED TOPICS 71
origin be chosen at that [Mint witli the x-axis tangent to the curves.
The Maclaurin developments are
y = /(^) = |/''(0)=«' + • • • + ^;^j a-'-ZC-'-XO) + 1 a=<->/<->(0) + . . .
y = ? (X) = ^ ?"(0) x» + . . . + ^^^^j ;«- V-" (0) + ^ ^!7<">(0) + . . . .
If these develoj)nients agree up to but not including the term in af, the
difference between the ordinates of the curves is
/W -!'(') = i? -^ [.''•"'(<>) - !/"'(0)] + • • . /'"'(O) * ?*"(0),
and is an infinitesimal of the nth order with respect to x. The curves
are then said to have contact of order n —1 at their point of tangency.
In general when two curves are tangent, the derivatives /"(O) and ^"(0)
are unequal and the curves have simple contact or contact of the first
order.
The problem may be stated differently. Let PM be a line which
makes a constant angle B with the ^--axis. Then, when P approaches 7',
if RQ be regarded as straight, the proportion
lim {PR : PQ) = lim (sin Z PQR : sin ZPRQ)=: sin 0 : 1
le 1
shows that PR and PQ are of the same order. Clearly also the lines
TAf and TN are of the same order. Hence if
PR PQ
Hence if two curves have contact of the (n — 1) st t'^^^ M, N
order, the segment of a line intercepted between ""j.
the two curves is of the »th order with respect to
the distance from the point of tangency to its foot. It would also be
of the 7ith order with respect to the peri)endicular TF from the point
of tangency to the line.
In view of these results it is not necessary to assume that the two
curves have a special relation to the axis. Let two curves y = f(x) and
1/ = g (a*) intersect when x = a, and assume that the tangents at that point
are not parallel to the ^-axis. Then
,j = yo + (.x- a)f(a) + ■ ■ •+ ^^ -_"j"''/(.->(a) + fe^V<"('')+ ' - '
y = y. + (x - «) <Aa) + ■ •• + ''("J'j^i"' /'-"(«) + ^^ ?"'(<•) + • • •
72 DIFFEKENTIAL CALCULUS
will be the Taylor developments of the two curves. If the difference
of the ordinates for equal values of a; is to be an infinitesimal of the
nth order with respect to x — a which is the perpendicular from the
point of tangency to the ordinate, then the Taylor developments must
agree up to but not including the terms in a;". This is the condition for
contact of order n — 1.
As the difference between the ordinates is
f(x) -g(x) = ^(x- ay [/(»>(«) - ^<»>(«)] + . . . ,
the difference will change sign or keep its sign when x passes through
a according as ti is odd or even, because for values sufficiently near to
X the higher terms may be neglected. Hence the curves will ci'oss each
other if the order of contact is even^ but will not cross each other if the
order of contact is odd. If the values of the ordinates are equated to find
the points of intersection of the two curves, the result is
0=^(x-a)-S[/<»>(a)-9<">(a)] + ...(
and shows that a; = a is a root of multiplicity n. Hence it is said that
two curves have in common as many coincident points as the order of
their contact plus one. This fact is usually stated more graphically
by saying that the curves have n consecutive points in common. It may
be remarked that what Taylor's development carried to n terms does, is
to give a polynomial which has contact of order n — 1 with the function
that is developed by it.
As a problem on contact consider the determination of the circle which shall
have contact of the second order with a curve at a given point (a, yo). Let
y = 1/0 + (X - a)r{a) + i(a; - «) V"(a) + • • •
be the development of the curve and let y" =f'{a) = tanr be the slope. If the
circle is to have contact with the curve, its center must be at some point of the
normal. Then if R denotes the assumed radius, the equation of the circle may be
written ea
(X - a)« + 2 K sin t (x - a) + (y - Vo)^ -2RcosT{y- yo) = 0,
where It remains to determine R so that the development of the circle will coincide
with that of the curve as far as written. Differentiate the equation of the circle.
dy _ R sin T + (a; — a)
dx fi cos T — (y — i/q)
m =
tanr =/'(a),
d^ _ [R cogr - (y - y^)]a -f- [« sin t + (x - a)]" /d^yX ^ 1
*5* [«C08T-(y-yo)]» * WVa,yo « COS» t'
R cos' T
TAYLOR'S FOKMULA; ALLIED TOPICS 78
U the development of the circle. The equation of the coefBcienU of (x — a)*,
-J_=r(a), given H = ??Slr = il±JAa)£L V
TbiH i8 the well known formula for the radius of curvature and shows that the cir-
cle of curvature ha« contact of at lea«t the second order with the curve. The circle
in Konietinies called the osculating circle instead of the circle of curvature.
39. Three theorems, one in geometry and two in kinematics, will
now U* proved to illustrate the direct application of the infinitesimal
methods to such problems. The choice will be :
1. The tangent to the ellipse is equally inclined to the focal radii
drawn to the point of contact.
2. The displacement of any rigid body in a plane may be regarded
at any instant as a rotation through an infinitesimal angle about some
point unless the body is moving parallel to itself.
3. The motion of a rigid body in a plane may be regarded as the
rolling of one curve upon another.
For the first problem consider a secant PP' which may be converted into a
tangent 7T' by letting the two points approach until they coincide. Draw the
focal ratlii to P and P" and strike arcs with F and F' as
centers. As F'P -^ PF= F'P" + P'F = 2 a, it follows
that NP = MP^. Now consider the two triangles PP*M
and l^PN nearly right-angled at M and N. The sides
P/*', P.V, PN, P'Af, P'N are all infinitesimals of the
same urder and of the same order as the angles at F and
F'. By proposition 4 of § 36 ' jr
MP' = PP'cosZPP'if + Cj, NP = PP'cosZP'Piyr + «,,
where Cj and c, are infinitesimals relative to MP* and NP or PP*. Therefore
lim [cos Z PP'M -r- cos Z P'PN] = cos Z TPF - cos Z TPF' = lim ^\~^ = 0,
and the two angles TPF' and T'PF are proved to be equal as desired.
To prove the second theorem note first that if a body is rigid, its position is com-
pletely determined when the position A B of any rectilinear segment of the body
is known. Let the points A and B of the body be de-
scribing curves -4^' and Blf so that, in an infinitesimal
interval of time, the line A B takes the neighboring posi-
tion A'W. Erect the perpendicular bisectors of the lines
A A' and BB' and let them intersect at 0. Then the tri-
angles ^0£ and A' OR have the three sides of the one
equal to the three sides of the other and are equal, and
the second may be obtained from the first by a mere rotation about O through the
angle ^0^'= BOB'. Except for infinitesimals of higher order, the magnitude of
the angle is AA'/OA or BB'/OB. Next let the interval of time approach 0 to that
A' approaches A and If approaches B. The perpendicular bisectors will approach
74 DIFFERENTIAL CALCULUS
the normalB to the arcs A A' and BW at A and 5, and the point O will approach
the intersection of those normals.
The theorem may then be stated that : At any instant of time the motion of a
rigid body in a plane may be considered as a rotation through an infinitesimal angle
about the interseclion of the normals to the paths of any two of its points at that in-
stant ; the amount of the rotation will be the distance ds that any point moves divided
by the distance of that point from the instantaneous center of rotation ; the angular
velocity about the instantaneous center will be this amount of rotation divided by the
interval of time dt^ that is, it will be v/r, where v is the velocity of any point of the body
and r ia its distance from the instantaneous center of rotation. It is therefore seen
that not only is the desired theorem proved, but numerous other details are found.
As ha^ been stated, the point about which the body is rotating at a given instant
is called the instantaneous center for that instant.
As time goes on, the position of the instantaneous center will generally change.
If at each instant of time the position of the center is marked on the moving plane
or body, there results a locus which is called the moving centrode or body centrode ;
if at each instant the position of the center is also marked on a fixed plane over
which the moving plane may be considered to glide, there results another locus which
is called the^ed centrode or the space centrode. From these definitions it follows
that at each instant of time the body centrode and the space centrode intersect at
the instantaneous center for that instant. Consider a series of
positions of the instantaneous center as P_ ^P-iP^iP^ Diarked
in space and Q-2Q-1QQ1Q2 marked in the body. At a given
instant two of the points, say P and Q, coincide ; an instant
later the body will have moved so as to bring Qj into coin-
cidence with P^ ; at an earlier instant Q_i was coincident with
P_i. Now as the motion at the instant when P and Q are together is one of
rotation through an infinitesimal angle about that point, the angle between PP^
and QQj is infinitesimal and the lengths PP^ and QQ^ are equal ; for it is by the
rotation about P and Q that Q^ is to be brought into coincidence with P^. Hence
it follows 1° that the two centrodes are tangent and 2° that the distances PP^ = QQi
which the point of contact moves along the two curves during an infinitesimal inter-
val of time are the same, and this means that the two curves roll on one another
without slipping — because the very idea of slipping implies that the point of con-
tact of the two curves should move by different amounts along the two curves,
the difference in the amounts being the amount of the slip. The third theorem
is therefore proved.
EXERCISES
1. If a finite parallelogram is nearly rectangled, what is the order of infinites-
imals neglected by taking the area as the product of the two sides ? What if the
figure were an isosceles trapezoid ? What if it were any rectilinear quadrilateral
all of whose angles differ from right angles by infinitesimals of the same order ?
2. On a sphere of radius r the area of the zone between the parallels of latitude
X and X -f d\ is taken as 2 Trr cos X • rd\, the perimeter of the base times the slant
height. Of what order relative to dX is the infinitesimal neglected ? What if the
perimeter of the middle latitude were taken so that 2 irr^ cos (X + J dX) dX were
UHunied?
TAYLOR^S FORMULA; ALLIED TOPICS 75
3. What is the order of the infinitesimal neglectetl in Uking iirr^dr a« the
Tohiine of a liollow gphere of interior radiiiH r and thickneM dr ? What if the mean
nuliu8 were taken instead of tlie interior ra^liuH ? Would any particular radius be
best?
4. Discuss the length of a space curve y =/(«), z = g{x) analytically ait the
length of the plane curve was discussed in the text.
5. Discuss proposition 2, p. 68, by Maclaurin's Formula and in particular show
that if the second derivative is continuous at the point of tangency, the infinite»-
imal in question is of the second order at least. How about the ca^e of the tractrix
a. a — Vo* — X* , ^/-z r
y = -log ■ -f va« - ««,
2 a + Va« - z*
and its tangent at the vertex x = a? How about 8{x) — c (x) of § 37 ?
6. Show that if two curves have contact of order n — 1, their derivatives will
have contact of order n — 2. What is the order of contact of the kth derivatives
k<n-l?
7. State the conditions for maxima, minima, and points of inflection in the
neighborhood of a point where /<'')(a) is the first derivative that does not vanish.
8. Determine the order of contact of these curves at their intersections:
V2(x2 + y2 + 2) = 3(x + y) ,^. r2 = a2co820 x« + y* = y
^ ' 5j-2-6xy+5|/2 = 8, ^^' y^ = ia{a-x), ^^' x» + y» = xy.
9. Show that at points where the radius of curvature is a maximum or mini-
mum the contact of the osculating circle with the curve must be of at least the
third order and must always be of odd order.
10. Let PN be a normal to a curve and P'N a neighboring normal. If 0 is the
center of the osculating circle at P, show with the aid of Ex. 6 that ordinarily the
perpendicular from O to P'N is of the second order relative to the arc PP and that
the distance OiV is of the first order. Hence interpret the statement : Consecutive
normals to a curve meet at the center of the osculating circle.
11. Does the osculating circle cross the curve at the point of osculation ? Will
the osculating circles at neighboring points of the curve intersect in real points ?
12. In the hyperbola the focal radii drawn to any point make equal angles with
the tangent. Prove this and state and prove the corresponding theorem for the
])arabola.
13. Given an infinitesimal arc AB cut at C by the perpendicular bisector of ita
chord AB. What is the order of the difference AC — BC ?
14. of what order is the area of the segment included between an infinitesimal
arc and its chord compared with the square on the chord ?
15. Two sides .45, ^C of a triangle are finite and differ infinitesimally ; the
angle $ a.i A is an infinitesimal of the same order and the side BC is either recti-
linear or curvilinear. What is the order of the nej^lected infinitesimal if the area
is assumed as \ ^Al^S ? What if the assumption ibIAB- AC'$?
76 DIFFERENTIAL CALCULUS
16. A cycloid is the locus of a fixed point upon a circumference which rolls on
% straight line. Show that the tangent and normal to the cycloid pass through the
highest and lowest points of the rolling circle at each of its instantaneous positions.
17. Show that the increment of arc As in the cycloid differs from 2 a sin \ BdB
by an infinitesimal of higher order and that the increment of area (between two
oonaecutive normals) differs from 3 a* sin^ \ Bd6 by an infinitesimal of higher order.
Hence show that the total length and area are 8 a and 3 Tra^. Here a is the radius
of the generating circle and 6 is the angle subtended at the center by the lowest
point and the fixed point which traces the cycloid.
18. Show that the radius of curvature of the cycloid is bisected at the lowest
point of the generating circle and hence is 4 a sin \ 6.
19. A triangle ABC is circumscribed about any oval curve. Show that if the
side BC is bisected at the point of contact, the area of the triangle will be changed
by an infinitesimal of the second order when BC is replaced by a neighboring tan-
gent B'C, but that if BC be not bisected, the change will be of the first order.
Hence infer that the minimum triangle circumscribed about an oval will have its
thiee sides bisected at the points of contact.
20. If a string is wrapped about a circle of radius a and then unwound so that
its end describes a curve, show that the length of the curve and the area between
the curve, the circle, and the string are
where B is the angle that the unwinding string has turned through.
21. Show that the motion in space of a rigid body one point of which is fixed
may be regarded as an instantaneous rotation about some axis through the given
point. To do this examine the displacements of a unit sphere surrounding the fixed
point as center.
22. Suppose a fiuid of variable density D{x) is flowing at a given instant through
a tube surrounding the x-axis. Let the velocity of the fluid be a function v{x) of x.
Show that during the infinitesimal time U the diminution of the amount of the
fluid which lies between x = a and x = a + ^ is
S[c(a + h)I){a-\- h)U-v{a)D{a)U\
where 8 is the cross section of the tube. Hence show that D (x) v (x) = const, is the
condition that the flow of the fluid shall not change the density at any point.
23. Consider the curve y = f{x) and three equally spaced ordinates at x = a — 5,
05 = o, « = a + J. Inscribe a trapezoid by joining the ends of the ordinates at
X = a ±i and circumscribe a trapezoid by drawing the tangent at the end of the
ordinate at z = a and producing to meet the other ordinates. Show that
S. = 2 if {a), 8 = 2 i^fia) + ^ /"(a) + ^ /(i-)(f ) j ,
TAYLOR'S FORMULA; ALLIED TOPICS 77
are the an-dH of the circiiinacribed trapezoid, the curve, the inacribed trapesold.
Hence infer that to compute the area under the curve from the in«cribed or cir>
cuin/icril)ed trapezoids introduces a relative error of the order d^, but that to com-
pute from tlie relation iS = ^ (2 6*9 + S^) introduces an error of only the order of I*.
24. lAit the interval from a to 6 be divided into an even number 2n of equal
parts a and let the 2 n + 1 ordinates yo« l/p * "1 Vsit &t the extremities of the inter-
vals be drawn to the curve y =/(jr). Inscribe trapezoids by joining the ends of
every other ortlinate beginning with i/^, y,, and going to y^,,. Circumscribe trape-
zoids by drawing tangents at the ends of every other ordinate yj, y,, • • ., Jftm-i'
Compute the area under the curve as
S=fy{z)dx = ^ [4(y, + y, + . . . + i/2._i)
+ 2 (i/o + y, + • • • + V2n] -Vo- V2«] + B
by using tha work of Ex. 23 and infer that the error R is less than (6— a) a</(»*)(f)/46.
This method of computation is known as SimpaorC8 Rule. It usually gives accu-
racy sufficient for work to four or even five figures when a = 0.1 and 6 — a = 1 ; for
/<**)(jr) usually is small.
25. Compute these integrals by Simpson's Rule. Take 2n = 10 equal intervals.
Carry numerical work to six figures except where tables must be used to find/(z) :
(rt) r '^ — = log 2 = 0.69315, ip) f * -^^ = tan-i 1 = 7 ir = 0.78536,
Ji X «/o 1 + X* 4
(7) r*'sinx(ix = 1.00000, («) f log,oxdx = 2 log^oX- 3f = 0.16776,
(e) r !^^<1±^ dx = 0.27220, (f) r ' l£L(l±^) dx = 0.82247.
»/o 1 + X* Jo X
The answers here given are the true values of the integrals to five places.
26. Show that the quadrant of the ellipse x = asin 0, y = 6 cos0 is
» = (if " Vl - e^ sin* 0 d<f>= \ira f V^{2-e^)-\- Jc^cosiru du.
Compute to four figures by Simpson's Rule with six divisions the quadrants of
the ellipses :
(a) c = i Vs, 8= 1.211 a, (^) e = i V2, « = 1.351 a.
27. Expand 8 in Ex. 26 into a series and discuss the remainder.
2 L \V \2-4/ 3 \2.4.6/ 6 \ 2-4. ..2n /2n-l J
1 /I • 8 • • • f2 n 4- lU' eS« + s
R^<—!-- (i-4 ^ M — SeeEx.l8,p.60,andPeirce's"Table8,"p.(».
l-ea\2.4...(2n + 2)/2n+l '^ ' * *^
Estimate the number of terms necessary to compute Ex. 26 (fi) with an error not
greater than 2 in the last place and compare the labor with that of Simpson's Rule.
28. If the eccentricity of an ellipse is j'f, find to five decimals the percentage
error nuule in taking 2ira as the perimeter. Anf>. 0.00694%
78 DIFFERENTIAL CALCULUS
29. If the catenary y = c cosh {x/c) gives the shape of a wire of length L sus-
pended between two points at the same level and at a distance I nearly equal to
X, find the first approximation connecting X, I, and d, where d is the dip of the
wire at its lowest point below the level of support.
SO. At its middle point the parabolic cable of a suspension bridge 1000 ft. long
between the supports sags 60 ft. below the level of the ends. Find the length of
the cable correct to inches.
40. Some differential geometry. Suppose that between the incre-
ments of a set of variables all of which depend on a single variable t
there exists an equation which is true except for infinitesimals of higher
order than M = dt, then the equation will be exactly true for the differ-
entials of the variables. Thus if
is an equation of the sort mentioned and if the coefficients are any func-
tions of the variables and if e^, e^,--- are infinitesimals of higher order
than dty the limit of
-^ At ^ At At At At At
or fdX'\-gdy-\-hd»-\-ldt=^0\
and the statement is proved. This result is very useful in writing
down various differential formulas of geometry where the approximate
relation between the increments is obvious and where the true relation
between the differentials can therefore be found.
For instance in the case of the differential of arc in rectangular coor-
dinates, if the increment of arc is known to differ from its chord by an
infinitesimal of higher order, the Pythagorean theorem shows that the
equation A«» = Ax^-fA/ or A^ = Ax" + Af + Az^ (7)
is true except for infinitesimals of higher order; and hence
d^ = dx'-\-d]/' or d^^dx'-^-df-^-d;^. (7')
In the case of plane polar coordinates, the triangle PP^N (see Fig.)
P'
M
has two curvilinear sides PP' and PN and is right-
angled at N. The Pythagorean theorem may be
applied to a curvilinear triangle, or the triangle may
be replaced by the rectilinear triangle PP'N with ^^^ ' "^
the angle at N no longer a right angle but nearly so. In either way of
looking at the figure, it is easily seen that the equation A^ = Ar* -f r^A^t^
TAYLOR'S FORMULA; ALLIED TOPICS
79
which the figure suggests differs from a true equation by an infinitesi-
null of higher order; and hence the inference that in polar oodrdinates
The two most used systems of co5rdinates
other than rectangular in space are the polar
or spherical and the rylindrical. In the first
the distance r=^OP from the pole or center,
the longitude or niftridional angle <^, and the
colatitude or polar angle 6 are chosen as coor-
dinates ; in the second, ordinary polar coordinates r = OM and ^ in
the a-y-plane are combined with the ordinary rectangular z for distance
from that plane. The formulas of transformation are
« = r cos By
z
0
p.
r,<p,z
z
y Y
x^
\
<r^
^
\M
z
y = rsindsin<^, ^ = C08"'ro . .^ . ^> W
x = r sin $ cos <f>, <^ = tan"
for polar coordinates, and for cylindrical coordinates they are
z = Zy y = r sin «^, x = r cos ^,
Formulas such as that
for the differential of
arc may be obtained for
these new coordinates by
mere transformation of
(7') according to the rules
for change of variable.
In both these cases,
however, the value of
ds may be found readily
by direct inspection of
the figure. The small
parallelepiped (figure
for polar case) of which
\s is the diagonal has
some of its edges and
faces curved instead of
straight; all the angles,
however, are right angles,
and as the edges are infinitesimal, the equations certainly suggested
holding except for infinitesimals of higher order are
80 DIFFERENTIAL CALCULUS
^^ = ^r^-^r'sm^e^<f>' + )''^e' and A*=^ = Ar^ + r^A<^^ + A«^ (10)
or d^ = di* + T^8\n^0dit^' + Aie^ and ds^ = dr" -^ r'difi^ -\- dz". (IC)
To make the proof complete, it would be necessary to show that noth-
ing but infinitesimals of higher order have been neglected and it might
actually be easier to transform Vc^* -|- di/^ -|- dz^ rather than give a
rigorous demonstration of this fact. Indeed the infinitesimal method is
seldom used rigorously ; its great use is to make the facts so clear to the
rapid worker that he is willing to take the evidence and omit the proof.
In the plane for rectangular coordinates with rulings parallel to the
y-axis and for polar coordinates with rulings issuing from the pole the
increments of area differ from
dA = ydx and dA = \ r^dt^ (11)
respectively by infinitesimals of higher order, and
A= fydx and A = i C 'r^d<l> (11')
are therefore the formulas for the area under a curve and between two
ordinates, and for the area between the curve and two radii. If the plane
is ruled by lines parallel to both axes or by lines issuing from the pole
and by circles concentric with the pole, as is customary for double inte-
gration (§§ 131, 134), the increments of area differ respectively by
infinitesimals of higher order from
dA = dxdy and dA = rdrdtfi, (12)
and the formulas for the area in the two cases are
A = \im^AA= j jdA = ijdxdy, (12')
^ = lim 2^ A^ = CCdA = fCrdrd<^,
where the double integrals are extended over the area desired.
The elements of volume which are required for triple integration
(§§ 133, 134) over a volume in space may readily be written down for
the three caAes of rectangular, polar, and cylindrical coordinates. In the
first case space is supposed to be divided up by planes x = a, y = />,
« sac perpendicular to the axes and spaced at infinitesimal intervals ; in
the second case the division is made by the spheres r = « concentric
with the ix)le, the planes <^ = /> through the polar axis, and the cones
^ =3 <j of revolution alx)ut the polar axis ; in the third case by the cylin-
ders r = fl, the planes </» = h, and the planes z = c. The infinitesimal
TAYLOR'S FORMULA; ALLIED TOPICS 81
volumes into which space is divided then differ from
dv = dxdydXf dv = t^ sin Bdrd^fxIO, dv = rdrd^dtt (13)
respectively by infinitesimals of higher order, and
CCCdxdydx, CCCt^^medrdf^W, CCCrdrd<f>^lz (IS*)
are the formulas for the volumes.
41. The direction of a line in space is represented by the three angles
whicli the line makes with the ])ositive directions of the axes or by the
cosines of those angles, the direction cosines of the line. From the defi-
nition and figure it appears that
dx dy dz
/ = C08a=— , m = co3p=-~y n = cosy = -j- (14)
are the direction cosines of the tangent to the arc at the point; of the
tangent and not of the chord for the reason
that the increments are replaced by the differ-
entials. Hence it is seen that for the direr-
Hon cosines of the tangent the proportion
I : ?n: n = dx : dt/ : dz (1^')
holds. The equations of a space curve are
^=/(0» y = 9(^), re = hit)
in terms of a variable parameter t* At the point (x^ y^ z^ where
t=^t^ the equations of the tamjent lines would then be
x-Xp^y-yo^z-ZQ x -x^ ^y -y^ ^z- z^
idx\ {dyX (dz\ fit;) /(g h\ty ^ ^
As the cosine of the angle B between the two directions given by the
direction cosines /, w, n and /', ///', n' is
cos $ = W -\- mm' -^ nn' , so //' -f 7/i»i' -|- nn' = 0 (16)
is the condition for the perpendicularity of the lines. Now if (x, y, «)
lies in the plane normal to the curve at x^ y^ z^ the lines determined
by the ratios x - x^\ y - y^i z — z^ ^nd. {dx\ : (dy\ : (dz\ will be per-
pendicular. Hence the efjuatlon of the normal plane is
(X - x^(<fa), + (y - yj(rfy), + (* - z^(<fa), = 0
«r /'(',)(* -^<)+5''(g(y-y,) + A'(0(«-«j = o. (17)
• For tlie sako of genpmllty the pnrnmetrio form in t Is assumed ; In a particular cam A
Kiiiiplitication nitKlit he inadp by talcin); one of tht> variables as t and one of the fii
/'. g', ft' would then he 1. ThuH in Ex. 8 (e), y should be taken as (.
82 DIFFERENTIAL CALCUXCTS
The tangent plane to the curve is not determinate ; any plane through
the tangent line will be tangent to the curve. If A be a pai-ameter, the
pencil of tangent planes is
There is one particular tangent plane, called the osculating plane ^which.
is of especial importance. Let
»'-=', = /'(<.)'■ + J/"(Or' + i/"'(f)T«, T = <-<„, t,<i<t,
with similar expansions for y and «, be the Taylor developments of
ar, y, z about the point of tangency. When these are substituted in the
equation of the plane, the result is
This expression is of course proportional to the distance from any point
ar, y, z of the curve to the tangent plane and is seen to be in general of
the second order with respect to t or ds. It is, however, possible to
choose for X that value which makes the first bracket vanish. The tan-
gent plane thus selected has the property that the distance of the curve
from it in the neighborhood of the point of tangency is of the third order
and is called the osculating plane. The substitution of the value of A gives
x-x^ y-y^ z-z^
^-^^ IZ-l/o ^-«o
f'(.h) ?'(«.) A'(g
= 0 or
W, W, (<^^)„
f'\h) 9'\h) h"(Q
(<p^\ (<Py\ (<?«)„
= 0 (18)
Oo
or (dyd^'z - dz^y\{x - x^) -f- (dzdh^ - dxd^z)fy - y^)
4- {dxdJ'y - dyd?x\{z -z^) = 0
as the equation of the osculating plane. In casef"(t^)=g"(t^) = h"(t ) = 0,
this equation of the osculating plane vanishes identically and it is neces-
sary to push the development further (Ex. 11).
42. For the case of plane curves the curvature is defined as the rate
at which the tangent turns compared with the description of arc, that
is, as d^/ds if d<ft denotes the differential of the angle through which
the tangent turns when the point of tangency advances along the curve
by ds. The radius of curvature R is the reciprocal of the curvature,
that is, it is ds/d<t>. Then
^ dx ds tkedt [l + y'«]» y" ^ >
TAYLOR'S FORMULA; ALLIED TOPICS 83
where accents denote differentiation with respect to x. For space curves
the same definitions are given. If /, w, n and l-^-dl^ yn + dm, n-f-^/n
are the direction cosines of two successive tangents,
coa d<^=: 1(1 + dl) -f m (;/i -|- dm) -|- n(n -f dn).
But /* -f m^ -f n« = 1 and (/ + (//)* -f (m + dm)* -h (» + dn)* = 1.
Hence <//"' + dm'^ + </n^ = 2 - 2 cos rf^ = (2 sin J dij,)*,
i- (2)"- ['-^^T- ^^^^^^!S±^ = "■ - "• * '■•■ <»'
where accents denote differentiation with respect to s.
The torsion of a space curve is defined as the rate of turning of the
oscuhiting phme compared with the increase of arc (that is, r/i^/</x, where
</^ is the differential angle the normal to the osculating jjlane turns
til rough), and may clearly be calculated by the same formula as the
curvature provided the direction cosines L, 3/, N of the normal to the
plane take the places of the direction cosines /, w, n of the tangent line.
Hence the torsion is
r' \ds)
dL' -f- dM* + d^y
= L'''-{-^f'^'-{^N'''; (20)
R* \ds/ ds^
and the radius of torsion R is defined as the reciprocal of the torsion,
where from the equation of the osculating plane
L M N
dydj^z — dzd^y dzd^x — dxd^^z dxd^y — dycthc
= , ^ =' (200
Vsum of squares
The actual computation of these quantities is somewhat tedious.
The vectorial disciLssion of curvature and torsion (§ 77) gives a better insight
into tlie principal directions connected witli a space curve. These are the direction
of the tangent, that of the normal in the osculating plane and directed towards
the concave side of the curve and called the principal normal, and that of the
normal to the osculating plane drawn upon that side which makes the three direc-
tions form a right-handed system and called the binormal. In the notations there
given, combined with those above,
r = jl + yi + zk, t = a + mj + nk, c = Xi + mJ + vk, n = Li + Mj + -Vk,
wliere X, /n, p are taken lus the direction cosines of the principal normal. Now dt
in parallel to c and tin is parallel to — c. Hence the results
m^dm^dn^d^ dL_dM_dN^_da .
X~/i~»'~iJ*° x~M~»~R ^'
84 DIFFERENTIAL CALCULUS
follow from dt/ds = C and dn/ds = T. Now dc is perpendicular to c and hence in
the plane of t and n ; it may be written as dc= (t.dc)t+ {n.dc)n. But as t.c = n.c=0,
t-dc = — c«dt and n.(ic = — cdn. Hence
dc=-(c.dt)t-(c.dn)ii=- Ctds+ Tnds = - -- ds + - ds.
dX I L dM m M dv _ n N
^^'"^ di=~^+R* d^="]K+^' dk — R^R' ^''^
Formulas (22) are known as FreneVa Formulas ; they are usually written with — R
in the place of R because a left-handed system of axes is used and the torsion, being
an odd function, changes its sign when all the axes are reversed. If accents denote
differentiation by s,
above formulas, - =
X' y' 7f
x" y" z"
t;" r' ^"
usual formulas, — = — ^^— — ^— — '^-— • (23)
X'
V'
z'
x"
V"
2"
X'"
r'
2"'
right-handeTR ^'^+y"^+^"^ left-handed ^ x-+rHz-
EXERCISES
1. Show that in polar coordinates in the plane, the tangent of the inclination
of the curve to the radius vector is rd<i>/dr.
2. Verify (10), (lO') by direct transformation of coordinates.
3. Fill in the steps omitted in the text in regard to the proof of (10), (lO') by
the method of infinitesimal analysis.
4. A rhumb line on a sphere is a line which cuts all the meridians at a constant
angle, say a. Show that for a rhumb line sin Od(f> = tan adO and ds = r sec ad6.
Hence find the equation of the line, show that it coils indefinitely around the
poles of the sphere, and that its total length is irr sec a.
5. Show that the surfaces represented by F(0, ^) = 0 and F(r, ^) = 0 in polar
coordinates in space are respectively cones and surfaces of revolution about the
polar axis. What sort of surface would the equation F{r, <p) = 0 represent ?
6. Show accurately that the expression given for the differential of area in
polar coordinates in the plane and for the differentials of volume in polar and
cylindrical coordinates in space differ from the corresponding increments by in-
finitesimals of higher order.
7. Show that — , r — , r sin ^ — are the direction cosines of the tangent to a
dH ds ds
■pace curve relative to the radius, meridian, and parallel of latitude.
8. Find the tangent line and normal plane of these curves.
(or) acy« = 1, y« = X at (1, 1, 1), (/3) x = cos t, y = sin«, z = kt,
(y) 2 ay = x\ QaH = x«, (S) x = t cos t, y = t sin «, z = kt,
(«) y = a;«, «« = 1 - y, (f) x^ + y* + «« = a^, x* + y^ + 2 ox = 0.
9. Find the equation of the osculating plane in the examples of Ex. 8. Note
that if 2 U* the Independent variable, the equation of the plane is
TAYLOB^S FORMULA; ALLIED TOPICS 85
10. A Hpace curve paaees through the origin, in tangent to the «-«xJs, and has
t z= 0 aaitH osculating; plane at the origin. Show that
X = t/\0) + i tT'(0) + . . . , y = i tV'(O) + . . . . « = J <»A'"(0) + . . .
will be the form of ite Maclaurin development if t = 0 gives x = y = x = 0.
11. If the 2d, 8d, • • •, (n ~ l)8t derivatives of /, g^ h vanish for ( = (^ but not
all tlu* nth derivatives vanish, show that there is a plane from which the curve
(IfparUi by an iiiiiniteKiinal of the (n + l)8t order and with which it therefore
has contact of order n. Such a plane is called a hyperosculating plane. Find its
equation.
12. At what points if any do the curves (^, (7), (*), (^), Ex. 8 have hyperoscu-
lating planes and what is the degree of contact in each case ?
13. Sliow that the expression for the radius of curvature is
where in the first case accents denote differentiation by s, in the second by t,
14. Show that the radius of curvature of a space curve is the radius of curva-
ture of its projection on the osculating plane at the point in question.
15. From Frenet's Formulas show that the successive derivatives of x are
^=^' x=i=-. X =_-_ = __-X- + — ,
where accents denote differentiation by s. Show that the results for y and z are
the same except that m, /x, M or n, k, N take the places of Z, X, L. Hence infer
that for the nth derivatives the results are
x<-) = /Pj + XPj + LP,, yCt) = mPi + AiPa + MP^, z^"^ = nP^ + rP, + NP^ ,
where Pj, P,, P, are rational functions of R and R and their derivatives by «.
16. Apply the foregoing to the expansion of Ex. 10 to show that
where R and R are the values at the origin where « = 0, / = m = ^=1, and the
(rthiT six direction cosines m, n, X, r, L, 3f vanish. Find 8 and write the expan-
sion of the curve of Ex. 8 (7) in this form.
17. Note that the distance of a point on the curve as expanded in Ex. 16 from
the sphere through the origin and with center at the point (0, R, R'R) ia
y/x'i + (y - ny^ + (2 - R'Ry^-^/R^ + iJ'«R«
(«« + y« - 2 /?!/ + 2« - 2 R'Rz)
vx« + (y - R)* + (« - R'R)* +Vi?« + /f'«R«
and consequently is of the fourth order. The curve therefore has contact of the
thinl onler with this sphere. Can the equation of this sphere be derived by a
limiting process like that of Ex. 18 as applied Vo the osculating plane
86
DIFFERENTIAL CALClJLUS
18. The osculating plane may be regarded as the plane passed through three
oonaecutive points of the curve ; in fact it is easily shown that
2 1
z. 1
Urn
appRMehO
Ax, Ar. aj
V
Vo
Xq + te Vo + ^y Zq-}- Sz 1
Xq + Ax Vo-^-^y Zq + Az 1
x — x
mo
0 y-Vo z-Zq
Wo Wo
'y)o {dH),
= 0.
19. Express the radius of torsion in terms of the derivatives of x, y, z hj t
(Ex. 10, p. 67).
20. Find the direction, curvature, osculating plane, torsion, and osculating
sphere (Ex. 17) of the conical helix x = tcost, y = tsint, z = kt Sitt = 27r.
21. Upon a plane diagram which shows As, Ax, Ay, exhibit the lines which
represent ds, dx, dy under the different hypotheses that x, y, or s is the independ-
ent variable.
CHAPTER IV
PARTIAL DIFFERENTIATION; EXPLICIT FUNCTIONS
43. Functions of two or more variables. The definitions and theo-
rems about finuitions of more than one independent varial)le are to a
huge extent similar to those given in Chap. II for functions of a single
variable, and the changes and difticulties which occur are for the most
part amply illustrated by the case of two variables. The work in the
text will therefore be confined largely to this case and the generalizar
tions to functions involving more than two variables may be left as
exercises.
If the value of a variable z is uniquely determined when the values
(ar, y) of two variables are known, z is said to be a function z =/(ic, y)
of the two variables. The set of values [(ir, y)] or of points P(aj, y) of
the x/z-plane for which z is defined may be any set, but usually consists
of all the points in a certain area or region of the plane bounded by
a curve which may or may not belong to the region, just as the end
points of an interval may or may not belong to it. Thus the function
1/Vl — x^ — y^ is defined for all points within the circle x* -}- ^ = 1,
but not for points on the perimeter of the circle. For most purposes it
is sufficient to think of the boundary of the region of definition as a
polygon whose sides are straight lines or such curves as the geometric
intuition naturally suggests, «
The first way of representing the function z =/(a', y) geometrically
is by the surface z =f(x, y), just as y =f(x) was represented by a curve.
This method is not available for u =/(j*, y, «), a function of three vari-
ables, or for functions of a greater number of variables ; for space has
only three dimensions. A second method of representing the function
«=/(ar, y) is by its contour lines in the a-y-plane, that is, the curves
/(j-, ?/) = const, are plotted and to each curve is attached the value of
the constant. This is the method employed on maps in marking heights
above sea level or depths of the ocean below sea level. It is evident that
these contour lines are nothing but the projections on the a'y-plane
of the curves in which the surface z =/(^, y) is cut by the planes
z = const. This method is applicable to functions u =/(ar, y, «) of
three variables. The contour surfaces u = const which are thus obtained
87
88
DIFFERENTIAL CALCULUS
''(a,b)+€
25
f(a,b)
are frequently called equlpoterUwZ surfaces. If the function is single
valued, the contour lines or surfaces cannot intersect one another.
The function z =f(Xy y) is continuoits for (a, h) when either of the
following equivalent conditions is satisfied :
1*. lim/(x, y) = /(rt, h) or lim/(«, y) = /(lim x, lim y),
no matter how the variable point P(Xj y) approoA^hes (a, i).
2*. If for any assigned c, a number 8 may be found so that
.*rti l.lA^y y) -/(«> ^) I < « ^^^^ \x-a\<h,\y-b\<h.
Geometrically this means that if a square with (a, b) as center and
with sides of length 2 8 parallel to the axes be drawn,
the portion of the surface z =f(x, y) above the
square will lie between the two planes z =f((if b)±€.
Or if contour lines are used, no line f(x, y) = const,
where the constant differs from f(a, b) by so much
as c will cut into the square. It is clear that in place
of a square surrounding (a, b) a circle of radius 8 or any other figure
which lay within the square might be used.
44. Continuity examined. From the definition of continuity just given and
from the corresponding definition in § 24, it follows that if /(x, y) is a continuous
function of x and y for (a, 6), then /(x, 6) is a continuous function of x f or x = a
and /(a, y) is a continuous function of y for y = h. That is, if / is continuous in
X and y jointly, it is continuous in x and y severally. It might be thought that
conversely if /(x, 6) is continuous f or x = a and /(a, y) for y = 6, /(x, y) would
be continuous in (x, y) for (a, b). That is, if / is continuous in x and y severally,
it would be continuous in x and y
jointly. A simple example will show ^ -v^ . x^.,--— 5 z
that this is not necessarily true. Con-
sider the case
'(.a,b.
/(O, 0) = 0
and examine z for continuity at
(0, 0). The functions /(x, 0)^2:,
and /(O, y)=v are surely continuous
in their respective variables. But the surface z =/(x, y) is a conical surface (except
for the points of the z-axis other than the origin) and it is clear that P(x, y) may
approach the origin in such a manner that z shall approach any desired value.
Moreover, a glance at the contour lines shows that they all enter any circle or
■quare, no matter how small, concentric with the origin. If P approaches the origin
along one of these lines, z remains constant and its limiting value is that constant.
In fact by approaching the origin along a set of points which jump from one con-
tour line to another, a method of approach may be found such that z approaches
no limit whatsoever but oscillates between wide limits or becomes infinite. Clearly
the oondltions of continuity are not at all fulfilled by z at (0, 0).
PARTIAL DIFFERENTIATION; EXPLICIT
89
Double limits. There often ariHe for coiutlderation expreMioiui like
limriim/(x,i/)-|,
Urn niin/iz,
.)],
(1)
where the limits exist whether x first approaclies its limit, and then y its limit, or
vice versa, and where the question ariscH as to whetlier the two limits thus obtained
are equal, that is, whether the order of taking the limits in the double limit may
be interchanged. It is clear that if tht* function /(x, y) is continuous at (a, 6), the
limits approached by the two expressions will be ecjual ; for the limit of /(x, y) is
/(tt, 6) no matter how (x, y) approaches (a, 6). If / is discontinuous at (a, 6), it
may still happen that the order of tl»e limits in the double limit may be inter-
changed, as was true in the case above where the value in either order was zero;,
but this cannot be affirmed in general, and special considerations must be applied
to each case when/ is discontinuous.
Varieties of regions.* For both pure mathematics and physics the classification
of regions according to their connectivity is important. Consider a finite region R
bounded by a curve which nowhere cuts itself. (For tlie present
purposes it is not necessary to enter upon the subtleties of the
meaning of "curve" (see §§ 127-128); ordinary intuition will
suffice.) It is clear that if any closed curve drawn in this region
liad an unlimited tendency to contract, it could draw together
to a point and disappear. On the other hand, if R' be a region
like R except that a portion has been removed so that /?' is
bounded by two curves one within the other, it is clear that
some closed curves, namely those which did not encircle the
portion removed, could shrink away to a point, whereas otlier
closed curves, namely those which encircled that portion, could
at most shrink down into coincidence with the boundary of that
portion. Again, if two portions are removed so as to give rise
to the region R", there are circuits around each of the portions
which at most can only shrink down to the boundaries of those
portions and circuits around both portions which can shrink down to the bounda-
ries anil a line joining them. A region like /?, where any closed curve or circuit
may be shrunk away to nothing is called a simply connected region ; whereas regions
in which there are circuits which cannot be shrunk away to nothing are called
multiply connected regions.
A multiply connected region may be made simply connected by a simple device
and convention. For suppose that in R' a line were drawn connecting the two
bounding curves and it were agreed that no curve or circuit drawn within R' should
cross this line. Then the entire region would be surrounded by a
single boundary, part of which would be counted twice. The figure
indicates the situation. In like manner if two lines were drawn in
R" connecting both interior lH)undaries to the exterior or connecting
the two interior boundaries together and either of them to the outer
b<iundary, the region would be rendered simply connecteti. The entire r^ion
would have a single boundary of which parts would be counted twice, and any
circuit which did not cross the lines could be shrunk away to nothing. The lines
• The discussion from tJn»
§§ 12^-126.
{Mtint to the end of § 4.^ may be connected with that of
90
DIFFERENTIAL CALCUL'US
thus drawn in the region to make it simply connected are called cuts. There is no
need that the region be finite ; it might extend off indefinitely in some directions
like the region between two parallel lines or between the sides of an angle, or like
the entire half of the xy-plane for which y is positive. In such cases the cuts may
be drawn either to the boundary or off indefinitely in such a way as not to meet
the boundary.
46. Multiple valued functions. If more than one value of z corresponds to the
pair of values (x, y), the function z is multiple valued, and there are some note-
worthy differences between multiple valued functions of one variable and of several
variables. It was stated (§ 23) that multiple
yalued functions were divided into branches
each of which was single valued. There are
two cases to consider when there is one vari-
able, and they are illustrated in the figure.
Either there is no value of x in the interval
for which the different values of the function
are equal and there is consequently a number
D which gives the least value of the difference
between any two branches, or there is a value of x for which different branches
have the same value. Now in the first case, if x changes its value continuously and
if /(x) be constrained also to change continuously, there is no possibility of passing
from one branch of the function to another ; but in the second case such change is
possible for, when x passes through the value for which the branches have the same
value, the function while constrained to change its value continuously may turn off
onto the other branch, although it need not do so.
In the case of a function z =/(x, y) of two variables, it is not true that if the
values of the function nowhere become equal in or on the boundary of the region
over which the function is defined, then it is impossible to pass continuously from
one branch to another, and if P (x, y) describes any ^
continuous closed curve or circuit in the region, the
value of /(x, y) changing continuously must return to
its original value when P has completed the descrip-
tion of the circuit. For suppose the function z be a
helicoidal surface z = atan-i(y/x), or rather the por-
tion of that surface between two cylindrical surfaces
concentric with the axis of the helicoid, as is the case
of the surface of the screw of a jack, and the circuit
be taken around the inner cylinder. The multiple num-
bering of the contour lines indicates the fact that the
function is multiple valued. Clearly, each time that
the circuit is described, the value of z is increased by the amount between the suc-
ceative branches or leaves of the surface (or decreased by that amount if the circuit
is described in the opposite direction). The region here dealt with is not simply
connected and the circuit cannot be shrunk to nothing — which is the key to the
situation.
Theorem. If the difference between the different values of a continuous mul-
tiple valued function is never less than a finite number D for any set (x, y) of
values of the variables whether in or upon the boundary of the region of defini-
tion, then the value /(«, y) of the function, constrained to change continuously,
0,2 IT
HB
PARTIAL DIFFERENTIATION; EXPLICIT 91
will return to it« Initial value when the point P(z, y), describing a cloaed cunre
which can be shrunk to nothing, completes the circuit and return* to ita starting
point.
Now owing to the continuity of / throughout the region, it i» possible to find a
nuinbtT a 80 that \/{x, y) — /{x', v')\< € when \x — x'\<8 and \y — y'\<8 no matter
what iMiintM of the rej^ion (x, y) and {x\ y') may be*. Hence the values of /at any
two points of a small region which lies within any circle of radius \ 8 cannot differ
by so much as the amount D. If, then, the circuit is so small
that it may be inclosed within such a circle, there is no possi-
bility of passing from one value of /to another when the circuit
is described and / must return to its initial value. Next let
there be given any circuit such that the value of / starting from V ^^
a given value /(x, y) returns to that value when the circuit has ^^-^
been completely described. Suppose that a modification were
introihiced in the circuit by enlarging or diminishing the inclosed area by a small
area lying wholly within a circle of radius ^ 8. Consider the circuit ABCDEA and
the modified circuit ABCDEA. As these circuits coincide except for the arcs BCI)
and BC'D^ it is only necessary to show that/ takes on the same value at D whether
D is reached from B by the way of C or by the way of C". But this is necessarily
so for the reason that both arcs are within a circle of radius \ 8.
Then the value of / must still return to its initial value /(x, y)
when the modified circuit is described. Now to complete the
prtxjf of the theorem, it suffices to note that any circuit which
can be shrunk to nothing can be made up by piecing together a
number of small circuits as shown in the figure. Then as the
change in /around any one of the small circuits is zero, the change must be zero
around 2, 3, 4, • • • adjacent circuits, and thus finally around the complete large
circuit.
lietlucibility of circuits. If a circuit can be shrunk away to nothing, it is said to
be reducible ; if it cannot, it is said to be irreducible. In a simply connected region
all circuits are reducible ; in a multiply connected region there are an infinity of
irreducible circuits. Two circuits are said to be equivalent or reducible to each
other when either can be expanded or shrunk into the other. The change in the
value of /on passing around two equivalent circuits from ^ to ^
is the same, provided the circuits are described in the same direc-
tion. For consider the figure and the equivalent circuits AC A
and AC A described as indicated by the large arrows. It is clear
that either may be modified little by little, as indicated in the
proof above, until it h;us been changed into the other. Hence the
change in the value of / around the two circuits is the same. Or, as another proof,
it may be oKserved that the combined circuit ACAC'A^ where the second Is
described as indicated by the small arrows, may be regarded as a reducible circuit
which touches itself at -4. Then the change of / around the circuit is zero and /
must lose as much on pa.ssing from .4 to ^ by C as it gains in passing from A to
A by C. Hence on passing from ^ to ^ by C in the direction of the large arrows
the gain in /must be the same as on passing by C
It is now possible to see that any circuit ABC may be reduced to circuits around
the portions cut out of the region combinal with lines going to and fntm A and the
boundaries. The figure shows this; for the circuit ABCBADCDA is dearij
92
DIFFERENTIAL CALCULUS
Inducible to the circuit AC A. It must not be forgotten that although the lines AB
and BA coincide, the values of the function are not necessarily the same on AB
as on BA but differ by the amount of change introduced in
/on paatdng around the irreducible circuit BC'B. One of the
cases which arises most frequently in practice is that in
which the successive branches of /(x, y) differ by a constant
amount as in the case z = tan- ^ {y/x) where 2 tt is the differ-
ence between successive values of z for the same values of the
variables. If now a circuit such as ABC'BA be considered, where it is imagined
that the origin lies within BC'B^ it is clear that the values of z along AB and
along BA differ by 2 Tr, and whatever z gains on passing from A to
B will be lost on passing from B to A^ although the values through
which 2 changes will be different in the two cases by the amount
2x. Hence the circuit ABC'BA gives the same changes for z as
the simpler circuit BC'B. In other words the result is obtained
that if the different values of a multiple valued function for the same
values of the variables differ by a constant independent of the values of
the variables^ any circuit may be reduced to circuits about the bound-
aries of the portions removed ; in this case the lines going from the point A to the
boundaries and back may be discarded.
EXERCISES
1. Draw the contour lines and sketch the surfaces corresponding to
(«)« =
« + y
2(0,0) = 0,
{P)z =
xy
z(0, 0) = 0.
y ' • ' ■ ^' ' x + y
Note that here and in the text only one of the contour lines passes through the
origin although an infinite number have it as a frontier point between two parts
of the same contour line. Discuss the double limits lim lim z, lim lim z.
x = 0 y==0 y = 0 a; = 0
2. Draw the contour lines and sketch the surfaces corresponding to
{a)z
x^ + y^-1
{P)z =
{y)z =
X2 -f 2 1/2 _ 1
2y "' X ^' 2a;2 + y2_i
Examine particularly the behavior of the function in the neighborhood of the
apparent points of intersection of different contour lines. Why apparent ?
3. State and prove for functions of two independent variables the generaliza-
tions of Theorems 6-11 of Chap. II. Note that the theorem on uniformity is proved
for two variables by the application of Ex. 9, p. 40, in almost the identical manner
as for the case of one variable.
4. Outline definitions and theorems for functions of three variables. In partic-
ular indicate the contour surfaces of the functions
(a)u
x + y + 2a;
(/S)u
_ X2+J/2 + 2;a
W" = f.
x-y-z ^ ' x-\-y-{-z
and discuss the triple limits as x, y, « In different orders approach the origin.
6. Let « = P(x, v)/Q{x, I/), where P and Q are polynomials, be a rational func-
tion of X and y. Show that if the curves P = 0 and Q = 0 intersect in any points,
all the contour lines of t will converge toward these points ; and conversely show
PARTIAL DIFFERENTIATION; EXPLICIT 98
that if two different contour lines of x apparently cut in tome point, all the contour
lines will converge toward that point, P and Q will there vanish, and x will be
undefined.
6. If D is the minimum difference between different values of a multiple Talaed
function, as in the text, and if the function returns to its initial value plus IX^D
when P describes a circuit, show that it will return to its initial value plus I/^D
when P describes the new circuit formed by piecing on to the given circuit a small
region which lien within a circle of radius \ 8.
7. Study the function z = tan-*(y/x), noting especially the relation between
contour lines and the surface. To eliminate the origin at which the function Is not
defined draw a small circle about the point (0, 0) and observe that the region of
the whole xy-plane outside this circle is not simply connected but may be made so
by drawing; a cut from the circumference off to an infinite distance. Study tlie
variation of the function as P describes various circuits.
8. Study the contour lines and the surfaces due to the functions
I —' X*
(a) X = tan- * «y, 03) « = tan- » , {y) z = sin- » (x — y).
Cut out the points where the functions are not defined and follow the changes In
the functions al)out such circuits as Indicated in the figures of the text. How may
the region of definition be made simply connected ?
9. Consider the function z = tan- *(P/Q) where P and Q are polynomials and
where the curves P = 0 and Q = 0 intersect in n points (Oj, 6j), {a^, b^), • • •, (a„, 6„)
but are not tangent (the polynomials have common solutions which are not mul-
tiple roots). Show that the value of the function will change by 2kTr if (x, y)
describes a circuit which includes k of the points. Illustrate by taking for P/Q
the fractions in Ex. 2.
10. Consider regions or volumes in space. Show that there are regions in which
some circuits cannot be shrunk away to nothing ; also regions in which all circuits
may be shrunk away but not all closed surfaces.
46. First partial derivatives. Let z=f(x,y) be a single valued
function, or one branch of a multiple valued function, defined for (a, b)
and for all points in the neighborhood. If y be given the value 6,
then z becomes a function f(x, b) of x alone, and if that function has a
derivative for x = a, that derivative is called the partial derivative of
z =f(xj y) with respect to x at (^/, b). Similarly, if x is held fast and
equal to a and if /(^, y) has a derivative when y = b, that derivative is
called the partial derivative of z with respect to y at («, b). To obtain
these derivatives formally in the case of a given function /(or, y) it is
merely necessary to differentiate the function by the ordinary rules,
treating y as a constant when finding the derivative with respect to x
and x as a constant for the derivative with respect to y. Notations are
g=|=/:=/.=*; = /v = /'^
U),
94 DIFFERENTIAL CALCUi:US
for the a--<ierivative with similar ones for the y-derivative. The partial
derivatives are the limits of the quotients
^.^J(a + h,l)-.f{a,b) ^.^fia,l, + lc)-f{a,b)^
ikAo h k=o k
provided those limits exist. The application of the Theorem of the
Mean to the functions /(«, b) and /(a, y) gives
f{a + h, h) - f{a, h) = hf,{a + BJi, b\ 0 < ^, < 1,
/(a, b + k) -f{a, b) = kf;{a, b -\- B^Jc), 0 < d, < 1, ^ ^
under the proper but evident restrictions (see § 26).
Two comments may be made. First, some writers denote the partial derivatives
by the same symbols dz/dx and dz/dy as if « were a function of only one variable
and were differentiated with respect to that variable ; and if they desire especially
to call attention to the other variables which are held constant, they affix them as
subscripts as shown in the last symbol given (p. 93). This notation is particularly
prevalent in thermodynamics. As a matter of fact, it would probably be impos-
sible to devise a simple notation for partial derivatives which should be satisfac-
tory for all purposes. The only safe rule to adopt is to use a notation which is
sufficiently explicit for the purposes in hand, and at all times to pay careful atten-
tion to what the derivative actually means in each case. Second, it should be noted
that for points on the boundary of the region of definition of /(x, y) there may be
merely right-hand or left-hand partial derivatives or perhaps none at all. For it
is necessary that the lines y = b and x = a cut into the region on one side or the
other in the neighborhood of (a, b) if there is to be a derivative even one-sided ;
and at a comer of the boundary it may happen that neither of these lines cuts
into the region.
Theorem. If f(x, y) and its derivatives /^ and f^ are continuous func-
tions of {Xj y) in the neighborhood of (a, ^), the increment A/ may be
written in any of the three forms
^f = f{<^ + h,b + k)-~f{a,b)
= hf:{a + eji, b) -f- V; (a + h,b-\- ejc)
= hf,{a -\-eh,b-{- Ok) 4- kf„(a -^ Oh, b + Ok) ^ ^
= V;K *) + fcf^ia, b) -f tji 4- y-,
where the ^s are proper fractions, the ^'s infinitesimals.
To prove tlie first form, add and subtract /(a + A, 6) ; then
V= [/(a + K b) -/(a, 6)] + [/(a ^ h,b -\- k)-f{a -^ h, &)]
= Vx'(a + B,h, b) + A:/; (a + A, 6 + O^k)
by the application of the Theorem of the Mean for functions of a single variable
(II 7, 26). The application may be made because the function is continuous and
the indicated derivatives exiKt. Now if the derivatives are also continuous, they
nuty be expressed as
/;(a + Vi b) =/;(«, '') + r,, /;(« + /I, ^ + e^k) =f;{a, b) -\- f^
PARTIAL DIFFERENTIATION; EXPLICIT 95
where f^, f, ^^y ^ ^^^ ^ ""^'^ ^ defiired by taking h and Jr milBcientJy nnall.
Hence the third form follows from the first. The second form, which hi gymmetric
in the increnientB A, it, may be obtained by writing x = a + (A and y = b ■{■ tk.
Then/(x, y) = *{t). Ab/Ib continuous in (2, y), the function ♦ U continuous in (
and it>i increment is
A* = /(a + t + Af A, ft + « + At*) -/(a + tA, 6 + t*).
This may be regarded as the increment of / taken from the point (x, y) with M • h
and A/ • Ar as increments in x and y. Hence A4> may be written as
A* = 6l'hf^{a + tA, 6 + <*) + At. */;(a + tA, 6 + 0:);+ f,A/ • A + r,At • *.
Now if A<l> be divided by At and At be allowed to approach zero, it is seen that
lim— = A/; (a + tA, 6 + tJfc) + kf^ia + tA, 6 + tJfc) = - .
At (U
The Theorem of the Mean may now be applied to ♦ to give ♦(!) — ♦(0) = 1 • *'{0)^
and hence
*(!) - «|.(0) =/(a + A, 6 + *) -/(a, b)
= A/= A/; (a + ^A, 6 + ^fc) + ik/;(o + tf A, 6 + Ok).
47. The jjartial differentials off may be defined as
dj^fj^, so that .70. = Aa-, ^ = ^'
rf f df (^
dj=f,^y, 80 that ^y = Ay, -;^ = ^>
where the indices x and y introduced in </^ and rf^/ indicate that ar and
\j respectively are alone allowed to vary in forming the corresponding
partial differentials. The total differential
which is the sum of the partial differentials, may be defined as that
sum ; but it is better defined as that part of the increment
A/= ^ A.r 4- 1^ Ay + C^Ar + t^y (7)
which is obtained by neglecting the terms ^jA.r -j- J^Ay, which are of
higher order than Ax and A//. The total differential may therefore be
fomputi'd by finding the partial derivatives, multiplying them respec-
tively by dx and di/y and adding.
The total differential of z =f(xy y) may be formed for (x^ y^ as
where the values x — x^ and y-^y^ are given to the independent differ-
entials dj- and <///, and df=^ dz is written as sr — «^. This, however, is
96
DIFFERENTIAL CALCULUS
the equation of a plane since x and y are independent. The difference
^f—df which measures the distance from the plane to the surface
along a parallel to the «-axis is of higher order than VAx* -\- Ay* ; for
^f-df
VAxN-Ap
i,^x -h l^y
<I^J + 1^.1 = 0.
V Ax* 4- Ay*
Hence the plane (8) will be defined as the tangent plane at {x^ y^, z^
to the surface z =f{x, y). The normal to the plane is
Ji =
y-Vo
(^Jo Vy\
-1
(9)
PP' = Ax,
PP" = Ay,
P"T"/PP"=f;,,
P'T' 4- P'T" = JV'r,
which will be defined as the normal to the surface at (x^, y^y z^). The
tangent plane will cut the planes y = y^ and x = x^ in lines of which
the slope is f^ and f^^. The surface will cut these planes in curves
which are tangent to the lines.
In the figure, PQSR is a portion of the
surface z =/(x, y) and PT'TT" is a cor-
responding portion of its tangent plane
at P(x^ y^y «o). Now the various values
may be read off.
PT' = 4/,
P^^R = A,/,
P'T" = c?,/,
iV'5 = A/,
N^T^df=dJJtdJ,
48. If the variables x and y are expressed as x = ^{f) and y = ^(^)
so that/(x, y) becomes a function of ^, the derivative of /with respect
to Ms found from the expression for the increment of/.
A/^§/Ax a/Ay Ax Ay
A^ dx ^t dy M ^» AiJ ^* A^
At = o A^ c?^ dx dt dy dt
The conclusion requires that x and y should have finite derivatives with
respect to t. The differential of /as a function of t is
and hence it appears that the differential has the same form as the total
differentUil. This result will be generalized later.
(10)
PARTIAL DIFFERENTIATION; EXPLICIT 97
As a particular case of (10) suppose tliat x and y are so related that
the point (ar, y) moves along a line inclined at an angle r to the s'-axis.
If 9 denote distance along the line, then
X = a;^ -f- * cos T, y = y^ + <8inT, dx =s cob rds, dyss sin rds (12)
The derivative (13) is called the directional derivative of /in the direc-
tion of the line. The partial derivatives /J, f^ are the particular direo-
tional derivatives along the directions of the r-axis and y-axis. The
directional derivative of / in any direction is the rate of increase of
/along that direction; if x =/(a', y) be inter-
preted as a surface, the directional derivative is
the slope of the curve in which a plane through
the line (12) and perpendicular to the ary-plane
cuts the surface. If /(a*, y) l^e represented by
its contour lines, the derivative at a point
{xy y) in any direction is the limit of the ratio
^f/As = AC/^s of the increase of/, from one contour line to a neigh-
boring one, to the distance between the lines in that direction. It is
therefore evident that the derivative along any contour line is zero and
that the derivative along tlie normal to the contour line is greater than
in any other direction because the element dn of the normal is less than
ds in any other direction. In fact, apart from infinitesimals of higher
Y
-^^
5^v
i^
1-
^
X
order.
An A/" A/* df df
Hence it is seen that the derivative along any direction may be found
by multiplying the derivative along the normal by the cosine of the angle
between that direction and the normal. The derivative along the normal
to a contour line is called the normal derivative of / and is, of course,
a function of (a?, y).
49. Next suppose that w =/(ar, y^z^-- •) is a function of any number
of variables. The reasoning of the foregoing paragraphs may be
repeated without change except for the additional number of variables.
The increment of/ will take any of the forms
V = /(« -f- A, * + A:, c + /, •••) -/(«, h c, ...)
= A/;(« 4- e^h, ft, r, . . .) + hf;{a ^ h, b -k- 6jc, c, - )
+ //;(« 4- ^,*-|-A-,c4-d,/, ••)+••
= Kfr 4- A/; -h //: 4- • • 4 t,A 4 tJc + (/ + • •,
98 DIFFERENTIAL CALCULUS
and the total differential will naturally be defined as
and finally if x, y, «, • • • be functions of t, it follows that
df^dldx dldy dldz
dt dx dt dy dt dz dt ^ ^
and the differential of /as a function of t is still (16).
If the variables x, y, z, '■- were expressed in terms of several new
variables r, s, • • • , the function / would become a function of those vari-
ables. To find the partial derivative of / with respect to one of those
variables, say r, the remaining ones, s, • • • , would be held constant and
/ would for the moment become a function of r alone, and so would a*,
y,Zj"-. Hence (17) may be applied to obtain the partial derivatives
dl^dldx^dldy^dldz^
dr dx dr dy dr dz dr *
df dfdx , dfdif . dfdz , ^ ^^^)
and a=a^ + -^a+ -i-T- H > etc.
cs ex OS oy OS cz cs
These are the formulas for change of variable analogous to (4) of § 2.
If these equations be multiplied by Ar, As, • • • and added,
^Ar + |^A. + ..- = ^(^A. + -A. + ...)+^(^Ar + . ..) + ..,
for when r, 5, • • • are the independent variables, the parentheses above
are dx, dy, dz, • • • and the expression on the left is df.
Theorem. The expression of the total differential of a function of
X, y, z, " ' OS df = f^dx -\- fydy -\-f^dz + • • • is the same whether x, y,
z, '•• are the independent variables or functions of other independent
variables r, s, • • • ; it being assumed that all the derivatives which occur,
whether of f hy x, y, z, • • • or oi x, y, z, " ■ by r, s, - ■, are continuous
functions.
By the same reasoning or by virtue of this theorem the rules
d(cu) = cdu, d(u -^v — w) = du-\- dv — dw,
d(uv) = udv + vdu, d{-]= > ^ ^
\vj v^
of the differential calculus will apply to calculate the total differential
of combinations or functions of several variables. If by this means, or
any other, there is obtained an expression
PARTIAL DIFFERENTIATION; EXPLICIT 99
d/=R(r, 8,t,- ')dr + S(r, *, ^, • • )tU + T(r, s,t," ^dt -h • • . (20)
for the total differential in which r, «, ^, • • • are independent variables,
the coefficients if, 5, 7*, • • • are the derivatives
For in the equation rf/= Rdr ^Sds-^Tdt-^--- -frdr -^f.dt -{-f.dt -(- • ,
the variables r, «,/,•• •, l)eing independent, may be assigned increments
al)Solutely at pUnisure and if the jmrticular choice rfr=l, rf» = </<=•== 0,
be made, it follows that R =J\,\ and so on. The single equation (20) is
thus equivalent to the equations (21) in number equal to the number of
the indej)endent variables.
As an example, consider the case of the function tan- ^ {y/x). By the rules (19)
J tan- ^ ^ = ^ ^^^'^^ = ^^^^ ~ y^/^* _zdy-ydx
X l + (y/x)» l + (y/x)« x« + y«
Then — tan- » ?^ = - — -^ . — tan- » ^ = —^—r , by (20)-(21).
ax X x* + y« ay X x« + y* J V ; \ ;
If y and x were expressed as y = sin h rs< and x = cosh rsi, then
_ J y _ xdy — ydx _ [stdr + rfd« + rsdt] [cosh*r«i — sinh*r«(]
X X* + y^ co8h*r8i + sinhVsi
df 8t df H a/ r«
and
ar cosh2r8t da cosh2r8< dt C06h2r8t
EXERCISES
1. Find the partial derivatives/^', /^ or/,', f^, /,' of these functions :
{a) log {x« + y2), 03) e* cos y sin z, (7) x* + 8 xy + y«,
y + sin»2),
\
<*> ^' <^> ^Ti^' <f) »og(«n* + «n*y + «»»•«)
(,) sin-1?^. - (^) ^ J, (0 tanh-iV2(^-^^: + ^V
^ ' » ^ ' X ^ ' \x« + ya + zV
2. Apply the definition (2) directly to the following to find the partial derivi^
lives at the indicated points :
(a) ^ at (1, 1), OS) x« + 3xy + y» at (0, 0), and (7) at (1, 1),
(8) ~ at (0, 0); also try differentiating and substituting (0, 0).
X + y
3. Find the partial derivatives and hence the total differential of :
enr
(«) c-*8iuy, («) e**8inhxy, (i) logtan/x + -yj,
(n)
0)' ^*>l^^ :o.)>o«C^-n/^>
100 DIFFERENTIAL CALCULUS
4. Find the general equations of the tangent plane and normal line to these
•urfaceB and find the equations of the plane and line for the indicated (x^, Vq) :
(a) the helicoid z = Jfc tan- ^ (y/x), (1, 0), (1, - 1), (0, 1),
(fi) the paraboloid ipz = jx^ + y^), (0, p), (2p, 0), (p, - p), _
(7) the hemisphere z = Va* - x^ - y^ (0, - J a), (J a, ^ a), (i Vs a, 0),
(a) the cubic xyz = 1, (1, 1, 1), (- h - h 4), (4, i, i).
6. Find the derivative with respect to tin these cases by (10) :
(a) /= X* + y*, X = acosi, y = 6sin«, (/3) tan-iA/-, y = coshi, x = sinht,
(7) sin- ^ (x — y), X = 3 <, y = 4 i', (5) cos 2 xy, x = tan- ^t,y = cot- ^ «.
6. Find the directional derivative in the direction indicated and obtain its
numerical value at the points indicated :
(a) x«y, T = 46°, (1, 2), (/3) sin^xy, r = 60°, (Vs, - 2).
7. (a) Determine the maximum value of df/ds from (13) by regarding t as
Tariable and applying the ordinary rules. Show that the direction that gives the
maximum is , ,
{p) Show that the sum of the squares of the derivatives along any two perpen-
dicular directions is the same and is the square of the normal derivative.
8. Show that (/; + y7y)/VT+y^ and (f^y' -/;)/Vn- y'2 are the deriva-
tives of / along the curve y = 4>{x) and normal to the curve.
9. If df/dn is defined by the work of Ex. 7 (or), prove (14) as a consequence.
10. Apply the formulas for the change of variable to the following cases :
(a) r = V^TV\ « = tan-.?. Find ^, '1. JFf+Wf-
X dx dy Wdx/ \dyl
(P) X = rco80, y = rsin^. Find ?^, ^, (^I)\ i/^V.
dr d<t> \dr/ r^ \d<f>/
(7) x=:2r~3«+ 7, y=-r + 8s-9. Find ?!^ = 4x+2y if u = x^^y^.
tr
r X = X' cos a - y' sin a, /?^\% /^V- (K\\ l?L\.
^'> 1 y = X' sin a + y cos a. ^^^^ U/ ^ W ~ W) + W)
(.) Prove ^ + ?^ = 0 if /(u,tj)=/(x-y,y-x).
(f) Let x = ax' -k- by' -\- cz", y = a'x' + 6y + cV, z - a"x' + h"y' + c"i', where
a, 6, c, a', 6', c', a", 6", c" are the direction cosines of new rectangular axes with
respect to the old. This transformation is called an orthogonal transformation. Show
©■*©•* ©•-©•*©vo'=o' -
11. Define directional derivative in space ; also normal derivative and estab-
U«h (14) for thla case. Find the. normal derivative otf=xyz at (1, 2, 3).
18. Find the total differential and hence the partial derivatives in Exs. 1, 3, and
(a) log(x« + y« + ,«), (^ y/x, (7) iciyery\ (J) xyz logxyz,
PARTIAL DIFFERENTIATION; EXPLICIT 101
(«) u = x* — y*, « = rco««i, y = «8lnrf. ¥ind du/dr, du/ da, tu/dt.
(f ) u = y/z, z = roos^sin^, y = r Bin 0 sin 0. Find u^', u^, u^'.
(ij) u = enr, 2 = logVH + «*, y = tan- > («/r). Find u^', u,'.
13. » ^ = ^ and ?^ =- ^. «how S^ = i*» »nd IV =_«!?« r. ^^pou,
axay ^y ftc ^rr«^ r d^ dr
coordinates and /, g are any two functions.
14. If p{x, y, z, t) in the premiire in a fluid, or p{x, y, z, t) is the density, depend-
ing on the jKiHltion in the fluid and on the time, and if u, c, lo are the velocities of
the particleH of the fluid along the axes,
dp ^ . ^ . ^ , ^ A ^P ^P , ^P , ^P . ^P
dt dx ty dz St dt dx dy dz dt
Explain the meaning of each derivative and prove the formula.
15. If z = xy, interpret z as the area of a rectangle and mark d^z, AyZ, Az on the
figure. Consider likewise u = xyz as the volume of a rectangular parallelepiped.
16. Small errors. If /(x, y) be a quantity determined by measurements on x
and y, the error in / due to small errors etc, dy in x and y may be estimated as
df = f^dx •{■ f^dy and the relative error may be taken as df -i-f= dlogf. Why
is this ?
(a) Suppose 5 = J a6 sin C be the area of a triangle with a = 10, 6 = 20, C = 80".
Find the error and the relative error if a is subject to an error of 0.1. Ans. 0.6, 1%.
(/3) In (a) suppose C were liable to an error of 10' of arc. Ans. 0.27, \%.
(7) If a, 6, C are liable to errors of 1%, the combined error in 5 may be 3.1%.
(a) The radius r of a capillary tube is determined from 13.6 rrr*/ = to by find-
ing the weight w> of a column of mercury of length I. If u; = 1 gram with an error
of 10-' gr. and I = 10 cm. with an error of 0.2 cm., determine the possible error
and relative error in r. Ans. 1.06%, 6 x 10-*, mostly due to error in I.
( e ) The formula c^ = a"^ •{■ b^ — 2 ab cos C is used to determine c where a = 20,
6 = 20, C = 60° with possible errors of 0.1 in a and b and 30' in C. Find the possible
absolute and relative errors inc. Ans. J, l\%.
(f) The possible percentage error of a product is the sum of the percentage
errors of the factors.
(tj) The constant g of gravity is determined from y = 2 si-* by observing a body
fall. If s is set at 4 ft. and t determined at about ^ sec, show that the error in g
is almost wholly due to the error in t, that is, that s can be set very much more
accurately than t can be determined. For example, find the error in t which would
make the same error in g as an error of | inch in s.
{(f) The constant g is determined by gt^ = irH with a pendulum of length I and
period (. Suppose t is determined by taking the time 100 sec. of 100 beats of the
pendulum with a stop watch that measures to ^ sec. and that I may be measured
as 100 cm. accurate to \ millimeter. Discuss the errors in g.
17. Let the coordinate x of a particle be x =/(9i, 9,) and depend on two inde-
>«*»d«nt variables qr,, q^. Show that the velocity and kinetic energy are
102 DIFFERENTIAL CALClJLUS
where dote denote differentiation by t, and a^j, a^g, a^2 are functions of (g^, q^).
Show — = —t < = 1. 2, and similarly for any number of variables q.
a*. ag<
18. The helix x = a cost, j/ = a sin t, 2 = at tan a cuts the sphere x^ + y^ ■{- z' =
a* sec^^ at sin- » (sin a sin /3) .
19. Apply the Theorem of the Mean to prove that /(x, y, z) is a consUnt if
/'=/'=/' = 0 is true for all values of x, y, z. Compare Theorem 16 (§ 27) and
make the statement accurate.
20. Transform f^= \(^)'+ (f^)^+ {%)' ^^ (^) cylindrical and (/S) polar
coordinates (§ 40).
21. Find the angle of intersection of the helix x = 2co»t, y = 28mt, z = t and
the surface xyz = 1 at their first intersection, that is, with 0 < t < ^ tt.
22. Let/, g, h be three functions of (x, y, z). In cylindrical coordinates (§ 40)
form the combinations F=f cos <p •}■ g sin <f>, G = — /sin <p + g cos <f>, H = h. Trans-
^ ' ax ey Sz ^' dy dz ^'' dx ^
to cylindrical coordinates and express in terms of F, G, H in simplest form.
23. Given the functions y^ and (z^)* and z(»^. Find the total differentials and
hence obtain the derivatives of x* and (x^)'*' and x(^).
50. Derivatives of higher order. If the first derivatives be again
differentiated, there arise four derivatives J^, f^^, y^i, f^"^ of the second
order, where the first subscript denotes the first differentiation. These
may also be written
r^-^ r^^^ r'^^ r'=?^
where the derivative of df/dy with respect to x is written d^fjdxdy
with the variables in the same order as required in D^D^f and opposite
to the order of the subscripts in fy^. This matter of order is usually of
no importance owing to the theorem : If the derivatives /^, /^ have
derivatives fj^, f'^ which are continuous in (x, y) in the neighborhood
of any point (x^ y^j the derivatives f^ and f^ are equal j that is,
The theorem may be proved by repeated application of the Theorem of the
Mean. For
[/{^o + ^^,1/0 + *)-/(a;o» Vo + k)] - [/(Xo + h, y^) -/(x^, y^)] = [0(2/0 + ^)- ^Wl
= [/(^o + ^,1/0 + *)-/(^o + '^, l/o)]- [/(a^o. Vo + k)-'f{Xo, Vo)] = [HH + ^)-H^o)]
where ^(y) stands for /{x^ + ^i, 2/)-/(Xo, y) and ^(x) for /(x, y^ + k) -/(x, y^).
Now
0(Vo + *) - 0 W = *0'(Vo + ^*) = *[/v'(^o + 'I, J/o + ^^) -/;(«o. Vo + ^^)]»
f (Xo + A) - ^ (Xo) = ^'(Xo + ^A) = A [/; (Xo + 0% Vo + k) - /; (Xo + 6>'/i, y^)]
PAKTIAL inFFEKENTIATION; EXPLICIT 108
by applying the Theorem of the Mean to ^{y) and ^(2) regarded aji fuuctioiw of a
tiiunle variable and then subKtittiting. The reKulte obtained are neceisarily equal
to each other ; but eiu^h of theMe Ik in form for another application of the theorem.
ki/^ixo + ^ Vo + ^*) -/^(a:©, Vo + ^^)] = ^^C^o + ^^ Vo + ^*).
H/^i^o + <^*» Vo + *) -/*(-«o + ^% Vo)] = W;;(Xo + ^A, J^o + n'*).
Hence /^{x^ + 17A, ^o + ffk) =/;;(Xo + ^A, y^ + v'k).
Ah the derivativcH/j^/^'^ are mipposed to exiHt and be continuous in the variables
(x, 2^) at and in the neighborhood of (Xq, y^), the limit of each Hide of the equation
exiHt8 Hs h±0^ k±0 and the equation is true in the limit. Hence
f^i^o^Vo)=f^{^o^Vo)'
The diiferentiatioii of the three derivatives/^,/;:^ =Jw'xfJ7i, will give
six derivatives of the third order. Consider f^ and f^ These inay
W writttMi as (/j)^ and (f^)yx and are equal by the theorem just proved
(provided the restrictions as to continuity and existence are satisfied).
A similar conclusion holds forf^^ and /^i; the number of distinct
derivatives of the third order reduces from six to four, just as the
number of the second order reduces from four to three. In like manner
for derivatives of any order, the value of the derivative depemh not on
the order in which the indiciiluat differentiations with respect to x and
y are performedy hut only on the total number of differentiations with
respect to eachy and the result may be written with the differentiations
collected as T^'n^nf
Analogous results hold for functions of any numl^er of variables. If
sevei-al derivatives are to he found and added together, a symbolic
form of writing is frequently advantageous. For example,
51. It is sometimes necessary to change the variable in higher deriv-
atives, particularly in those of the second order. This is done by a
rei)eated application of (18). Thus f^ would be found by differentiat-
ing the first equation with respect to r, and f^ by differentiating the
first by s or the second by r, and so on. Compare p. 12. The exercise
below illusti-ates the method. It may l)e remarked that the use of higher
differentials is often of advantage, although these differentials, like the
higher differentials of functions of a single variable (Exs. 10, 16-19,
p. 67), have the disadvantage that their form dei)ends on what the
independent variables are. This is also illustrated below. It should be
particularly l)orne in mind that the great value of the first differential
104 DIFFERENTIAL CALCULUS
lies in the facts that it may be treated like a finite quantity and that
its fonn is independent of the variables.
To change the variable in r^^ + f ^ to polar coordinates and show
Sh) c*v _d^ ^?5.i^ fa; = rco80, y = rsin<f>,
®° ftB~ar^ d^dx' dy^drdy d<t> tiy
by applying (18) directly with sc, y taking the place of r, s, • • • and r, 0 the place
of «, y, «, • • • . These expressions may be reduced so that
ar _ a» X ac — y _ ?E ? 4. ^ — y
at ~ ar Vx^ + y* d<l>x^-\-y^~ drr d<f> f^ '
Sh d dv d dv dr d dv d<f>
Next — = = 1
8x* dxdx drdx dx d<pdx dx
[a^rx e»^x a'^o — y ap a — yix
dr^r drdrr drd<p r^ d<f>dr r^ jr
The differentiations of x/r and — y/r^ may be performed as indicated with respect to
r, 0, remembering that, as r, 0 are independent, the derivative of r by 0 is 0. Then
dH_x^dH y^dv ^xy dH ^xy dv y^ dH
ax2 ~ r2 dr-^ r^ dr r^ drd<f> r^ d<p r* d<p^ '
In like manner dH/dy^ may be found, and the sum of the two derivatives reduces
to the desired expression. This method is long and tedious though straightforward.
It is considerably shorter to start with the expression in polar coordinates and
transform by the same method to the one in rectangular coordinates. Thus
dv dv dx , dv dy dv dv . 1 /dv dv
— = H - = — COS0 + — sin0— '
dr dxdr dy dr dx dy
l/dv . dv \
r\dx dy }
( dv\ (d^ ^ , dH . \ / dH dH . \ dv dv .
(^W = (aT^'"'^^^'^"^)^-'(^'''^+a-^""T"'a-i'''^ + S""^'
dv dv dx dv dy dv . dv dv dv
— = H = rsin0H r cos 0 = y -\ x.
dip dx dip dy di> dx dy dx dy
1 a»t /dH . ^ dH . \ / dH . dH \
- =-* = I in; "n 0 cos 0 ) y + ( sin 0 H cos 0 )x
Td4^^ \ax2 ^ ^dx I \ dxdy dy^ V
dv dv .
cos 0 sin 0.
ax ^ ay '^
Then
dr\dr) rd4>^~ W^ dyV
or ?!? + ?!^ = li/r??Ul?^ = ^-^^?!!4.i^ m^
bi* dy^ r dr\ dr/ r^ d<p^ dr^ r dr r^ diP^' ^ '
The definitions dlf = f^dx\ d^yf = f^dxdy, d^/ = f^'^dy^ would naturally be
glf en for particU differentiaU of the second order, each of which would vanish if /
reduced to either of the Independent variables x, y or to any linear function of
them. Thu« the second differentials of the Independent variables are zero. The
PARTIAL DIFFERENTIATION; EXPLICIT 106
second total differential would be obtained by differentiating the first total differ-
ential.
J-y=d(^/=d(^dx + ^dy) = d^dx + d^dy + ?^d«x + ^d«y;
d^ = ^.dX+^dv. d^ = ^d. + ^d.,
bz ax* ayax ay axay ay*
and fri/=^dx« + 2^d«iy + ^di/=» + ^d»x + ^d«y. (14)
ax" bzty dv^ dx dy
The last two terms vanish and the total differential reduces to the first three terms
if I and y are the independent variables ; and in this case the second derivatives,
fr^fx^^fy'r ^^^ ^''^ coefticients of dx^, 2dxdy, dy'^, which enables those derivatives
t<» be found by an exttMiHion of the method of finding the first derivatives (§ 49).
The nietlioti is particularly useful when all the second derivatives are needed.
The problem of the change of variable may now be treated. Let
j«, = ?!?dx^ + 2^^^dxdy + ^dy^
ax=» ax« &y^
= ^dr« + 2-^drd^ + ^d0« + ?^d*r+ ^d«^,
where x, y are the independent variables and r, 0 other variables dependent on
them — in this case, defined by the relations for polar coordinates. Then
dx = cos <f>dr — r sin <pd<p, dy = sin <f>dr + r cos <f>d<f>
or dr = cos 0dx ■\- sin 0dy, rd<p = — sin <pdx + cos <f>dy. (25)
Then d*r = (— sin <pdx + cos <pdy) d<p = rd<pd<f> = rd<f>^,
drdtp + nP0 = — (cos 0dx + sin 0dy) d<p = — drd<f>,
where the differentials of dr and rd<f> have been found subject to d*x = d*y = 0.
Hence dh = rd<p^ and rd^<f> = — 2 drd^. These may be substituted in d*i> which
becomes
d2o = — -dr« + 2( )drd0 +( — - + r — |d^*.
^ \drd<t> rdif>l \a02 trj
Next the values of dr^^ drd<p, dtpl^ may be substituted from (26) and
^ [dH . 2/a8» I dv\ . , /dH , aoXsinVl^ •
dh = \ —-co8^<t>--{ |cos08in0 + ( — : + r — )—-^\dx*
. « r^^o . / ^*« 1 ^\ co8*0 - sin20 a^p cos 0 sin ^"l , ,
+ 2 — -co8A8in0 + ( 1 ■^ — -Idxdy
. r^o I «. . 2/a«u 1 ar\ , . /a«c . aoXcoeVl^ •
Thus finally the derivatives t^^ t^, tj^^ are the three brackets which are the
coefficients of dx*, 2dxdy, dy*. The value of v^ + v^'^ is as found before.
52. The condition f'J^=f'^ which subsists in accordance with the
fundamental theorem of § 50 gives the condition that
M(x, y)dx 4- Nix, y)dy = ^dx + |^ ^y = df
106 DIFFERENTIAL CALCULUS
be the total differential of some function f(x, y). In fact
d df dM dN _ d df
dydx dy dx cxdy
dM dN (dM\ (dN\
The second form, where the variables which are constant during the
differentiation are explicitly indicated as subscripts, is more common in
works on thermodynamics. It will be proved later that conversely if
this relation (26) holds, the expression Mdx + Ndy is the total differ-
ential of some function, and the method of finding the function will
also be given (§§ 92, 124). In case Mdx -\- Ndy is the differential of
some function f(x, y) it is usually called an exact differential.
The application of the condition for an exact differential may be
made in connection with a problem in thermodynamics. Let S and U
be the entropy and energy of a gas or vapor inclosed in a receptacle of
volume V and subjected to the pressure p at the temperature T. The
fundamental equation of thermodynamics, connecting the differentials
of energy, entropy, and volume, is
dU=TUS-pdv; and (f )^= - (|)^ (27)
is the condition that dU he 2i total differential. Now, any two of the
five quantities U, 5, v, T, p may be taken as independent variables. In
(27) the choice is S, v; if the equation were solved for dS, the choice
would he Uy v; and U, S if solved for dv. In each case the cross differ-
entiation to express the condition (26) would give rise to a relation
between the derivatives.
If p, T were desired as independent variables, the change of variable
should be made. The expression of the condition is then
{fr[<i-'mr{i[^L%-'m.]},
\dp/T dTdp dTdp dp8T \dT/„ ^ dpdT
where the differentiation on the left is made with p constant and that on the right
with T constant and where the subscripts have been dropped from the second
derivatives and the usual notation adopted. Everything cancels except two terms
which give
PARTIAL DIFFERENTIATION; EXPLICIT 107
The in>iM)rtHijce of the test for an exact differential lie« not only in the r< hitiuns
obtalntMl between the derivativen as above, but also in the fact that in :ijii.li..i
niatlienuiticH a great many expreHsionH are written a*i dif!erential8 wliii h ;u<- imi
tlie total differentialHof any functionH and which must be distinguiHhol fiMm «x:i< t
differentialH. For instance if dll denote the infinitesimal portion of h* ;it ;i<i<l< .1
to the gas or vajK^r alxive considered, the fundamental equation is expn-Ks^Mi ah
dll = dU •\- pdv. That is to say, the amount of heat added is equal to the increaM
in the energy plus the work done by the gas in expanding. Now dll is not the dif-
ferential of any function H{U^ v) ; It Is dS = dll/T which is the differential, and
this is one reason for introducing the entropy S. Again if the forces X, Y act on a
particle, the work done during the displacement through the arc ds = y/dx* + dy*
is written d \V = Xdx + Ydy. It may happen that this Is the total differential of
some function ; indeed, if
dW=-^dV{x,y), Xdx -^ Ydy = - dV, X=-— , F=-— ,
dx by
where the negative sign Is introduced in accordance with custom, the function V is
called the potential energy of the particle. In general, however, there Is no poten-
tial energy function I", and dW is not an exact differential ; this is always true
when part of the work is due to forces of friction. A notation which should dis-
tinguish between exact differentials and those which are not exact is much more
needed than a notation to distinguish between partial and ordinary derivatives ;
but there appears to be none.
Many of the physical magnitudes of thermodynamics are expressed as deriva-
tives and such relations as (26) establish relations between the magnitude& Some
definitions :
specific heat at constant volume is C„ = ( — ) = t( — ),
specific heat at constant pressure
latent heat of expansion
coefficient of cubic expansion
modulus of elasticity (isothermal)
modulus of elasticity (adiabatic)
53. A polynomial is said to be homogeneous when each of its terms
is of the same order when all the variables are considered. A defini-
tion of homogeneity which includes this case and is applicable to more
general ciises is : A function /{Xy i/y Zy - • •) of ant/ number of variables is
called homogeneous if the function is multiplied by somepowerofX when
all the variables are multij)lied by X; and the power of X which faotore
Cp
Ut/p
'(i).
K
\dt/T
'a
- .w
Es
I
108 DIFFERENTIAL CALCULUS
out is called the order of homogeneity of the function. In symbols the
oondition for homogeneity of order n is
f(Xx, \y, \z,'") = Xy(a:, y,z,-- •). (29)
^^ a^l^ f + tan-^, ^J= (29')
are homogeneous functions of order 1, 0, — 1 respectively. To test a
function for homogeneity it is merely necessary to replace all the vari-
ables by A times the variables and see if X factors out completely. The
homogeneity may usually be seen without the test.
If the identity (29) be differentiated with respect to A., with x'=\x, etc.,
('" i + y ^' + "^ 4 ■•" ■ • •)'^^^' ^^' ^' ■ ■ '^ = "^""'/(»^' y- «' • • •)•
A second differentiation with respect to A, would give
or (x'^+ 2:..y^ + 2,'^,+ • • •)/= n{n - l)X-y(a,, y,.,- ■).
Now if X be set equal to 1 in these equations, then x' = x and
-% + y% + '^t + — nfi.,y,.,...), (30)
'^S-*-2-2'ft + ^i + 2x.g + ... = »(»-l)/(x,^, .,...).
In words, these equations state that the sum of the partial derivatives
each multiplied by the variable with respect to which the differentia-
tion is performed is n times the function if the function is homogeneous
of order n ; and that the sum of the second derivatives each multiplied
by the variables involved and by 1 or 2, according as the variable is
repeated or not, is n (n — 1) times the function. The general formula
obtained by differentiating any number of times with respect to X may
be expressed symbolically in the convenient form
(xD, + yD^ + «i>. -f • • •)*/= n(n-l)-.-{n-k + 1)/ (31)
This is known as Euler^a Formula on homogeneous functions.
It li worth while noting that in a certain sense every equation which represents
a gAometric or physical relation is homogeneous. For instance, in geometry the
magnitudes that arise may be lengths, areas, volumes, or angles. These magni-
tudes are expreased as a number times a unit ; thus, V2 ft., 3 sq. yd., ir cu. ft.
PARTIAL DIFFERENTIATION; EXPLICIT 109
In adding and subtracting, the teraiH must be like quantitiea; length* Added to
lengths, areas to areaji, etc. The fundamental unit is taken aa length. The unita of
area, volume, and angle are derived therefrom. Thua the area of a rectangle or
the volume of a recUngular parallelepiped la
A = aft. X 6 ft. =a6ft.2 = a6 8qft., F = aft. x6ft. x eft. = abeii* = abcca.ft.,
and the units sq. ft., cu. ft. are denoted aa ft.*, ft.* juat aa if the simple unit ft.
ha<l been treated as a literal (juantity and included in the multiplication. An area
or volume is therefore considered as a compound quantity consisting of a number
which gives ita magnitude and a unit which gives its quality or dimensiona. If L
denote length and [L] denote "of the dimensions of length," and if similar nota-
tions be introduced for area and volume, the equations [A] = [L]* and [F] = [L]*
state that the dimensions of area are squares of length, and of volumes, cubes of
lengths. If it be recalled that for purposes of analysis an angle is measured by the
ratio of the arc subtended to the nwlius of the circle, the dimensions of angle are
seen to be nil, as the definition involves the ratio of like magnitudes and must
therefore be a pure number.
When geometric facts are represented analytically, either of two altematiyes ia
open : 1°, the equations may be regarded as existing between mere numbers ; or
2°, as between actual magnitudes. Sometimes one method is preferable, sometimes
the other. Thus the equation x* + j/^ _ ^ of a circle may be interpreted as 1% the
sum of the squares of the coordinates (numbers) is constant ; or 2**, the sum of the
squares on the legs of a right triangle is equal to the square on the hypotenuse
(Pythagorean Theorem). The second interpretation better sets forth the true
inwardness of the equation. Consider in like manner the parabola y^ = 4px. Gen-
erally y and x are regarded as mere numbers, but they may equally be looked
upon as lengths and then the statement is that the square upon the ordinate equals
the rectangle upon the abscissa and the constant length 4p; this may be inter-
preted into an actual construction for the parabola, because a square equivalent
tt) a rectangle may be constructed.
In the last interpretation the constant p was assigned the dimensions of length
so as to render the equation homogeneous in dimensions, with each term of the
dimensions of area or [L]^. It will be recalled, however, that in the definition of
the parabola, the quantity p actually has the dimensions of length, being half the
distance from the fixed p<jint to the fixed line (focus and directrix). This is merely
another corroboration of the initial statement that the equations which actually
arise in considering geometric problems are homogeneous in their dimensions, and
must be so for the reason that in stating the first equation like magnitudes most
be compared with like magnitudes.
The question of dimensions may be carried along through such processes aa
differentiation and integration. For let y have the dimensions [y] and z the dimen-
sions [x]. Then Ay, the difference of two y's, must still have the dimensions [y]
and Ax the dimensions [x]. The quotient Ay/ Ax then has the dimensions [y]/[x].
For example the relations for area and for volume of revolution.
-.. S"-. "•• e]-sj-<« [a-s-™--
and the dimensions of the left-hand side check with those of the right-hand side.
As integration is the limit of a sum, the dimensions of an integral are the product
110 DIFFERENTIAL CALCULUS
of the dimensions of the function to be integrated and of the differential dx
Thus if , ^
J'** dx 1 ^ .X ,
— = - tan-i - + c
0 a2 + x2 a a
were an integral arising in actual practice, the very fact that a^ and x^ are added
would show that they must have the same dimensions. If the dimensions of x
be [L], then
and this checks with the dimensions on the right which are [i]-^, since angle has
no dimensions. As a rule, the theory of dimensions is neglected in pure mathe-
matics ; but it can nevertheless be made exceedingly useful and instructive.
In mechanics the fundamental units are length, mass, and time ; and are denoted
by [L], [3f], [T]. The following table contains some derived units :
velocity ^^ » acceleration -^-^ , force - — i-L_i
J [T] [r]2 [r]2
areal velocity ^^-^ , density - — i , momentum ^ — ^-t— ^ ,
[T] [Lf [T]
angular velocity , moment - — -* *- -* , energy . i — -"- -* .
With the aid of a table like this it is easy to convert magnitudes in one set of
units as ft., lb., sec, to another system, say cm., gm., sec. All that is necessary is
to substitute for each individual unit its value in the new system. Thus
g = 32J -^ , 1 f t. = 30.48 cm., g = 32| x 30.48 -^^ = 980i -^^ .
sec.=* sec. 2 '' sec.^
EXERCISES
1. Obtain the derivatives/^, /^, /;;, /;; and verify /;; =/;;.
(a) sin-i I , ip) log ?i±i^' , (7) Jy) + rp (xy).
*> xy \x/
2. Compute dH/dy^ in polar coordinates by the straightforward method.
3. Show that a2 — = — if r =/(x + at) + 0 (x - at).
4. Show that this equation is unchanged in form by the transformation :
g + 2xy2| + 2(y-y»)|+xV/=0; u = xy, v = l/y.
5. In polar coordinates 2 = r cos ^, x = r sin ^ cos <t>,y-r sin 6 sin 0 in space
The work of transformation may be shortened by substituting successively
z = rjCO80, y = rjSin0, and 2 = rcos^, ri = rsintf.
6. Let X, I/, z, ( be four independent variables and x = r cos^, y = r sin 0, z = z
the equations for transforming x, y, z to cylindrical coordinates. Let
PARTIAL DIFFERENTIATION i EXPLICIT 111
^eE^x IfyH tx* dy* tyti UH
■bow Z = i?^. Jrco80+ l^Bin^ = --?5. Frin^- Oco80 = i?3.
r ^ r H r U
where r- »Q = a//ar. (Of importance for the Hertz oficlllator.) Take y/e^ = 0.
7. Apply the test for an exact differential to each of the following, and write
by inspection the functions corresponding to the exact differentials :
(a) Sawtc + y^dy, (/3) 3xydx + x«d|/, (7) x*ydx + y«dy,
ardjc + ydy zdx - ydy ydx - zdy
(n) (4x» + 3z«y + y*)dx + (x« + 2xi/ + 3ir')dy, (^) xV(dx + dy).
8. Express the conditions that P(x, y^z)dX'\- Q(x, y, «)di/ + iif(x, v, z)ds be
an exact differential dF{x, y, z). Apply these conditions to the differentials :
(a) SxV^dx + 2xVdy + x^yHz, (/3) (y + 2)dx + (x + 2)dj/ + (x + y)dz.
9.
Obtain ( — | = ( — ] and ( — | = ( — ] from (27) with proper variables.
VdrA UvIt \ds)p \dp)s ^ ' *^ *^
10. If three functions (called thermodynamic potentials) be defined as
show d}^ = - SdT- pdv, dx = TdS + vdp, df=-SdT-^ vdp,
and express the conditions that d^, dx, df be exact. Compare with Ex. 9.
11. State in words the definitions corresponding to the defining formulas, p. 107.
12. If the sum (3fdx + Ndy) + (Pdx + Qdy) of two differentials is exact and one
of the differentials is exact, the other is. Prove this.
13. Apply Euler's Formula (31), for the simple case A: = 1, to the three func-
tions (20') and verify the formula. Apply it for A: = 2 to the first function.
14. Verify the homogeneity of these functions and determine their order :
(a) y Vx + X (log X - log y), (/9) _^^ , (^) ^^z
Vx2 + 1/2 ax-\-by-\'Cz
(«)x|/elr» + ^^ (e)V^cot-i^ (D ,^ .^
z Vx + vy
15. State the dimensions of moment of inertia and convert a unit of moment of
inertia in ft.-lb. into its equivalent in cm.-gm.
16. Discuss for dimensions Peirce's formulas Nos. 93, 124-125, 220, 300.
17. Continue Ex. 17, p. 101, to show = — and — mt — H
di^i dqt dtd<ii dqt dqi
18. If Pi = -— in Ex. 17, p. 101, show without analysis that 2 T = y,p, + y,p,.
cm
If T' denote T' = T, where T' is considered as a function of p,, p, while Tis con-
sidered as a function of y^ 7^, prove from T = v,Pi + 9,p, — T that
tpi " cV?i "~ dqi
112 DIFFERENTIAL CALCULUS
19. If (x,, y,) and (x,, y,) are the coordinates of two moving particles and
are the equations of motion, and if Xj, y^ Xg, yj are expressible as
»l =/l(gi, 9«, q's)* V\ = ffli^V ^21 ^a)' ^2 =/2(9l, g'2' 93)» ^2 = 92{Qv ^21 ^s)
in termB of three independent variables q^, q^, ^g, show that
^'- '^^^ '^^^ 'Wi^ 'dq,-dtSq, ag/
where T = ^ (m^vf + m^v}) = T{q^, q^, g,, q^, q^, q^) and is homogeneous of the
second degree in ^p q^, ^3. The work may be carried on as a generalization of
Ex. 17, p. 101, and Ex. 17 above. It may be further extended to any number of
particles whose positions in space depend on a number of variables g.
20. In Ex. 19 if p, = — , generalize Ex. 18 to obtain
dqi
. _sr ar'__ar _dp^ dT
dpi* dqi dqi' ^ dt dq^
The equations Q< = and Of = — H are respectively the Lagran-
^ dtdiii dqi ^ dt dqi . °
gian and Hamiltonian equations of motion.
21. If rr' = k^ and 0' = 0 and r' (r', <p')=v (r, 0), show
^ 1 50^ 1 dH'
dr^ r'dr' r^
I.
22. If rr' = k^, 0' = 0, 0' = 0, and tj'(r', 0', $') = -tj(r, 0, ^), show that the
expression of Ex. 5 in the primed letters is kr^/r'^ of its value for the unprimed
letters. (Usefulin § 198.)
23. If z = z<t>(^^ ^.^plt], show x2?!^ + 2xy-^ + y2— = 0.
X/ \x/ dx^ dxdy dy^
24. Make the indicated changes of variable :
«S-S-(S*S[©-*(I)1--"
au at) atj au
25. For an orthogonal transformation (Ex. 10 (f), p. 100)
^,^,^_^ a^iD a^o
ax" ay2 a22~ax'2"^ay^"^iP2*
54. Taylor's Formula and applications. The development of /(a-, y)
18 found, aa was the Theorem of the Mean, from the relation (p. 95)
PARTIAL DIFFERENTIATION; EXPLICIT 118
A/=*(l)-*(0) if *(t)=:/(a-^th,b-^tk).
If ^(t) be expanded by Maclaurin's Formula to n tenns,
♦(0 - *(0) = ^*'(0) + ^ ♦"(0) + . . . 4- ^^^-^ ♦<-»(0) + J ♦<->(^).
The expressions for ♦'(^) and *'(0) may be found as follows by (10) :
♦'(0 = ^/; + % *'(0) = [/^/; + A-y;],..,
then ♦"(0 = A (A/;; + A/-) -h A: (/*/; + kf^)
= ;iy- + 2 Hf^ + A-y;; = (az), + kD,)%
^%t) = (AZ), + hD^Yf, 4/0(0) = [(/,z)^ + A-/>,)y],.«.
And /(a + A, ^^ + A;) -/(«, b)=:^f= *(1) - *(0) := (AZ), 4- a'/>»)/(«, *)
4- ^ (Ai), 4- Ai),)V(a, i) + . . 4- ^^^^^ (AZ>, 4- kD^Y-^f(a, b)
4- -^ (AD, 4- kD^yf(a 4- dA, ^^ 4- Ok). (32)
In this expansion, the increments A and k may be replaced, if de-
sired, by 5c — « and y — b and then f(x, y) will be expressed in terms
of its value and the values of its derivatives at (a, b) in a manner
entirely analogous to the case of a single variable. In particular if the
point («, h) about which the development takes place be (0, 0) the
development becomes Maclaurin's Formula for /(x, y).
/(', y) =/(0, 0) + (xD, + yD,)/(0, 0) + |j (xD, + yD,yf(0, 0) + . . .
+ (^riyr (^^^ + y^^)" "'/(o- «) + ^ (^^^ + y«,)"/(»-^. «.'/)• (32')
Whether in ^Faolaurin's or Taylor's Formula, the successive terms are
homogeneous pol^'nomials of the 1st, 2d, • • •, (71 — l)st order in j*, y or
in a; — a, y — b. The formulas are unique as in § 32.
Rupjwse Vl — X* — y* is to be developed about (0, 0). The successive deriva-
tivi's are
Vl - x" - y« Vl - x2 - y*
/" = _^ll±J^, ^/^ ^ a;y ^.. ^ ~ 1 4 x«
r" = Hi-y^)3; ^... _ y«-2xy«-y
Mid Vl - x« - y« = 1+ (Ox 4 0 y) 4 H- a;" -f Oxy - y«) 4 i (Ox« 4 • •) 4 • • •,
'■ Vl - x« - y* = 1 - |(x« + y*) + tenns of fourth order 4 • • • •
In this case the expansion may be found by treating x< 4 y^ as a single term and
expanding by the binomial theorem. Tlie result would be
114 DIFFERENTIAL CALCtJLUS
[1 - (X» + y«)]* = l-l{X^ + y^)-l (X* + 2x22/2 + y4) _ ^l^(x2 + y2)« .
That the development thus obtained is identical with the Maclaurin development
that might be had by the method above, follows from the uniqueness of the devel-
opment. Some such short cut is usually available.
55. The condition that a function z =f(x, y) have a minimum or
maximum at (a, h) is that A/> 0 or A/< 0 for all values of h = Ao;
and k — ^y which are sufficiently small. From either geometrical or
analytic considerations it is seen that if the surface z —f{x, y) has a
minimum or maximum at (a, h), the curves in which the planes y = b
and X = a cut the surface have minima or maxima at a; = a and y = b
respectively. Hence the partial derivatives /^ and /^ must both vanish
at (a, b), provided, of course, that exceptions like those mentioned on
page 7 be made. The two simultaneous equations
/; = o, /; = o, (33)
corresponding to f'(^) = 0 in the case of a function of a single varia-
ble, may then be solved to find the positions (x, y) of the minima
and maxima. Frequently the geometric or physical interpretation of
z =f(Xf y) or some special device will then determine whether there
is a maximum or a minimum or neither at each of these points.
For example let it be required to find the maximum rectangular parallelepiped
which has three faces in the coordinate planes and one vertex in the plane
x/a + V/^ + z/c = 1. The volume is
V = xyz = cxyll V
\ a b/
^ = -2-xy-^y2 + c2/ = 0 ?^^_2^xy--x2 + cx = 0.
cz a 0 cy 0 a
The solution of these equations is x = J a, y = ^ 6. The corresponding z is ^c and
the volume Fis therefore a6c/27 or ^ of the volume cut off from the first octant by
the plane. It is evident that this solution is a maximum. There are other solutions
of Fj = Fy = 0 which have been discarded because they give F = 0.
The conditions/; =/; = 0 may be established analytically. For
Now as fj, 1^^ are infinitesimals, the signs of the parentheses are deter-
mined by the signs oif^,/^ unless these derivatives vanish; and hence
unless /; = 0, the sign of A/ for Ax sufficiently small and positive and
Ay = 0 would be opposite to the sign of A/ for Ax sufficiently small and
negative and Ay = 0. Therefore for a mlnlnium or rnaxlmum /J = 0;
and in like manner f'^ = 0. Considerations like these will serve to
•stablish a criterion for distinguishing between maxima and minima
PARTIAL DIFFERENTIATION; EXPLICIT 115
analogous to the criterion furnished by /"(x) in the case of one vari-
able. Forif/;=/; = 0, then
by Taylor's Formula to two terms. Now if the second derivatives are
continuous functions of (a*, y) in the neighborhood of (a, 6), each deriv-
ative at (a -^ $hf b -j- $k) may be written as its value at (a, b) plus an
infinitesimal. Hence
A/ = J (AV^ + 2 hk/;;, + kx,\.. ») + i Q'% + 2 /'*i, + /.-«o.
Now the sign of A/ for sufficiently small values of A, k must be the
same as the sign of the first j)arenthcsis j)rovided that parenthesis does
not vanish. Hence if the quantity
(/ry;;-h2M/;; + /:y;;x,.,)
> 0 for every (A, A;), a minimum
< 0 for every (A, Ar), a maximum.
As the derivatives are taken at the point (a, 6), they have certain constant
vahies, say A^ li^ C. The question of distinguishing between minima and maxima
therefore reduces to the discussion of the possible signs of a quadratic form
AK^ ■\- 2Bhk-{- Ck^ for different vahies of h and A:. The examples
show that a quadratic form may be : either 1°, positive for every (A, k) except (0, 0) ;
or 2*^, negative for every (/i, k) except (0, 0) ; or 3°, positive for some values (A, it)
and negative for others and zero for others ; or finally 4°, zero for values other than
(0, 0), but either never negative or never positive. Moreover, the four possibilities
here mentioned are the only cases conceivable except 6°, that A — B = C = (i and
the form always is 0. In the first case the form is called a definite positive form, in
the second a d^Hnite negative form, in the third an ind(finite form, and in the fourth
and fifth a singular form. The first case assures a minimum, the second a maxi-
mum, the third neither a minimum nor amaxinmm (sometimes called a minimax) ;
but the case of a singular form leaves the question entirely undecided just as the
condition /"(/) = 0 did.
The conditions which distinguish between the different possibilities may be ex-
pressed in terms of the coefficients A^ B^ C.
l°po8. def., Jfl<AC, ^,0 0; 8° indef ., B^>AC\
2°neg. def., li^<AC, ^, C<0; 4° sing., & = AC.
The conditions for distinguishing between maxima and minima are :
^' = ^\f"*^f" f" J^^ ^^^^ minimum ;
r.^Or" '^"" t /^ /;; < O maximum ; ^ '
It may be noted that in applying these conditions to the case of a definite form it
is sufficient to show that either /^^ "^.^y ^* ixwitive or negative because they neces-
sarily have the same sign.
116 DIFFERENTIAL CALCULUS
EXERCISES
1. Write at length, without symbolic shortening, the expansion of /(oj, y) by
Taylor's Formula to and including the terms of the third order in x — a, y — b.
Write the formula also with the terms of the third order as the remainder.
2. Write by analogy the proper form of Taylor's Formula for/(x, y, z) and
prove it. Indicate the result for any number of variables.
3. Obtain the quadratic and lower terms in the development
(a) of xy^ + sin xy at (1, I v) and (/S) of tan-i {y/x) at (1, 1).
4. A rectangular parallelepiped with one vertex at the origin and three' faces
in the coordinate planes has the opposite vertex upon the ellipsoid
x^a^ + yV^ + zVc^ = 1.
Find the maximum volume.
5. Find the point within a triangle such that the sum of the squares of its
distances to the vertices shall be a minimum. Note that the point is the intersec-
tion of the medians. Is it obvious that a minimum and not a maximum is present ?
6. A floating anchorage is to be made with a cylindrical body and equal coni-
cal ends. Find the dimensions that make the surface least for a given volume.
7. A cylindrical tent has a conical roof. Find the best dimensions.
8. Apply the test by second derivatives to the problem in the text and to any
of Exs. 4-7. Discuss for maxima or minima the following functions :
(or) x^y + xy^ ~ x, (/3) x^ -{■ y^ - x^y^ - i {x^ + y\
(e) a;8 + 2/*-9«y + 27, (f) x'^ ■}■ y* - 2x^ + 4xy - 2y^.
9. State the conditions on the first derivatives for a maximum or minimum of
function of three or any number of variables. Prove in the case of three variables.
10. A wall tent with rectangular body and gable roof is to be so constructed as
to use the least amount of tenting for a given volume. Find the dimensions.
11. Given any number of masses m^, nig, • • •, m„ situated at (Xj, y^), {x^, y^, . . .,
(Xn, Vn)- Show that the point about which their moment of inertia is least is their
center of gravity. If the points were (Xj, y^, z^)^ • • • in space, what point would
make Zmr^ a minimum ?
12. A test for maximum or minimum analogous to that of Ex. 27, p. 10, may
be given for a function /(x, y) of two variables, namely : If a function is positive
all over a region and vanishes upon the contour of the region, it must have a max-
imum within the region at the point for which /; =/; = 0. If a function is finite
all over a region and becomes infinite over the contour of the region, it must have
a minimum within the region at the point for which f^ =f^ = 0. These tests are
•ubject to the proviso that/^' =fy = 0 has only a single solution. Comment on the
test and apply It to exercises above.
18. If a, 6, c, r are the sides of a given triangle and the radius of the inscribed
circle, the pyramid of altitude h constructed on the triangle as base will have itjs
maximum surface when the surface is J (a + 6 + c) VH + A^.
CHAPTER V
PARTIAL DIFFERENTIATION ; IMPLICIT FUNCTIONS
56. The simplest case ; F(x, y) = 0. The total differential
dF= F'^dx -h F'^dy = dO = 0
. ,. . dy F' dx F' ,^.
mdicatee 1 = '^/ T^'K ^
as the derivative of y by a*, or of x by y, where y is defined as a function
of a", or a* as a function of y, by the relation F(x, y) = 0 ; and this method
of obtaining a derivative of an ifnplu'U function without solving expli-
citly for the function has probably been familiar long before the notion
of a i)artial derivative was obtained. The relation F(x, y) = 0 is pictured
as a curve, and the function y = <^(a-), which would be obtained by solu-
tion, is considered as nmltiple valued or as restricted to some definite
portion or bi-anch of the curve F(ic, y) = 0. If the results (1) are to
be applied to find the derivative at some point
('o' y©) ^^ ^^^^ curve F(x, y) — 0, it is necessary
that at that ix)int the denominator F'^ or F'^ should
not vanish.
These pictorial and somewhat vague notions
may be stated precisely as a theorem susceptible
of proof, namely : Let x^ be any real value of x
such that 1*, the equation Fix^^ y) = 0 has a real solution y^ ; and 2*, the
function F(Xy y) regarded as a function of two independent variables
(x, y) is continuous and has continuous first partial derivatives F^, F^ in
the neighborhood of (j*^, y^ ; and 3", the derivative F^(x^y ^0)"^ ^ ^^^^
not vanish for (x^^ y^ ; then F{x, y) = 0 may be solved (theoretically)
as y=ifi(x) in the vicinity of x = x^ and in such a manner that
yo~ ^(•''o)» ^^^^ ^W ^^ continuous in a*, and that <i>(x) has a derivative
^'(ar) = — F^/F^ ; and the solution is unique. This is the fundamental
theorem on implicit functions for the simple case, and the proof follows.
By the conditions on F^, F^, the Theorem of the Mean U applicable. Henoe
F(x, y) - F(jro, y^) = F(x, y) = (AF; + tF;)^ + ,*. ^ + ^. («)
Furthermore, in any square |A|<a, \k\<8 surrounding {Zq, Vq) and sufBciently
small, the continuity of F^ insures |F^|<2f and the continuity of F^ taken with
117
-0
118
DIFFERENTIAL CALCULUS
Y
X
/i/4-S
S
1
1
( ,
r i/firv
8
^
.<
- V s
1
i/o 0
0
2a
m
W
X
the fact that F'^{Xq, y^ ^ 0 insures |Fy|>m. Consider the range of x as further
restricted to values such that \x — Xq\< mh/M \t m<M. Now consider the valua
of F(x, y) for any x in the permissible interval
and for y = Vo + « or y = 2/0 — *• ^^ \kF'^\>'mZ
but |(x — Xo)-^x|<'^*i ^^ follows from (2) that
F(x, Vo + «) has the sign of 5F; and F(x, y^ - 3)
has the sign of — JF^ ; and as the sign of F^ does
not change, F(x, y^ + «) and F(x, ^o - ^) ^^ave
opposite signs. Hence by Ex. 10, p. 45, there is
one and only one value of y between y^—^ and
y^ + « such that F(x, y) = 0. Thus for each x in
the interval there is one and only one y such
that F(x, y) = 0. The equation F(x, y) = 0 has a
unique solution near (x^, y^). Let y = <f>{x) denote the solution. The solution is
continuous at x = Xq because | y — y^ | < 5. If (x, y) are restricted to values y = ^ (x)
such that F(x, y) = 0, equation (2) gives at once
Jc^y-yp^^y^ K{^ + BKy + Bk) dy ^ Ki^o^Vo)
h x-Xo Ax F;{x + eh,y-\-ek)' dx F^ix^^y^
As F^, F^ are continuous and F^ 7^ 0, the fraction k/h approaches a limit and the
derivative 0'(Xq) exists and is given by (1). The same reasoning would apply to
any point x in the interval. The theorem is completely proved. It may be added
that the expression for <f>'{x) is such as to show that 4>'{x) itself is continuous.
The values of higher derivatives of implicit functions are obtainable
by successive total differentiation as
F- + 2 F-y' + F-2j" 4- F^y" = 0, (3)
etc. It is noteworthy that these successive equations may be solved for
the derivative of highest order by dividing by F^ which has been assumed
not to vanish. The question of whether the function y = <f,(x) defined
implicitly by F(x, y) = 0 has derivatives of order higher than the first
may be seen by these equations to depend on whether F(Xj y) has
higher partial derivatives which are continuous in (x, y).
57. To find the maxima and minima of y = <f> (x)^ that is, to find the
ix)ints where the tangent to F(x, y) = 0 is parallel to the a:-axis, observe
that at such points y' = 0. Equations (3) give
f; = o,
+ Fy!/"=0.
(4)
Henoe always under the assumption that FJ ^ 0, there are maxima at
the inteneetiona of F= 0 and f;,= 0 if f;; and f; have the same sign,
and mininui at the intersections for which F^ and F^ have opposke signs;
the case F^ = 0 still remains undecided.
PARTIAL DIFFERENTIATION; IMPLICIT 119
For example if F{x, y) = x* + y* — ^axy = 0, the derivatives are
8(x« - ay) + 8 {y* - oi) j^ = 0,
6x - aa/ + 6yy'« + 3(i/2 - ar)y" = 0,
dy _ je* — gy
dz y* — ox
d*y 2a*zy
dx« (y« - axY
To find the maxima or minima of y as a function of z, aolve
F^rrOrrz^-ay, F = 0 = z» + y»-8 oxy, F^ ^ 0.
The fftU solutions of F^ = 0 and F = 0 are (0, 0) and (jy^2a, Via) of which the
first must be discarded because F^' (0, 0) = 0. At (v^2a, Via) the derivative*
F^ and F'Jj^ are positive ; and the point is a maximum. The curve F = 0 i« the
folium of Descartes.
The r61e of the variables x and y may be interchanged if Fj =^ 0 and
the equation F(xy y) = 0 may be solved for x = ^(y), the functions ^
and i/^ being inverse. In this way the vertical tangents to the curve
F = 0 may be discussed. For the points of F = 0 at which both F^ = 0
and Fy = 0, the equation cannot be solved in the sense here defined.
Such points are called singular ^joints of the curve. The questions of
the singular points of F = 0 and of maxima, minima, or minimax (§ 55)
of the surface z = F(xj y) are related. For if F^ = F^ = 0, the surface
has a tangent i>lane parallel to « = 0, and if the condition 2; = F = 0 is
also satisfied, the surface is tangent to the a-y-plane. Now if « = F(ar, y)
has a maximum or minimum at its point of tangency with « = 0, the
surfa(H^ lies entirely on one side of the plane and the point of tangency
is an isolated point of F{Xj y) = 0 ; whereas if the surface has a mini-
max it cuts through the plane z = 0 and the point of tangency is not
an isolated point of F(j*, y) = 0. The shape of the curve F = 0 in the
neighborhood of a singular point is discussed by developing F(x, y)
about that point by Taylor's Formula.
For example, consider the curve F(jr, y) = z* + y* — x*y* — J (z* + y*) = 0 and
the surface z = F{Xj y). The conunon real solutions of
F; = 8za-2zy«-z = 0, F; = Sy" - 2z2y - y = 0, F(x, y) = 0
are the sinprular pojntj?. The real solutions of F^ = 0, F^ = 0 are (0, 0), (1, 1),
(J, \) and of these the first two satisfy F(z, y) = 0 but the last does not. The
singidar points of the curve are therefore (0, 0) and (1, 1). The test (34) of § 55
shows that (0, 0) is a maximum for z = F(x, y) and hence an isolated point of
F{x^ y) = 0. The test also shows that (1, 1) is a minimax. To diacaas the corre
F{x, y) = 0 near (1, 1) apply Taylor's Formula.
0 = F{x, y) = i (3 A« ~ 8 AJt + 8 ilr2) + J (6 A« - 12 h^k - 12 AA* + fl A-*) + remainder
= i (8 cos^* 0 — 8 sin 0 cos 0 + 8 sin* 0)
+ r (cos* 0 — 2 cos* 0 sin 0—2 cos ,f> si n- ip + sin' 0) + • • • ,
120 DIFFERENTIAL CALCULUS
If polar coordinates A = r cos^, Jk = r sin 0 be introduced at (1, 1) and r^ be can-
celed. Now for very small values of r, the equation can be satisfied only when
the first parenthesis is very small. Hence the solutions of
8 - 4 sin 2 0 = 0, sin 2 0 = }, or 0 = 24° liy, 65° 42J',
and 0 + X, are the directions of the tangents to F{x, y) = 0. The equation F=Oia
0 = (1^ - 2 sin 2 0) + r (cos <f> + sin 0) (1 - IJ sin 2 0)
if only the first two terms are kept, and this will serve to sketch F{x, y) = 0 for
very lanAll "value* of r, that is, for 0 very near to the tangent directions.
58* It is important to obtain conditions for the maximum or minimum
of a function z =f(x, y) where the variables x, y are connected by a
relatioui F(x, ?/) = 0 so that z really becomes a function of x alone or y
alone. For it is not always possible, and frequently it is inconvenient,
to solve F(xy y) = 0 for either variable and thus eliminate that variable
from z = f(x^ y) by substitution. When the variables x, y\\\z= f(x, y)
are thus connected, the minimum or maximum is called a constrained
minimum or maximum ; when there is no equation F(x, y) = 0 between
them the minimum or maximum is called free if any designation is
needed.* The conditions are obtained by differentiating z =f{x, y)
and Fix J y) = 0 totally with respect to x. Thus
^ = ^4.^^ = 0 ^ = ^ + ^^ = 0
dx ex dy dx ' dx dx dy dx '
where the first equation arises from the two above by eliminating dy/dx
and the second is added to insure a minimum or maximum, are the con-
ditions desired. Note that all singular points of F(x, y) = 0 satisfy the
first condition identically, but that the process by means of which it
was obtained excludes such points, and that the rule cannot be expected
to apply to them.
Another method of treating the problem of constrained maxima and
minima is to introduce a multiplier and form the function
% = ^(x, y)=f(Xyy)-\-kF{x,y\ X a multiplier. (6)
Now if this function z is to have a free maximum or minimum, then
*; =/; + af; = 0, *; =f^ + xf; = o. (7)
These two equations taken with F = 0 constitute a set of three from
which the three values a, y, X may be obtained by solution. Note that
•The adjective "relative" is sometimes used for constrained, and "absolute" for
free; but the term "absolute" is best kept for the greatest of the maxima or least of
the minima, and the term " relative " for the other maxima and minima.
PARTIAL DIFFERENTIATION; IMPLICIT 121
X cannot be obtained from (7) if both F^ and F^ vanish ; and hence this
method also rejects the singular points. That this method really deter-
mines the constrained maxima and minima of /(j*, y) subject to the
constraint F(x, y) = 0 is seen from the fact that if X \)e eliminated from
(7) the condition/;/'^ —f^F^ = 0 of (5) is obtained. The new method
is therefore identical with the former, and its introduction is more a
matter of convenience than necessity. It is possible to show directly
that the new method gives the constrained maxima and minima. For
the conditions (7) are those of a free extreme for the function ♦(jr, y)
which depends on two indej)endent variables (jt, y). Now if the equa-
tions (7) be solved for (a*, y), it appears tliat the position of the maximum
or minimum will be expressed in terms of X as a parameter and that
consequently the point (^(X), y(X)) cannot in general lie on the curve
F(Xy y) = 0 ; but if X be so determined that the jx)int shall lie on this
curve, the function ^(r, //) has a free extreme at a point for which
F=0 and hence in particular must have a constrained extreme for the
particular values for which F(Xj y) = 0. In speaking of (7) as the con-
ditions for an extreme, the conditions which should be imposed on
the second derivative have been disregarded.
For example, suppose the maximum radius vector from the origin to the folium
of Descartes were desired. The problem is to render/(x, y) = z* + y^ maximum
subject to the condition F(x, y) = x^ + 2/* — 3 dxy = 0. Hence
2x + 3\(z2- a2/) = 0, 2y + 3X(y2_aj:) = o, x» + y»-3axy = 0
or 2x-3(y2_ ax)-2y-3(x*-ay) = 0, x* + y« - 3axi/ = 0
are the conditions in the two cases. These equations may be solved for (0, 0),
(IJ a, 1^ a), and some imaginary values. The value (0, 0) is singular and X cannot
be determined, but the point is evidently a minimum of x* + y* by inspection. The
point (1 J a, 1 J a) gives X = — IJ a. That the point is a (relative constrained) maxi-
mum of x^ 4- y'^ is also seen by inspection. There is no need to examine cP/. Ini
most practical problems the examination of the conditions of the second orderj
may be waived. This example is one which may be treated in polar coordinates
by the ordinary methods ; but it is noteworthy that if it could not be treated that
way, the method of solution by eliminating one of the variables by solving the
cubic F{x, y) = 0 would be unavailable and the methods of constrained maxima
would be required.
EXERCISES
1. By total differentiation and division obuiln dy/dx in the« caaes. Do not
substitute in (1), but use the method by which it was derived.
(a) ax2 + 26ary + cy«- 1 =0, (/3) x* + y* = 4 oa«y, (7) (cos*)*- (alny)* = 0,
(«) (J* + y^y^ = a«(x« - y«), (,) e' -I- e» = 2xy, (f) x-*y-* = tan-»xy.
2. Obtain the second derivative d^/dx* in Ex. 1 (a), OS), («), (i") by differen-
tiating the value of dy/dx obtained above. Compare with um of (8).
122 DIFFERENTIAL CALCULUS
4. Find the radius of curvature of these curves :
(a) x? + y 5 = riS, R = 8 (axy)i, (/3) x^ + y' = ai, R = 2 V(x + y)ya,
(7) 6=^x2 + c'i/- = a262, (3) ary2 = a2 (a - x), (c) (ax)2 + (6y)f = 1.
5. Find /, y", y"' in case x^ -\- y^ - S axy = 0.
6. Extend equations (3) to obtain y'" and reduce by Ex. 3.
7. Find tangents parallel to the x-axis for {x^ + y-)^ = 2 a^ (x^ - y^).
8. Find tangents parallel to the y-axis for (x^ + y^ + ax)^ = a^ (x^ + y2).
9. If 62 < ac in ax2 + 2 &xy + cy2 4-/x + 9'y + ^ = 0, circumscribe about the
curve a rectangle parallel to the axes. Check algebraically.
10. Sketch x» + y' = x2y2 + ^ (x2 + y2) near the singular point (1, 1).
11. Find the Angular points and discuss the curves near them :
(a) x3 + y8 = 3 axy, (/3) (x2 + y2)2 = 2 a2 (x^ _ yi)^
(7) x* + y* = 2(x - y)2, (5) y^ + 2xy2 = x2 + y^
12.*Make these functions maxima or minima subject to the given conditions.
Discuss the work both with and without a multiplier :
a 6 , . sinx u
la) 1 . a tan x + o tan y = c. An». - — = - .
ucosx ccosy siny v
iP) x'^ + y2, ax2 + 26xy + cy2 =/. Find axes of conic.
(7) Find the shortest distance from a point to a line (in a plane).
13. Write the second and third total differentials of F(x, y) = 0 and compare
with (3) and Ex. 6. Try this method of calculating in Ex. 2.
14. Show that F^dx + F^dy = 0 does and should give the tangent line to
F(Xy y) = 0 at the points (x, y) if dx = { — x and dy = ri — y^ where ^, 17 are the
coordinates of points other than (x, y) on the tangent line. Why is the equation
inapplicable at singular points of the curve ?
59. More general cases of implicit functions. The problem of
im])licit functions may be generalized in two ways. In the first place
a greater number of variables may occur in the function, as
F{x, y, z) = 0, F(x, y, «,..., w) = 0 ;
and the question may be to solve the equation for one of the variables
in terms of the others and to determine the partial derivatives of the
chosen dependent variable. In the second place there may be several
equations connecting the variables and it may be required to solve the
equations for some of the variables in terms of the others and to
determine the partial derivatives of the chosen dependent variables
PARTIAL DIFFERENTIATION; IMPLICIT 128
with respect to the independent variables. In both cases the formal
differentiation and attempted formal solution of the equations for the
derivatives will indicate the results and the theorem under which the
solution is pioj>er.
Consider the case JF*(a?, y, «) = 0 and form the differential.
dF(x, y, X) = F^dx -h F'^dy -f F'^dz = 0. (8)
If « is to be the dependent variable, the partial derivative of x by a; is
found by setting rfy = 0 so that y is constant. Thus
dx
dx
■©r-g - |-(l).-g <"
are obtained by ordinary division after setting dy — d and dlx = 0 re-
sj)ecrively. If this division is to be legitimate, F'^ must not vanish at
the point considered. The immediate suggestion is the theorem : If,
when real vahu'S (x^, y^ are chosen and a real value z^ is obtained
from F(«, x^, y^) = 0 by solution, the function F(ar, y, z) regarded as
a function of three indcjMMident variables (j-, y, z) is continuous at
and near (x^y y^, z^ and has continuous first partial derivatives and
Kip^Qi y^i ^o)"^^* ^^^^ ^(^y y» «) = 0 may be solved uniquely for
z = if}(xy y) and <^ (x, y) will be continuous and have partial derivatives
(9) for values of (x, y) sufficiently near to (x^f y^.
The theorem is again proved by the Law of the Mean, and in a similar manner.
F(x, V, z) - F{x^, 2/o, 2o) = F(x, y, z) = QiF'^ + kF'^ + ii^^x, + »*. r. + •*,«. + #1.
As F^, F^, F,' are continuous and Fj(Xo, y^, z^ ^ 0, it is possible to take a so
small that, when \h\<i,\k\<i,\l\<i, the derivative \F'^\>m and | F^ | < m, | F^ | < M.
Now it is desired so to restrict A, k that ± SF^ shall determine the sign of the
parenthesis. Let
|x-XoI<imVM, ly-l/oKi*^//** **^en \hF;^-\- kF^\<mS
and the signs of the parenthesis for (j, y, Zq + i) and (x, y, Zq — 8) will be opposite
since I F,' I > m. Hence if (x, y) be held fixed, there is one and only one value of t
for which the parentliesis vanishes l)etween Zo + * *"^ ^o ~ *• Thus z is defined as a
single valued function of (x, y) for sufficiently small values of A = x — x<„ fc = y — y^.
Also
1 function of (x, y) for sufficiently small values of A = x —
I ^ Ki^o + »h,y^-\-$k,zo-¥et) i^ f;(...)
h F^{x^-\-Bh,y^^Ok,z,-\-ei)' k f;(...)
h respectively are assigned the values 0. The limits exist
when k and A respectively are assigned the values 0. The limits exist when A = 0 or
k ± 0. Hut in the first case Z = Az = A^^ is the increment of z when x alone varies,
and in the second case I = Az =ApZ. The limits are therefore the desired partial
derivatives of z by x and y. The proof for any number of variables would be
similar.
124 DIFFERENTIAL CALClTLUS
If none of the derivatives F^,F^, F^ vanish, the equation F(x, y, «) = 0
may be solved for any one of the variables, and formulas like (9) will
express the partial derivatives. It then appears that
\dx)Xdz)^ dxdz f:f', ■"' ^^^>
/dz\ /dx\ (dy\ _^^_xdy__
\dx)Xdy)\dz)- dx dydz- "■ ^^^^
and
in like manner. The first equation is in this case identical with (4)
of § 2 because if y is constant the relation F(cc, y, z) = 0 reduces to
G (Xy z) = 0. The second equation is new. By virtue of (10) and simi-
lar relations, the derivatives in (11) may be inverted and transformed
to the right side of the equation. As it is assumed in thermodynamics
that the pressure^ volume, and temperature of a given simple substance
are connected by an equation F(pj v, T) = 0, called the characteristic
equation of the substance, a relation between different thermodynamic
magnitudes is furnished by (11).
60. In the next place suppose there are two equations
F(x, y, u, v) = 0, G{x, y, u, v) = 0 (12)
between four variables. Let each equation be differentiated.
c^F = 0 = F'Jx + F'^dy + F'Ju + F'^dv,
dG = 0= G'^dx H- G'^dy + G'J.u + G'„dv. (13)
If it be desired to consider u^ v as the dependent variables and x, y sis
independent, it would be natural to solve these equations for the differ-
entials du and dv in terms of dx and dy ; for example,
^^^ (f^g: - f:g:^) dx + (f:g:, - f:g:.) dy
f:g:,-f:g: ^^''^^
The differential dv would have a different numerator but the same de-
nominator. The solution requires F'^ GJ — F'^ G^ 4^ 0. This suggests the
desired theorem : If {u^^ v^ are solutions of F = 0, 6^ = 0 corresponding
*o (*o' y©) ^"^^ if K^v — F^g;, does not vanish for the values (a:^, y^, u^, v^,
the equations F = 0, C? = 0 may be solved for w = <^(x, y), v = i^(a!, y)
and the solution is unique and valid for (ar, y) sufficiently near (ar^, y^
— it being assumed that Fand G regarded as functions in four variables
are continuous and have continuous first partial derivatives at and near
(*o» yo» **o» ^d » moreover, the total differentials du^ dv ai-e given by (13')
and a similar equation.
PARTIAL DIFFERENTIATION; IMPLICIT 126
The proof of this theorem may be deferred (f 64). Some observations
should be made. The equations (13) may be solved for any two vari-
ables in terms of the other two. The partial derivatives
du(x^^ du{x,v)^ dx(u,v)^ dx{u, y)
dx dx du du ^ '
of u by a* or of a- by u will naturally depend on whether the solution
for u is in terms of (ar, y) or of (a-, r), and the solution for x is in (m, v)
or (m, y). Moreover, it must not be assumed that du/dx and dx/du are
reciprocals no matter which meaning is attached to each. In obtaining
relations between the derivatives analogous to (10), (11), the values of
the derivatives in terms of the derivatives of F and G may be found or
the equations (12) may first be considered as solved.
Thus if u = 0 (x, y), du - <f>^dx + 0^dy,
tj = ^ (X, y), dv = yp'^dx + ^y'dy.
yff^du — ifi'dv , — rj/'du + <t>jdv
Then dx = ^—, — ^,, dy = —^^ fV
^x^y - *y^x ^x^» - *y^x
dx f « dx - 0y
and — = , , ' — — - , — = — -- — =—— , etc.
XT du dx , dv dx , ., _,
Hence 1 = 1 , (16)
dx du dx dv
as may be seen by direct substitution. Here u, v are expressed in terms of x, y for
the derivatives u^, v^ ; and x, y are considered as expressed in terms of u, c for the
derivatives x^, x^.
61. The questions of free or constrained maxima and minima, at any
rate in so far as the determination of the conditions of the first order is
concerned, may now be treated. If F(x, y, z) = 0 is given and the max-
ima and minima of « as a function of (x, y) are wanted,
K (^, y, ^) = 0, f; (a-, y, z) = 0, F(x, y,z) = 0 (16)
are three equations which may be solved for a*, y, z. If for any of these
solutions the derivative F^ does not vanish, the surface « = ^ (x, y) has
at that point a tangent plane parallel to z = 0 and there is a maximum,
minimum, or minimax. To distinguish between the possibilities further
investigation must be made if necessary ; the details of such an investi-
gation will not be outlined for the reason that special methods are
usually available. The conditions for an extreme of u as a function of
(a*, y) defined implicitly by the equations (13') are seen to be
FX-n'G?;=o, f;(?; - f;<?; = 0, f=o, g = o. (it)
The four equations may be solved for x, y, m, v or merely for x, y.
126 DIFFERENTIAL CALCULUS
Suppose that the maxima, minima, and minimax of u =f(x, y, z) sub-
ject either to one equation F{x, y,z) = 0 or two equations F(x, y, z) = 0,
G (xj y, «) = 0 of constraint are desired. Note that if only one equation
of constraint is imposed, the function u = f(x, y, z) becomes a function
of two variables ; whereas if two equations are imposed, the function u
really contains only one variable and the question of a minimax does
not arise. The method of multipliers is again employed. Consider
*(a^,y,«)=/+^^ or ^=f-h\F-\-fiG (18)
as the case may be. The conditions for a free extreme of $ are
^; = 0, ^; = 0, ^: = 0. (19)
These three equations may be solved for the coordinates x, y, z which
will then be expressed as functions of X or of X and /x according to the
case. If then X or X and fi be determined so that (x, y, z) satisfy F = 0
or F = 0 and G = 0, the constrained extremes of u =f(x, y, z) will be
found except for the examination of the conditions of higher order.
As a problem in constrained maxima and minima let the axes of the section of
an ellipsoid by a plane through the origin be determined. Form the function
* = x2 + y2 + 2;2 + x/^ + ^ + ?! _ l\ + ^(te + mi/ + nz)
by adding to x^ + y2 ^ j>i^ which is to be made extreme, the equations of the ellipsoid
and plane, which are the equations of constraint. Then apply (19). Hence
taken with the equations of ellipsoid and plane will determine a;, y, 2, X, fi. If the
equations are multiplied by x, y, z and reduced by the equations of plane and
ellipsoid, the solution for X is X =— r^ =— (x^ + y2 ^ ^2). The three equations
then become
1 iikfl 1 fi.mJtP' 1 nrx'^
®® ^TT^'^^^TZr^'^^^^Zr^""^ determines r2. (20)
The two roots for r are the major and minor axes of the ellipse in which the plane
cute the ellipsoid. The substitution of «, y, z above in the ellipsoid determines
Now when (20) is solved for any particular root r and the value of fi is found by
(21), the actual coordinates x, y, z of the extremities of the axes may be found.
PARTIAL DIFFERENTIATION; IMPLICIT 127
BZBRCI8£8
1. Obtain the partial derivatives of x by z and y directly from (8) and not by
substitution in (9). Where does the solution fail ?
2> V* t* 1
«')^= + ^ + c^ = »' Wx + v + « = _.
(7) (Jc* + y* + «*)« = a«xa + 6V + cU\ («) ^z = c.
2. Find the second derivatives in Ex. 1 (a), (/3), (3) by repeated differentiation.
3. State and prove the theorem on the solution of F(x, y, «, u) = 0.
4. Show that the product a,,Er of the coefficient of expansion by the modulus
of ehuiticity (§ 52) is equal to the rate of rise of pressure with the temperature if
tlu' volume is constant.
5. Establish the proportion Es:ET=Cp: C„ (see % 62).
o T* T,. V /v u dudxdydz ^ dudx ,
6. Iiy(,.y,^„) = o.8how---- = 1. --=1.
7. Write the equations of tangent plane and normal line to F{x, y, z) = 0 and
find the tangent planes and normal lines to Ex. 1 (/3), (8) at a; = 1, y = 1.
8. Find, by using (13), the indicated derivatives on the assumption that either
X, y or u, V are dependent and the other pair independent :
(a) u6 + r« + x« - 32/ = 0, m' + tr' + 2/3 + 3x = 0, u^, u;, m^, v^
(/9) X + 2/ + u + c = a, x2 + 2/2 + u2 + c* = 6, xj, <, r^, r^^
(7) Find d2/ »" ^^^^ cases if x, v are independent variables.
9. Prove — ^ + —^ = 0 if F(x, 2/, u, t?) = 0, G(x, y, u, c) = 0.
ex du ex er
10. Find du and the derivatives u^, m^, u^' in case
z^ + 2/2 + z* = WW, Z2/ = M^ + v^ + w2^ 3.^2; = uvw.
11. If F(x, y, «) = 0, (?(x, y, 2) = 0 define a curve, show that
x-rXf, _ y-yp _ z-Zq
{f'^g:^f:gx (f;g;-f;(?Oo {Kg',-f'^g:x
is the tangent line to the curve at (Xg, 2/0 » Zq)- Write the normal plane.
12. Formulate the problem of implicit functions occurring in Ex. 11.
13. Find the perpendicular distance from a point to a plane.
14. The sum of three positive numbers is x + y + z = jV, where N is given.
Determine x,y,zm that the product x'^z'" shall be maximum if p, q, r are given.
Ans. x:y:z:N = p:q:r:{p-\-q'¥r).
15. The sum of three positive numbers and the sum of their squares are both
-iven. Make the product a maximum or minimum.
16. The surface (x« + y^+z*)« = ax«+f>y3+c2« is cut by the plane ix + my+n2=0.
Xp
— = 0.
r« — a
128 DIFFERENTIAL CALCULUS
17. In case F(x, y, u, c) = 0, G (x, y, u, r) = 0 consider the differentials
dt = — dx + — dy, dx = — du + — dtj, dy = -^du-\- —dv.
dz dy du dv du dv
Sabstitute in the first from the last two and obtain relations like (16) and Ex. 9.
18. If /(x, y, z) is to be maximum or minimum subject to the constraint
F(x, Vy z) = 0, show that the conditions are that dx :dy:dz = 0 :0:0 are indeter-
minate when their solution is attempted from
f^dx-^f^dy-\-f^dz = 0 and F;^dx + F^dy + F^dz = 0.
From what geometrical considerations should this be obvious ? Discuss in connec-
tion with the problem of inscribing the maximum rectangular parallelepiped in
the ellipsoid. These equations,
dx:dy:dz ^f^ - fzK '-fzK -fLK --KK 'KK = 0:0:0,
may sometimes be used to advantage for such problems.
19. Given the curve F(x, y, z) = 0, (?(x, y, z) = 0. Discuss the conditions for
the highest or lowest points, or more generally the points where the tangent is
parallel to z = 0, by treating u =/(x, y, z) = z as a maximum or minimum sub-
ject to the two constraining equations F = 0, G = .0. Show that the condition
F^Gy = F'^G'^ which is thus obtained is equivalent to setting dz = 0 in
F^dx + Fydy -\- F'^dz = 0 and G^dx -h G'^dy + G^dz = 0.
20. Find the highest and lowest points of these curves :
(a) x2 4- y2 = 2;2 + 1^ X + y + 2z = 0, (p) ±4-^- + -=1, Ix -\- my + nz = 0.
a^ tr c^
21. Show that F'^dx + F^dy -h F'^dz = 0, with dx = ^ - x, dy = -n - y, dz = ^ - z,
is the tangent plane to the surface F(x, y, z) = 0 at (x, y, z). Apply to Ex. 1.
22. Given F(x, y, u, v) = 0, Cr(x, y, u, v) = 0. Obtain the equations
?^ j_ ^ ?!f _L ?Z ^^ - 0 ^^ ^^^^ ^^ ^" _ 0
dx du dx dv dx~ ' dy du dy dv dy~ '
dx du dx dv dx~ '' dy du dy dv dy~ ^
and explain their significance as a sort of partial-total differentiation of F = 0
and G = 0. Find u^ from them and compare with (13'). Write similar equations
where x, y are considered as functions of (u, v). Hence prove, and compare with
(16) and Ex. 9,
Suay aB8y_ ^^ . ^??_o
dydu dy aw ~ ' dy du dy dv ~
23. Show that the differentiation with respect to x and y of the four equations
under Ex. 22 leads to eight equations from which the eight derivatives
dhi ^ d^ dhi dH dH
ftt«* bxty' dydx' dy'^' ^' "*' ^
vaaj be obtained. Show thus that formally u^' = u".
PARTIAL DIFFERENTIATION; IMPLICIT 129
62. Functional determinants or Jacobians. Let two functions
« = *(a;,y), t; = ^(x,y) (22)
of two independent variables be given. The continuity of the functions
and of their first derivatives is assumed throughout this discussfon
and will not be mentioned again. Suppose that there were a relation
F(Uf v) = 0 or F(^f ^) = 0 between the functions. Then
The last two equations arise on differentiating the first with respect to
jc and y. The elimination of F^ and F^ from these gives
4>: r.
i»^w - <ki^x =
<^; ^;
^(^, y) Vyy/ ^ ^
The determinant is merely another way of writing the first expression ;
the next form is the customary short way of writing the determinant
and denotes that the elements of the determinant are the first deriva-
tives of u and v with respect to x and y. This determinant is called the
functional determinant or Jacobian of the functions w, v or <f>, tff with
resi)ect to the variables x, y and is denoted by J. It is seen that : If
there is a functional reUition F(<f>, ^) = 0 hetiveen two functionSj the
Jacobian of the functions vanishes identically , that is, vanishes for all
values of the variables (a*, y) under considei-ation.
Conversely, if the Jacobian vanishes identically over a two-dimensional
region for (x, y), the functions are connected by a functional relcUion.
For, the functions u, v may be assumed not to reduce to mere constants
and hence there may be assumed to be points for which at least one of
the partial derivatives 4>^j <^y, tf/^, ^^ does not vanish. Let <f>^ be the
derivative which does not vanish at some particular point of the region.
Then n = <^(j', y) may be solved as ar = x(w, y) in the vicinity of that
point and the result may be substituted in v.
by (11) and substitution. Thus ^r/^y = J/<t»xy ^^^ i^ /=0, then
dv/dy = 0. This relation holds at least throughout the region for which
^^ =^ 0, and for points in this region cv/dy vanishes identically. Hence
V does not depend on y but becomes a function of u alone. This es-
tablishes the fact that v and u are functionally connected.
130 DIFFERENTIAL CALCULUS
These considerations may be extended to other cases. Let
u = it>(x, y, z), V = tlf{x, y, z), w = x{^, V, «)•
If there is a functional relation F(uy v, w) = 0, differentiate it.
or
d{x,y,z) d{x,y,z)
<^x
r.
Xx
^y
^y
xl,
<!>:
^:
X.
0.
(25)
(26)
The result is obtained by eliminating FJ, F'^, F^ from the three equations.
The assumption is made, here as above, that F^, FJ, F^ do not all vanish ;
for if they did, the three equations would not imply J = 0. On the
other hand their vanishing would imply that F did not contain w, v, w,
— as it must if there is really a relation between them. And now con-
versely it may be shown that if / vanishes identically, there is a func-
tional relation between Uj v, w. Hence again the necessary and sufficient
conditions that the three functions (25) he functionally connected is that
their Jaxiohian vanish.
The proof of the converse part is about as before. It may be assumed that at
least one of the derivatives of w, v, w or 0, ^, x by x, y, z does not vanish. Let
0^ 5ii 0 be that derivative. Then m = 0 (x, y, z) may be solved as x = w (u, y, z)
and the result may be substituted in v and w as
v = f(x, y, 2;) = ^(w, y, z), w = x(i», y, 2) = xK y, 2).
Next the Jacobian of v and w relative to y and z may be written as
dv dw
dy dy
dz ~dz
if ,1^; Xy
-*t>y/<t>x Xy
- '^zhx Xz
Xy K
Xz *t>z
^^'a
+ Xx
+ X.
Vy -<t>;K\
Vz -'t'z/M
<Py "f^y]]^!,
A« J vanishes identically, the Jacobian of v and w expressed as functions of y, z,
al»o vanishes. Hence by the case previously discussed there is a functional rela-
tion F(t>, 10) = 0 independent of y, z ; and as t>, u? now contain u, this relation may
be considered as a functional relation between u, r, w.
63. If in (22) the variables ?/, v be assigned constant values, the
equations define two curves, and if u, v be assigned a series of such
values, the equations (22) define a network of curves in some part of the
PARTIAL DIFFERENTIATION; IMPLICIT
181
ary-plane. If there is a functional relation u = F(v), that is, if tlie
tJatMjbian vanishes identically, a constant value of v implies a constant
value of u and hence the locus for which v is constant is also a locus
for wh'u'.h u is constant ; the set of v-curves coincides with the set of
M-curves and no true network is formed. This
(rase is uninteresting. Let it l3e assumed that
the Jacohian does not vanish identically and
even tliat it does not vanisli for any point (x^ y)
of a certain region of the ;r//-plane. The indi-
cations of § 60 are that the equations (22) may
then 1x3 solved for a*, y in terms of i/, v at any
IK)int of the region and that there is a pair of
the curves through each point. It is then proper to consider (m, v) as
the coordinates of the points in the region. To any point there corre-
spond not only the rectangular coordinates (x, y) but also the curvi-
linear coordinates (w, v).
The equations connecting the rectangular and curvilinear coordinates
may be taken in either of the two forms
u = ^ (jr, y), V = ^ (a-, y) or X = /(w, v), y = g(Uy r), (22')
each of which are the solutions of the other. The Jacobians
\x, yj \m, v)
(27)
are reciprocal each to each ; and this rela-
tion may be regarded as the analogy of
the relation (4) of § 2 for the case of
the function y = tf>(jr) and the solution
X =f{y) = ^~*(y) in the case of a single
variable. The differential of arc is
(u, v+dv)
(x+dx. i/+dy)
(u-i-du, v-fdv)
v+dv
{x-t-duX.V-i-dMV)
u-fdu
(u+du, V)
ds* = dj-^ -\-dy^= Edu* -f 2 Fdudv + Gdt^y
X
(28)
=(s)"-©"
dxdx
dudv
dy dy
du dv
7^. G
'(iHi)'
The differential of area included between two neighboring u-curves and
two neighboring v-curves may be written in the form
dA = j(^^^\ dudv = dudv -4- j('^^\
These statements will now be proved in detail.
(29)
132
DIFFERENTIAL CALCULUS
To prove (27) write out the Jacobians at length and reduce the result.
tiiH^y-
du tv
dx dx
dx dy
du du
du dv
dy dy
dx dy
dv dv
dudx dvdx dudy dvdy
dxdu dx dv dxdu dx dv
1 0
dudx
dydu
dv dx dudy dv dy
dydv dydu dy dv
0 1
= 1,
where the rule for multiplying determinants has been applied and the reduction
has been made by (15), Ex. 9 above, and similar formulas. If the rule for multi-
plying determinants is unfamiliar, the Jacobians may be written and multiplied
without that notation and the reduction may be made by the same formulas as
before.
To establish the formula for the differential of arc it is only necessary to write
the total differentials of dx and dy, to square and add, and then collect. To obtain
the differential area between four adjacent curves consider the triangle determined
by (u, r), (u -\- du, c), (u, c + dw), which is half that area, and double the result.
The determinantal form of the area of a triangle is the best to use.
dA = 2'
d,^ d^
dvX dvy
dx , dy ,
— du — du
du du
dx , dy ^
— dv — dv
dv dv
dx dy
du du
dx dy
dv dv
dudv.
The subscripts on the differentials indicate which variable changes ; thus duX, d^y
are the coSrdinates of (u + du, v) relative to (m, v). This method is easily extended
to determine the analogous quantities in three dimensions or more. It may be
noticed that the triangle does not look as if it were half the area (except for infin-
itesimals of higher order) in the figure ; but see Ex. 12 below.
It should be remarked that as the differential of area dA is usually
considered positive when du and dv are positive, it is usually better to
replace J in (29) by its absolute value. Instead of regarding (^^, v) as
curvilinear coordinates in the ay-plane, it is possible to plot them in
their own wr-plane and thus to establish by (22') a transfoTmatlon of
the xy-plane over onto the wv-plane. A small area in the cc^z-plane then
becomes a small area in the wv-plane. If J > 0, the transformation is
called direct ; but if / < 0, the transformation is called perverted. The
significance of the distinction can be made clear only when the ques-
tion of the signs of areas has been treated. The transformation is called
conformal when elements of arc in the neighborhood of a point in the
xy-plane are proportional to the elements of arc in the neighborhood of
the corresponding point in the wv-plane, that is, when
d»« = rfx« + dy« = A; (du" -f- dv^ = kda-\ (30)
PARTIAL DIFFERENTIATION; IMPLICIT 138
For in this case any little triangle will be transformed into a little tri-
angle similar to it, and hence angles will be unchanged by the transfor-
mation. That the transformation be conformal requires that F = 0 and
E = G. It is not necessary that E = G — k be constants ; the ratio of
similitude may Ixi different for different points.
64. There remains outstanding the proof that equations may be solved
in the neighlwrliocKl of a point at which the Jacobian does not vanish.
The fact was indicated in § 60 and used in § 63.
Theorem. Let p equations in n -\- j) variables be g^ven, say,
F.i^v ^v • • ', ^.+p) = 0, F, = 0, . . ., F, = 0. (31)
Let the p functions be soluble for x^^ a-.^, • •, x^ when a particular set
^(p+i)o» '"> ^(••+p)o °^ ^^^^ other n variables are given. Let the functions
and their first derivatives be continuous in all the n -^ p variables in the
neighborhood of (a-^^, x^j • • •, af(,+px,)- Let the Jacobian of the functions
with respect to a^j, x^, • • •, x^,
dF^ dFp
dx, dx.
dF^ dj^
ox^ dx.
^ 0, (32)
» •'^(•t+P)*
fail to vanish for the particular set mentioned. Then the p equations
may be solved for the p variables a?j, a-^, • • •, x^^ and the solutions will be
continuous, unique, and differentiable with continuous first partial
derivatives for all values of x^+i, •••, ic^+p sufficiently near to the
values x^^^,^, •.•, x^,^^^.
Theorem. The necessary and sufficient condition that a functional
relation exist between p functions of p variables is that the Jacobian
of the functions with respect to the variables shall vanish identically,
that is, for all values of the variables.
The proofs of these theorems will naturally be given by mathematical indaction.
Each of the theorems haj« been proved in the simplest cases and it remains only to
show that the theorems are true for p functions in case they are for p <- 1. Expand
the determinant J.
For the first theorem J t^ 0 and hence at least one of the minors J^, • • •, /^ moat
fail to vanish. Let that one be Jj, which is the Jacobian of F,, • • •, F, with
to X,, • • •, Zp. By the assumption that the theorem holds for the case j> ~ 1,
p — 1 equations may be solved for x,, • • •, a^ in terms of the » + 1 varlablM x,,
134 DIFFERENTIAL CALCULUS
a^+ii • * •♦ ^+p» *"^ *^*'® results may be substituted in F^. It remains to show tha.
Fj = 0 is soluble for x^. Now
iIl = ^Il + ?Il^ + ... + 'll'^ = J/J^^O. (32-)
dXi dx^ axj ^1 ^i* ^^1
For the derivatives ofx^^-'-iXp with respect to x^ are obtained from the equations
axj ax, axj dxpdx^' * ax^ axg exj axp ax^
resulting from the differentiation of Fg = 0, • • •, Fp = 0 with respect to x^. The
derivative dXi/dx^ is therefore merely Ji/Ji , and hence dF^/dx^ = J/J^ and does
not vanish. The equation therefore may be solved for x^ in terms of Xp + i, • • •,
x» +p, and this result may be substituted in the solutions above found for Xg, • • •, Xp.
Hence the equations have been solved for Xj, Xg, • • •, Xp in terms of Xp +i , • • • , x„ +p
and the theorem is proved.
For the second theorem the procedure is analogous to that previously followed.
If there is a relation F{u^, . ., iip) = 0 between the p functions
Uj = 01 (Xj, • • •, Xp), • • •, Up=z <f>p{x^, • • •, Xp),
differentiation with respect to x^ , • • • , Xp gives p equations from which the deriva-
tives of F by Mj, • • •, Wp may be eliminated and j( ^^ ' ^| = 0 becomes the
dition desired. If conversely this Jacobian vanishes identically and it be assumed
that one of the derivatives of ui by xj, say du^/dx^, does not vanish, then the solution
Xj = w(Ui, Xg, • • •, Xp) may be effected and the result may be substitiited in u^,
. • •, Up. The Jacobian of Wg, • • •, Up with respect to Xg, • • •, Xp will then turn out
to be / -^ auj/axj and will vanish because J vanishes. Now, however, only p — 1
functions are involved, and hence if the theorem is true for p — 1 functions it must
be true for p functions.
EXERCISES
1. If u = ax + by ■\- c and v = a'x + h'y + c' are functionally dependent, the
lines u = 0 and tj = 0 are parallel ; and conversely.
2. Prove x + y + 2, xy + yz + zx, x^ + y2 _|_ ^a functionally dependent.
3. If u = ox + fey + cz + d, D = a'x + 6'y + c'z + d', w = a"x + h"y + c"z + d"
are functionally dependent, the planes m = 0, r = 0, iy = 0 are parallel to a line.
4. In what senses are — and 4i' of (24') and — ^ and — ^ of (32') partial or total
ay »-» ^ ' dxi axi ^ ^ ^
derivatives ? Are not the two sets completely analogous ?
con-
5. Given (26), suppose
tute in u = 0, and prove tni/dx — J
* !^ I T' 0. Solve V = ^ and lo = x f or y and «, substi-
^« Xz
Yy Xy
^* Xz\
6. If w = u (z, y), t = t> (x, y), and » = x (f , ii),y = y (f , 17), prove
State the extension to any number of variables. How may (27') be used to prove
(27) ? Again state the extension to any number of variables.
PARTIAL DIFFERENTIATION; IMPLICIT 185
7. Vro\e dV = J (^^^-^^] dudvdto = dudvdw -^ J (^^^-^^\ U the element of
\u, V, 10/ \x, y, x/
volume in Hpace with curvilinear coordinates u, v, to = constn.
8. In what parts of the plane can u = x* -k- j/^^ v = xy not be used m curri-
linear coiirtlinateB ? Express cW for these co^irtlinates.
9. Trove that 2 u = x^ — i/^, t = jrj/ is a confomutl transformation.
10. Prf>ve that x = » y = -; is a conformal transformation.
11. Define confonnal transformation in space^. If the transformation
X — au -{■ bv •{■ cw^ V — a'u + b'v + c'lo, z = a"u + b"t) + cf'w
is conformal, is it orthogonal ? See Ex. 10 (f), p. 100.
12. Show that the areas of the triangles whose vertices are
(M, t), (u + du, r), (u, V + d») and (u + d", t + dr), (a + du, »), (u, o + dc)
are intinitesiniaU of the same order, as suggested in § 03.
13. Would the condition F= 0 in (28) mean that the set of curves u = const,
were perpendicular to the set r = const. ?
14. Express J^, F, G In (28) in terms of the derivatives of u, v by jr, y.
15. If a; = 0(s, t), y = V(«» 0» 2: = x(*i 0 *^® ^^® parametric equations of a
surface (from which s, t could be eliminated to obtain the equation between
sc, y, z), show
^^jlXil\^jltl±\ andfind ?^.
&x \ s, < / \s,tj dy
65. Envelopes of curves and surfaces. Let the equation F(a-, y, a) = 0
be considered as lepieseiiting a family of curves where the dififerent
curves of the family are obtained by assigning different values to the
parameter a. Such families are illustrated by
(xr-ay-hf=l and ax-\-y/a=lj (33)
which are circles of unit radius centered on the a;-axis and lines which
cut off the area ^ a^ from the first quadrant. As a changes, the circles
remain always tangent to the two lines y = ± 1 and
the point of ttmgency ti-aces those lines. Again, as ^'
a changes, the lines (33) renuiin tangent to the hyper-
bola xi/ = k, owing to the j)rop€»rty of the hyi>erlx)la
that a tangent forms a triangle of constant area with
the asymptotes. The lines y = ± 1 are called the -
envelope of the system of circles and the hyj^rbola
xy = A* the envelope of the set of lines. In general, if there is a
to which the curves of a famUy F(x, y, a) = 0 are tangent and \f the
point of tanyency describes that curve as a varies^ the curve is called
136 DIFFERENTIAL CALCULUS
t?ie envelope (or part of the envelope if there are several such curves)
of the family F(Xj y, a) = 0. Thus any curve may be regarded as the
envelope of its tangents or as the envelope of its circles of curvature.
To find the equations of the envelope note that by definition the
enveloping curves of the family F(Xf y, a) = 0 are tangent to the envelope
and that the point of tangency moves along the envelope as a varies.
The equation of the envelope may therefore be written
x = <k(a), y = ^{cc) with F(4>,xf;,a) = 0, (34)
where the first equations express the dependence of the points on the
envelope upon the parameter a and the last equation states that each
point of the envelope lies also on some curve of the family F(x, y, a) = 0.
Differentiate (34) with respect to a. Then
F;,<I>'(<^) + F^xl^Xa) + f; = 0. (36)
Now if the point of contact of the envelope with the curve JP = 0 is an
ordinary point of that curve, the tangent to the curve is
K(^ - ^o) + K(y - 2/o) = 0 ; and • f;<^' + f;^' = 0,
since the tangent direction dy:dx = if/' : <^' along the envelope is by
definition identical with that along the enveloping curve ; and if the
point of contact is a singular point for the enveloping curve, F^ = F^ = 0.
Hence in either case F^ = 0.
Thus for points on the envelope the two equations
F(x,y,a) = 0, f:(x, y, a) = 0 (36)
are satisfied and the equation of the envelope of the family F = 0 may
be found by solving (36) to find the parametric equations x = <^(a),
y z= \f/(a) of the envelope or by eliminating a between (36) to find the
equation of the envelope in the form $ (x, y) = 0. It should be remarked
that the locus found by this process may contain other curves than the
envelope. For instance if the curves of the family F= 0 have singular
points and if a; = <^(a), y = \l/(a) be the locus of the singular points
as a varies, equations (34), (35) still hold and hence (36) also. The
rule for finding the envelope therefore finds also the locus of singular
points. Other extraneous factors may also be introduced in performing
the elimination. It is therefore important to test graphically or analyt-
ically the solution obtained by applying the rule.
As a first example let the envelope of {x — a)a + y« = 1 be found.
F(x, y, a) = (X - a)« + 2/2 - 1 = 0, F^ = - 2 (x - a) = 0.
The elimination of a from these equations gives y^ — 1 = 0 and the solution
for a gives X = a, y = ±l. The loci indicated as envelopes are y = ± 1. It Is
PARTIAL DIFFERENTIATION; IMPLICIT 137
geometrically evident that tbeae are really envelopes and not extraneoua factora
But aa a second example conaider oz -f y/<r = 1. Here
F{x, y, or) = ax + y/a -1 = 0, K = ^- V/a* = <>•
The8olutioni8y = ar/2,z = 1/2 a, which gives a;y = ^. This is the envelope ; it could
not be a locus of singular points of F = 0 as there are none. Suppose Uie elimina-
tion of a be made by Sylvester's method as
- y/a^ + O/a + x + Oa = 0
0/a* --y/a +0 + xa = 0 ^^^
y/a* — \/a + x + Oo: = 0
0/a* + y/a - 1 + xa = 0
— y 0 z 0
0 -y Ox
y -I X 0
0 y -I X
= 0;
the reduction of tlie determinant gives xy{4xy— 1) = 0 as the eliminant, and con-
tains not only the envelope 4zi/ = 1, but the factors x = 0 and y = 0 which are
obviously extraneous.
As a third problem find the envelope of a line of which the length intercepted
between the axes is constant. The necessary equations are
- + ?^ = 1, a^-\-tf^ = K\ ^da + ^dp = 0, ada-\-pdfi = 0,
a § a* p*
Two parameters a, p connected by a relation have been introduced ; both equations
have been differentiated totally with respect to the parameters ; and the problem
is to eliminate nr, /9, da, d/3 from the equations. In this case it is simpler to carry
both parameters than to introduce the radicals which would be required if only
one parameter were used. The elimination of da, dp from the last two equations
gives X : y = a* : /S* or y/x : y/y = a:p. From this and the first equation,
1111
« xKxt + y*) P yKx* + y*)
and hence xt + y^ = K^.
66. Consider two neighboring curves of F(x, y, a) = 0. Let (x^j yj
be an ordinary point of a = a^ and (x^ -h dxy y^ -f- dy) of a^ -{- da. Then
F{x^ ^dx,y^ + dy, a^ -f- da) - Fix^, y^, a^
= F'Jx + F'^dy + Fjia = 0 (37)
holds except for infinitesimals of higher order. The distance from the
point on a^ -h da to the tangent to a^ at (x^, y^ is
±Vf^f;' Vf^^Tf^
except for infinitesimals of higher order. This distance is of the first
order with da, and the normal derivative da/dn of § 48 is finite except
when F^ = 0. The distance is of higher order than da, and da/dn is
infinite or dn/da is zero when F^ — 0. It appears therefore that the
enfelope is the locus of points at whirh the distance between two neigK-
boring curves is of higher order than da. This is also apparent geomet-
rically from the fact that the distance from a point on a curve to the
138 DIFFERENTIAL CALCULUS
tangent to the curve at a neighboring point is of higher order (§ 36).
Singular points have been ruled out because (38) becomes indetermi-
nate. In general the locus of singular points is not tangent to the
curves of the family and is not an envelope but an extraneous faxjtor ;
in exceptional cases this locus is an envelope.
If two neighboring curves Fix, y, a) = 0, F(x, y, or + Aa) = 0 inter-
sect, their point of intersection satisfies both of the equations, and hence
also the equation
^ [F(a;, y, a + Aa) - F(x, y, at)] = F', (x, y, a + BAa) = 0.
If the limit be taken for Aa = 0, the limiting position of the intersec-
tion satisfies F^ = 0 and hence may lie on the envelope, and will lie on
the envelope if the common point of intersection is remote from singular
points of the curves F(x, y, a) = 0. This idea of an envelope as the
limit of points in which neighboring curves of the family intersect is
valuable. It is sometimes taken as the definition of the envelope. But,
unless imaginary points of intersection are considered, it is an inade-
quate definition ; for otherwise y = (x — ay would have no envelope
according to the definition (whereas y = 0 is obviously an envelope) and
a curve could not be regarded as the envelope of its osculating circles.
Care must be used in applying the rule for finding an envelope. Otherwise not
only may extraneous solutions be mistaken for the envelope, but the envelope may
be missed entirely. Consider
y — sin ax = 0 or a — x-i sin-i y = 0, (39)
where the second form is obtained by solution and contains a multiple valued
function. These two families of curves are identical, and it is geometrically clear
that they have an envelope, namely y = ± 1. This is precisely what would be
found on applying the rule to the first of (39) ; but if the rule be applied to the
second of (39), it is seen that 2?^ = 1, which does not vanish and hence indicates no
envelope. The whole matter should be examined carefully in the light of implicit
functions.
Hence let F(x, y, a) = 0 be a continuous single valued function of the three
variables (x, y, a) and let its derivatives F^, F^, F^ exist and be continuous. Con-
sider the behavior of the curves of the family near a point (Xq, y^) of the curve for
a = a^ provided that (x^,, y^) is an ordinary (nonsingular) point of the curve and
that the derivative F^(Xq, y^, a^) does not vanish. As F^ ;«i 0 and either F^ ^t 0
or F^ ^ 0 for (x^, y^, «„), it is possible to surround (x^, y^) with a region so small
that F(x, y, a) = 0 may be mUed for a =/(x, y) which will be single valued and
differentiable ; and the region may further be taken so small that F^ or F^ remains
different from 0 throughout the region. Then through every point of the region
there is one and only one curve a =/(x, y) and the curves have no singular points
within the region. In particular no two curves of the family can be tangent to
each other within the region.
PARTIAL DIFFERENTIATION; IMPLICIT 189
Furthermore, in such a region there is no envelope. For let any curve which
traverses the region be » = ^(t), y = ^(Q. Then
Along any curve a =/(x, y) the equation f^ +/^dy = 0 holds, and if x = ^(Q,
y z=f{t) be tangent to this curve, (/y = dx = f : 0' and a'(l) = 0 or a = const.
Hence the only curve which has at each point the direction of the curve of the
family through that \)oiut is a curve which coincides throughout with some curve
of the family and is tangent to no other member of the family. Hence there is no
envelope. The result in that an envelope can be present only when F^ = 0 or when
F'^ = F'^ = 0, and this latter case has been seen to be included in the condition
F^ = 0. H F(x, y, a) were not single valued but the branches were separable, the
same conclusion would hold. Hence in case F(x, y, a) is not single valued the loci
over which two or more values become inseparable must be added to those over
which F'^ - 0 in order to insure that all the loci which may be envelopes are taken
into account.
67. The preceding considerations apply with so little change to other
cases of envelopes that the facts will merely be stated without proof.
Consider a family of surfaces F{x^ y, «, a, fi) = 0 depending on two
I)arameters. The envelope may \ye defined by the property of tangency
as in § 65 ; and the co7ulitlo?ui for an envelope ivoiild be
F(x,y,z,a,P) = 0, f;=0, f; = 0. (40)
These three equations may be solved to express the envelope as
parametrically in terms of a, ^ ; or the two parameters may be elimi-
nated and the envelope may be found as * (a*, y, z) = 0. In any case
extraneous loci may be introduced and the results of the work should
therefore be tested, which generally may l^e done at sight.
It is also possible to determine the distance from the tangent plane
of one surface to the neighboring surfaces as
-^Tf+F^Tl^ -yw^in^TTF^
and to define the envelope as the locus of points such that this distance
is of higher order than \da\ + \dp\. The equations (40) would then also
follow. This definition would apply only to ordinary points of the sur-
faces of the family, that is, to points for which not all the derivatives
F^, F^y F,' vanish. But as the elimination of a, )8 from (40) would give
an equation whi(;h included the loci of these singular points, there
would l)e no danger of losing such loci in the rare instances where they,
too, happened to be tangent to the surfaces of the family.
140 DIFFERENTIAL CALCUXUS
The application of implicit functions as in § 66 could also be made in this case
and would show that no envelope could exist in regions where no singular points
occurred and where either F^ or F^ failed to vanish. This work could be based
either on the first definition involving tangency directly or on the second definition
which involves tangency indirectly in the statements concerning infinitesimals of
higher order. It may be added that if F{x, y, z, a, /3) = 0 were not single valued,
the surfaces over which two values of the function become inseparable should be
added as possible envelopes.
A family of surfaces F(x, y, «, a) = 0 depending on a single param-
eter may have an envelope, and the envelope is found from
F(x,y,z,a) = 0, f:(x, y, z, a) = 0 (42)
by the elimination of the single parameter. The details of the deduction
of the rule will be omitted. If two neighboring surfaces intersect, the
limiting position of the curve of intersection lies on the envelope and
the envelope is the surface generated by this curve as a varies. The
surfaces of the family touch the envelope not at a point merely but
along these curves. The curves are called characteristics of the family.
In the case where consecutive surfaces of the family do not intersect
in a real curve it is necessary to fall back on the conception of imagi-
naries or on the definition of an envelope in terms of tangency or
infinitesimals; the characteristic curves are still the curves along
which the surfaces of the family are in contact with the envelope and
along which two consecutive surfaces of the family are distant from
each other by an infinitesimal of higher order than da.
A particular case of importance is the envelope of a plane which
depends on one parameter. The equations (42) are then
Ax + Bi/-^Cz + D = 0, A'x + B'y + C'z + D' := Oy (43)
where Aj Bj C, D are functions of the parameter and differentiation
with respect to it is denoted by accents. The case where the plane
moves parallel to itself or turns about a line may be excluded as trivial.
As the intersection of two planes is a line, the characteristics of the
system are straight lines, the envelope is a ruled surface, and a j^lane
tangent to the surface at one point of the lines is tangent to the surface
throughout the whole extent of the line. Cones and cylinders are exam-
ples of this sort of surface. Another example is the surface enveloped
by the osculating planes of a curve in space ; for the osculating plane
depends on only one parameter. As the osculating plane (§ 41) may be
regarded as passing through three consecutive points of the curve, two
oonBeoutive osculating planes may be considered as having two consecu-
tive points of the curve in common and hence the characteristics are
PARTIAL DIFFERENTIATION; IMPLICIT 141
the tangent lines to the curve. Surfaces which are the envelopes of a
plane which depends on a single parameter are called developable eurfaeeM,
A family of curves dependent on two parameters as
n^^ y, «, «, )8) = 0, G (x, i/,z,a,fi) = 0 (44)
is called a congruence of curves. The curves may have an envelope, that
is, there may be a surface to which the curves are tangent and which
may Ije regarded as the locus of their points of tangency. The envelope
\sk obtained by eliminating a, fi from the equations
F=0, C = 0, F:G;-FiG: = 0. (45)
To see this, suppose that the third condition is not fulfilled. The equa-
tions (44) may then be solved as er = f(x, y, z), fi = g (a;, y, z). Reason*
ing like that of § 66 now shows that there cannot possibly be an
envelope in the region for which the solution is valid. It may therefore
be inferred that the only possibilities for an envelope are contained in
the equations (45). As various extraneous loci might be introduced in
the elimination of a, p from (45) and as the solutions should therefore
be tested individually, it is hardly necessary to examine the general
question further. The envelope of a congruence of curves is called the
focal surface of the congruence and the points of contact of the curves
with the envelope are called the focal points on the curves.
EXERCISES
1. Find the envelopes of these families of curves. In each ca«e test the answer
or its individual factors and check the results by a sketch :
(a) y = 2ax + a\ (/3) y^ = ^(x - a), (7) 2/ = a^ + k/a,
(«) a(y + a)2 = x», (e) y = a(x + a)«, (r) y^ = a{x- a)\
2. Find the envelope of the ellipses x^/a* + y«/6« = 1 under the condition that
(a) the sum of the axes is constant or (/3) tlie area is constant.
3. Find the envelope of the circles whose center is on a given parabola hiwI
which pass through the vertex of the parabola.
4. Circles pass through the origin and have their centers on x* — y* = c*. Find
their envelope. Ana. A lemniscate.
6. Find the envelopes In these cases :
(or) X + xya = sin- »xy, (/3) x + a = vers- ^ y + V2 y — y*,
(7) y + a = Vl-l/x.
6. Find the envelopes In these cases :
(a) aa5 + /5y + a/Sz = l, (^) - + ^ + ; — ^— i = l>
<'^>:;5 + 5 + ^ = ^ witha/»y = t«.
a* p* 7'
7. F<nd the envelopes in Ex. 6 (a), (/S) if <r = /9 or if a = — /9.
142 DIFFERENTIAL CALCULUS
8. Prove that the envelope of F(x, y, z, or) = 0 is tangent to the surface along
the whole characteristic by showing that the normal to F{x, y, z, or) = 0 and to the
eliminant of F = 0, F^ = 0 are the same, namely
K^F;:F: and p; + i.^|:F; + ^^g: F; + ^i^.
where a(x, y, z) is the function obtained by solving Fa = 0. Consider the problem
alao from the point of view of infinitesimals and the normal derivative.
9. If there is a curve x = <t>{a), y = ^(a), z = x(«) tangent to the curves of
the family defined by F(x, y, z, a) = 0, G{x, y, z, or) = 0 in space, then that curve
is called the envelope of the family. Show, by the same reasoning as in § 65 for
the case of the plane, that the four conditions F = 0, C? = 0, F^ = 0, G« = 0 must
be satisfied for an envelope ; and hence infer that ordinarily a family of curves in
space dependent on a single parameter has no envelope.
10. Show that the family F{x, y, z, a) = 0, F^{x, y, z, or) = 0 of curves which
are the characteristics of a family of surfaces has in general an envelope given by
the three equations F = 0, Fa =0, F^a = 0.
11. Derive the condition (45) for the envelope of a two-parametered family of
curves from the idea of tangency, as in the case of one parameter.
12. Find the envelope of the normals to a plane curve y =f{x) and show that
the envelope is the locus of the center of curvature.
13. The locus of Ex. 12 is called the evolute of the curve y =/(x). In these cases
find the evolute as an envelope :
(a) y = x\ (/3) x = a sin t, y = b cos i, (7) 2xy = a^,
ls)y^ = 2mx, (e) x = a{0--smO), y = a{l— cosO), (f) y = coshx.
14. Given a surface z =/(x, y). Construct the family of normal lines and find
their envelope.
15. If rays of light issuing from a point in a plane are reflected from a curve in
the plane, the angle of reflection being equal to the angle of incidence, the envelope
of the reflected rays is called the caustic of the curve with respect to the point.
Show that the caustic of a circle with respect to a point on its circumference is a
cardioid.
16. The curve which is the envelope of the characteristic lines, that is, of the
rulings, on the developable surface (43) is called the ciispidal edge of the surface.
Show that the equations of this curve may be found parametrically in terms of the
parameter of (43) by solving simultaneously
^x + By + Cz + D = 0, A'x + B'y + C'z + ZK = 0, A''x + B''y + Cz + IX' = 0
for X, y, z. Consider the exceptional cases of cones and cylinders.
17. The tenn " developable " signifies that a developable surface may be developed
or mapped on a plane in such a way that lengths of arcs on the surface become equal
lengths in the plane, that is, the map may be made without distortion of size or
shape. In the case of cones or cylinders this map may be made by slitting the cone
or cylinder along an element and rolling it out upon a plane. What is the analytic
sUtement in this case ? In the cawe of any developable surface with a cuspidal
edge, tlie developable surface being the locus of all tangents to the cuspidal edge,
PARTIAL DIFFERENTIATION; IMPLICIT 148
the length of arc upon the surface may be written as dr* = (dt -f ds)* + f«d*»/i?*,
where 8 denoteH arc measured ahing the cuspidal edge and t denotes dist*noe along
the ungent line. Thin furin uf d<r^ may be obtained geometrically by in<inltj>rftn«i
analysis or analytically from the equations
X =/(«) + (/"(«), y = flr(») + <(f'(«). « = A(«) + tA'(*)
of the developable surface of which x =/(«), V = 9(«), z = A(s) is the cuspidal edge.
It i8 thuH Keen that da'^ is the same at corresponding points of all developable sur-
faceH for which tlie radius uf curvature R of the cuspidal edge Is the same function
of 8 witliout rej^ard to the torsion ; in particular the torsion may be tero and the
developable may reduce to a i)lane.
18. Let the line x = az + 6, y = C2 + d depend on one parameter so as to gen-
erate a ruled surface. By identifying this form of the line with (48) obtain by
substitutiun tlie conditions
^a + Be + C = 0, A'a-\- li'c + C = 0 ^a' + Be' = 0 la' c'
^6 + Bd + Z) = 0, A'b ■\- Rd -^^ IT = 0 ^"^ ^6' + Bd'=0 °^ W d'
a.s the condition that the line generates a developable surface.
68. More differential geometry. The representation
F{x,y,z) = (i, or z=f{x,y) (46)
or X = <\>{u, v), y = if;(u, r), z = x(u, v)
of a surface may be taken in the unsolved, the solved, or the iKinimetric
form. The parametric form is equivalent to the solved form provided
Uf V he taken as x, y. The notation
-^ ,=^ r = — , « = — , t = —
^ dx cy dx^ dxdy dy^
is adopted for the derivatives of z with respect to x and y. The applica-
tion of Taylor's Formula to the solved form gives
A« ^ph -{-qk-\- \{rh^ 4- 'ishk -\- tk^ + . • • (47)
with h = Aa-, k = A//. The linear terms ph -|- qk constitute the differ-
ential dz and represent tliat part of the increment of z which would be
obtained by replacing the surface by its tangent plane. Apart from
infinitesimals of the third order, the distance from the tangent plane up
or down to the surface along a jxirallel to the «-axis is given by the
quadratic terms \ (rh^ -\- 2 shk -f tk^.
Hence if the quadratic terms at any point are a positive definite form
(§ 65), the surface lies above its tiingent i)lane and is concave up; but
if the form is negative definite, the surface lies below its tangent plane
and is concave down or convex up. If the form is indefinite but not
singular, the surfiu'o lies imrtly above and partly below its tiingent
])lane and may be called concavo-convex, that is, it is saddle-sha{KHl. If
tlu' form is singular nothing can be definitely stated. These statements
(48)
144 DIFFERENTIAL CALCULUS
are merely generalizations of those of § 55 made for the case where the
tangent plane is parallel to the ary-plane. It will be assumed in the
further work of these articles that at least one of the derivatives r, s, t
is not 0.
To examine more closely the behavior of a surface in the vicinity of
a particular point upon it, let the xy-plane be taken in coincidence with
the tangent plane at the point and let the point be taken as origin.
Then Maclaurin's Formula is available.
z = \ (rx^ H- 2 sxy -h tif) -h terms of higher order
= J p*(r cos* 6 -{-2 s sin 6 cos 0 -{- 1 sin* 6) -{- higher terms,
where (p, 6) are polar coordinates in the £cy-plane. Then
| = rcos*^ + 2«sin^cosd + ^sin*d = ^,^ri + ^^Yy (49)
is the curvature of a normal section of the surface. The sum of the
curvatures in two normal sections which are in perpendicular planes
may be obtained by giving $ the values 6 and ^ -f ^ tt. This sum
reduces to r + ^ and is therefore independent of 0.
As the sum of the curvatures in two perpendicular normal planes is
constant, the maximum and minimum values of the curvature will be
found in perpendicular planes. These values of the curvature are called
the principal values and their reciprocals are the principal radii of
curvature and the sections in which they lie are the principal sections.
If « = 0, the principal sections are ^ = 0 and ^ = ^ tt ; and conversely
if the axes of x and i/ had been chosen in the tangent plane so as to be
tangent to the principal sections, the derivative s would have vanished.
The equation of the surface would then have taken the simple form
« = ^ (rx* + ti/^ -\- higher terms. (50)
The principal curvatures would be merely r and t, and the curvature
in any normal section would have had the form
1 cos^d , sirv'O ,^ . ,^
- = -77- + —^— = r cos* e -h t sm* 0.
If the two principal curvatures have opposite signs, that is, if the
signs of r and t in (50) are opposite, the surface is saddle-shaped.
There are then two directions for which the curvature of a normal sec-
tion vanishes, namely the directions of the lines
d = ±tan->V-/^,//?, or Vpjic =± V|7|y.
These are called the astjmptotic directions. Along these directions the
surface departs from its tangent plane by infinitesimals of the third
PARTIAL DIFFERENTIATION; IMPLICIT 145
order, or higher order. If a curve is drawn on a surface so that at each
point of the curve the tangent to the curve is along one of the asymp-
totic directions, the curve is called an asymptotic curve or line of the
surfax'e. As the surfiice departs from its tangent plane by infinitesimals
of higher order than the second along an asymptotic line, the tangent
plane to a surface at any point of an asymptotic line must be the oscu-
lating plane of the asymptotic line.
The character of a point upon a surface is indicated by the Dupin
indicatrix of the point. The indicatrix is the conic
^ + jf^ = l, cf.« = i('-x" + <A (61)
which has the principal directions as the directions of its axes and the
square roots of the absolute values of the principal radii of curvature
as the magnitudes of its axes. The conic may be regarded as similar to
the conic in which a plane infinitely near the tangent plane cuts the
surface when infinitesimals of order higher than the second are neg-
lected. In case the surface is concavo-convex the indicatrix is a hyper-
bola and should be considered as eitlier or both of the two conjugate
hyperbolas that would arise from giving z positive or negative values
in (51). The point on the surface is called elliptic, hyperbolic, or
paral>olic according as the indicatrix is an ellipse, a hyperbola, or a pair
of lines, as happens when one of the principal curvatures vanishes.
These classes of points correspond to the distinctions definite, indefinite,
and singular applied to the quadratic form rJi^ -\- 2 shk -\- tk\
Two further results are noteworthy. Any curve drawn on the surface
differs from the section of its osculating plane with the surface by
infinitesimals of higher order than the second. For as the osculating
plane passes through three consecutive points of the curve, its inter-
section with the surface passes through the same three consecutive
points and the two curves have contact of the second order. It follows
that the i-adius of curvature of any curve on the surface is identical
with that of the curve in which its osculating plane cuts the surface.
The other result is Meiisnier^s Theorem : The radius of curvature of an
oblique section of the surface at any point is the projection upon the
plane of that section of the radius of curvature of the normal section
which passes through the same tangent line. In other words, if the
radius of curvature of a normal section is known, that of the oblique
sections through the same tiingent line may be obtained by multiplying
it by the cosine of the angle between the plane normal to the surface
and the plane of the oblique section.
146 DIFFERENTIAL CALCITLUS
The proof of Meusnier's Theorem may be given by reference to (48). Let the
z-axis in the tangent plane be taken along the intersection with the oblique plane.
Neglect infinitesimals of higher order than the second. Then
y = 0(x)= Jaj;2, z = l{rz^-\-28xy + ty^) = irx* (480
will be the equations of the curve. The plane of the section is as; — ry = 0, as may
be seen by inspection. The radius of curvature of the curve in this plane may be
found at once. For if u denote distance in the plane and perpendicular to the
X-axis and if i* be the angle between the normal plane and the oblique plane
oz — ry = 0,
u = z sec r = y esc v = \r sec vx^ = \a esc v • x^.
The form u = \r sec v • x"^ gives the curvature as r sec v. But the curvature in the
normal section is r by (48'). As the curvature in the oblique section is sec v times
that in the normal section, the radius of curvature in the oblique section is cos v
times that of the normal section. Meusnier's Theorem is thus proved.
39. These investigations with a special choice of axes give geometric proper-
ties of the surface, but do not express those properties in a convenient analytic
form ; for if a surface z = /(x, y) is given, the transformation to the special axes
is difficult. The idea of the indicatrix or its similar conic as the section of the
surface by a plane near the tangent plane and parallel to it will, however, deter-
mine the general conditions readily. If in the expansion
Az - dz = :^{rh^ + 28hk + tk^) = const. (52)
the quadratic terms be set equal to a constant, the conic obtained is the projection
of the indicatrix on the xy-plane, or if (52) be regarded as a cylinder upon the
xy-plane, the indicatrix (or similar conic) is the intersection of the cylinder with
the tangent plane. As the character of the conic is unchanged by the projection,
the point on the surface is elliptic if s^ < rt, hyperbolic if s^ > rt, and parabolic if
8^ = rt. Moreover if the indicatrix is hyperbolic, its asymptotes must project into the
asymptotes of the conic (52), and hence if dx and dy replace h and k, the equation
rdx2 + 2 sdxdy + tdy^ = 0 (53)
may be regarded as t?ie differential equation of the projection of the asymptotic lines
on the xy-pUme. If r, s, t be expressed as functions/^, /^, f^ of (x, y) and (53) be
factored, the integration of the two equations 3f(x, y)dx-\- iV(x, y)dy thus found
will give the finite equations of the projections of the asymptotic lines and, taken
with the equation z =/(x, j/), will give the curves on the surface.
To find the lines of curvature is not quite so simple ; for it is necessary to deter-
mine the directions which are the projections of the axes of the indicatrix, and
these are not the axes of the projected conic. Any radius of the indicatrix may
be regarded as the intersection of the tangent plane and a plane perpendicular to
the xy-plane through the radius of the projected conic. Hence
« - 2o = P(» - «o) + ?(y - ^o)' (^ - «o)^ = (y - Vq) ^
are the two planes which intersect in the radius that projects along the direction
determined by A, k. The direction cosines
h:k.ph-\-qk
VA« + A;2 + {ph + qk)^
and A: A;: 0 (54)
PARTIAL DIFFERENTIATION; IMPLICIT 147
are therefore those of the radius in the indicatriz and of lU projection and thej
(leu^rinine tlie cotdne of the angle 0 between the radius and iU projection. The
square uf the radius in (52) is
fi^ + k^, and (A« + A«) sec* ^ = A« + *• + (p* + qk)*
is tiierefore tlie square of the corresponding radius in the indicatrix. To deter-
mine the axes of the indicatrix, this radius is to be made a maximum or minimum
subject to (62). With a multiplier X,
A + pA + 7* + X(rA + «*) = 0, * + pA + y* + X(«A + 0:) = 0
are the conditions required, and the elimination of X gives
A'^[«(l +p'^)-P7r] + hk[t(l-\-p^)-r{l + q')]-k"[t{l + q'')-pqt] = 0
as the equation that detennines the projection of the axes. Or
(1 -\-})-i)dx-\-pqdy _ pqdx -^ {I ■\- q*) dy
rdx -{■ sdy adx •\- idy
XM the differential equation of the projected lines of curvature.
In adilition to the asymptotic lines and lines of curvature the geodesic or shortest
lines on the surface are important. These, however, are better left for the metiiods
of the calculus of variations (§ 159). The attention may therefore be turned to
finding the value of the radius of curvature in any nonnal section of the surface.
A reference to (48) and (49) shows that the curvature is
1 _ 2z _ rh* ■{■2shk-\-tk^ _ r A« -j- 2 sAJk -t- <ik»
in the special case. But in the general case the normal distance to the surface if
(Az — dz) cos 7, with sec 7 = Vl + p'^ + q'^, instead of the 2 z of the special caee, and
the radius p'^ of the special case becomes p* sec*0 = A* + A:* + {ph + qk)^ in the
tangent plane. Hence
l^ _ 2{Az-dz)co8y _ rP -f 2 slm + tm«
/J - Ai 4. A:2 + (pA + 9Ac)« ~ VH- p3 + 9«
where the direction cosines /, m of a radius in the tangent plane have been intro-
duced from (54), is the general expression for the curvature of a normal section.
The form
1_ ^ rh^-{-2shk-\-tk^ 1 .^..
U - Aa + fca + (pA + (?ifc)« VlTpM^ *
where the direction A, k of the projected radius remains, is frequently more con-
venient than (56) which conUins the direction cosines /, m of the original direction
in the tangent plane. Meusnier's Theorem may now be written in the form
cos p rP + 2 slm + fm* .,-.
— — — = , » l***/
^ Vl + p* + 9*
where p is the angle between an oblique section and the tangent plane and wliere
f, m are the <lirection cosines of the intersection of the planes.
The work here given has depended for iti* relative simplicity of sUtement upon
the jussumption of the surface (4rt) in solved form. It is merely a problem In
implicit partial differentiation U) pass from p, q, r, «, f to their equivalence in terms
of F^, F^y F,' or the derivatives of ^, ^, x ^7 ^» P-
148 DIFFERENTIAL CALCULUS
EXERCISES
1. In (49) show — = ^-i— + cos 2 ^ + « sin 2 ^ and find the directions of
^ ' R 2 2
maximum and minimum R. If R^ and R^ are the maximum and minimum values
of Ry show
i- + i- = r + t and ^^^ = rt-^.
Half of the sum of the curvatures is called the mean curvature ; the product of the
curvatures is called the total curvature.
2. Find the mean curvature, the total curvature, and therefrom (by construct-
ing and solving a quadratic equation) the principal radii of curvature at the origin :
(a) z = xy, (p) z = x^ -\- xy + y^, (7) z = x{x-\-y).
3. In the surfaces (a) z = xy and (/3) z = 2x^ + y^ find at (0, 0) the radius of
curvature in the sections made by the planes
(a) jc + y = 0, ip) x + y + z = 0, (7) a; + y + 22 = 0,
(8)x-2y = 0, {€) x-2y + z = 0, (f) a; + 2y + J2; = 0.
The oblique sections are to be treated by applying Meusnier's Theorem.
4. Find the asymptotic directions at (0, 0) in Exs. 2 and 3.
5. Show that a developable surface is everywhere parabolic, that is, that ri — s^ = 0
at every point ; and conversely. To do this consider the surface as the envelope of
ite tangent plane z- p^p^- q^y = Zq- p^fc^ - q^y^, where p^, q^, Xq, y^, Zq are func-
tions of a single parameter a. Hence show
j(^J = 0 = (r(-«^)„ and j(?oiIo^Is^lMoJ = y^^,. _ ri)„.
The first result proves the statement ; the second, its converse.
6. Find the differential equations of the asymptotic lines and lines of curvature
on these surfaces :
(or) z = xy, iP) z = tan-i(y/x), (7) 22 ^_ ^2 _ cosh a;, (5) xyz = l.
7. Show that the mean curvature and total curvature are
2U1 V 2(l+p2 + 52)t ' R^R^ (H.p2+g2)2-
8. Find the principal radii of curvature at (1, 1) in Ex. 6.
9. An umbilic is a point of a surface at which the principal radii of curvature
(and hence all radii of curvature for normal sections) are equal. Show that the
r 8 t
conditions are :; = — = for an umbilic, and determine the umbilics of
1 + p* pg 1 + g2 '
the ellipsoid with semiaxes a, 6, c.
CHAPTER Vf
COMPLEX NUMBERS AND VECTORS
70. Operators and operations. If an entity u is clianged into an
entity c by some law, the change may be regarded as an operation per-
formed upon w, the operand^ to convert it into v ; and if / be introduced
as the symbol of the oj^eration, the result may be written as v =.fu.
For brevity the symbol / is often called an operator. Various sorts
of operand, operator, and result are familiar. Thus if u is a positive
number n, the application of the operator V gives the square root ; if u
represents a range of values of a variable Xy the expression /(a-) or fx
denotes a function of a; ; if w be a function of x, the operation of dif-
ferentiation may be symbolized by D and the result Du is the derivar
tive; the symbol of definite integration j {*)d* converts a function
u (x) into a numl^er ; and so on in great variety.
The reason for making a short study of operators is that a consider-
able number of the concepts and rules of arithmetic and algebra may
be so defined for operators themselves as to lead to a calculus of opera-
tions which is of frequent use in mathematics ; the single application to
the integration of certain differential equations (§ 95) is in itself highly
valuable. The fundamental concept is that of a product : If u is oper-
ated upon by f to givefu = v and ifv is operated upon by g to give gv = Wj
so that ' j^ J- /i\
fu = r, gv = gfu = Wy gfu = m>, (1)
then the operation indicated as gf which converts u directly into to is
called the product off by g. If the functional symbols sin and log be
regarded as operators, the symbol log sin could be regarded as the
product. The transformations of turning the a-y-plane over on the
X-axis, 80 that x' = x, y' = — y, and over the y-axis, so that x' = — x,
y' = y^ may be regarded as operations ; the combination of these opera-
tions gives the transformation x' = — x, y' z= — y, which is equivalent
to rotating the plane through 180® about the origin.
The products of arithmetic and algebiu satisfy the commutdttrt^ mw
'jf = fgy that is, the products of g by /and of / by g are equal. This
is not true of operators in general, as may be seen from the ^t that
140
150 DIFFERENTIAL CALCtlLUS
log sin X and sin log x ai-e different. Whenever the order of the factors
is immaterial, as in the case of the transformations just considered, the
operators are said to be commutative. Another law of arithmetic and
algebra is that when there are three or more factors in a product, the
factors may be grouped at pleasure without altering the result^ that is,
h{gf) = (hg)f=hgf. (2)
This is known as the associative law and operators which obey it are
called associative. Only associative operators are considered in the
work here given.
For the repetition of an operator several times
//=/", fff=f, rr =/"*', (3)
the usual notation of powers is used. The law of indices clearly holds ;
for /*•''■" means that / is applied m -\- n times successively, whereas
f^f* means that it is applied n times and then m times more. Not
applying the operator /at all would naturally be denoted by/**, so that
J^u = u and the operator /® would be equivalent to multiplication by 1 ;
the notation /** = 1 is adopted.
If for a given operation / there can be found an operation g such
that the product fg =f^ — 1 is equivalent to no operation, then g is
called the inverse of / and notations such as
/? = 1, ?=/-' = !' //-'=/y = l (4)
are regularly borrowed from arithmetic and algebra. Thus the inverse
of the square is the square root, the inverse of sin is sin~\ the inverse
of the logarithm is the exponential, the inverse of D is i . Some oper-
ations have no inverse; multiplication by 0 is a case, and so is the
square when applied to a negative number if only real numbers are
considered. Other operations have more than one inverse ; integra-
tion, the inverse of D, involves an arbitrary additive constant, and the
inverse sine is a multiple valued function. It is therefore not always
true that /~ ^f = 1, but it is customary to mean by /~ ^ that particular
inverse of /for which f-^f=:ff-^ = l. Higher negative powers are
defined by the equation /" " = (/~ ^)", and it readily follows that
/^/~'' = 1, as may be seen by the example
The law of indices f^f* =y>" + »« also holds for negative indices^ except
in 80 far as /" ^f may not be equal to 1 and may be required in the
reduction of/"/" to/*+".
COMPLEX NUMBERS AND VECTORS 151
If Uf Vj and u -^ v are o])erand8 for the operator /and if
f(u + r)=fu^fv, (5)
80 that the operator applied to the sum gives the same result as the
sum of the results of oi)ei-ating on each ojx'nmd, then the operator
/ is called linear or duttrUntthe. If / denotes a function such thai
f(x 4- y) =/(x) 4-/(y), it has been seen (Ex. 9, p. 46) that / must be
e(liiivalent to multiplication by a constant and fx = Cx. For a less
siMM'iali/.cd iiihTprctiitiun this is not so; for
JJi^if -f- r) = I)u 4- Dv and I {a -f- r) = / u -f / «'
are two of the fundamental formula,s of calculus and show operators
which are distributive and not equivalent to multiplication by a constant
Nevertheless it does follow by the same reasoning as used before (Ex. 9,
p. 45), that/nw = nfa if /is distributive and if n is a rational number.
Some operators have also the proi)erty of addition. Supjx)8e that u
is an oi)erand and/ g are oj)ei-ators such that/w and gu are things tliat
may be added together as/^ -|- gu^ then the sum of the oj)erators, / + y,
is defined by the equation (f -\- g) u = fa -\- gu. If furthermore the
oj)erators / ^, h are distributive, then
h{f-\-g) = hf^-hg and {f+g)h^fh + gh, (6)
and the multiplication of the oi)erators becomes itself distributive. To
prove this fact, it is merely necessary to consider that
and (/ -h g) (hu) = fhu + ghu.
Operators which are associative^ commutative, distributive, and which
admit addition may be treated algehraicalhj^ in so far as pohjnomials are
Concerned J by the ordinary algori^fns of algebra ; for it is by means
of the associative, commutative, and distributive laws, and the law of
indices that ordinary algebraic polynomials are rearranged, multiplied
out, and ftictored. Now the oi)erations of multiplication by constants
and of differentiation or partial differentiation as applied to a function
of one or more variables x, y, «, • • • do satisfy these laws. For instance
c(I>u)=D (cu), D^D^u = D^D^u, (D, + D^) D,n = D,D,u + Dfi.u. (7)
lTi*nr'.'. fnr example, if y be a function of or, the expression
LTy 4- afi^'^y H + a^_^Dy + «^,
where the coefficients a are constants, may be written as
(/>• -f flj/)-» + • . -f «.-i/> + «.)y (8)
152 DIFFERENTIAL CALCULUS
and may then be factored into the form
[(Z) - a^D -a^---{D- a,_,)(D - «„)]y, (8')
where a^, a^y ", a^ are the roots of the algebraic polynomial
x" + a^x"-^ H h a„_iX + a„ = 0.
EXERCISES
1. Show that (fgh)-'^ = h-^g-^-^, that is, that the reciprocal of a product of
operations is the product of the reciprocals in inverse order.
2. By definition the operator gfg-^ is called the transform of / by g. Show
that (a) the transform of a product is the product of the transforms of the factors
taken in the same order, and (/3) the transform of the inverse is the inverse of the
transform.
3. If s ?£ 1 but «2 = 1, the operator s is by definition said to be involutory. Show
that (a) an involutory operator is equal to its own inverse ; and conversely (/3) if
an operator and its inverse are equal, the operator is involutory ; and (7) if the
product of two involutory operators is commutative, the product is itself involu-
tory ; and conversely (5) if the product of two involutory operators is involutory,
the operators are commutative.
4. If /and g are both distributive, so are the products /gr and gf.
5. If /is distributive and n rational, show/nw = nfu.
6. Expand the following operators first by ordinary formal multiplication and
second by applying the operators successively as indicated, and show the results
are identical by translating both into familiar forms.
(a) (D-l)(D-2)y, Ans. fl- S^ + 2y,
ax^ ax
(/8) (i)-l)2)(D + l)y, (7) i)(D-2)(D+l)(D + 8)y.
7. Show that (D— a) e«^ fe-'*^Xdx = X, where JT is a function of a;, and
hence infer that e^ Ce-"^{*) dx is the inverse of the operator (D — a) (#).
8. Show that D{e^) =z e^{D ■\- a)y and hence generalize to show that if
P(D) denote any polynomial in D with constant coefficients, then
P{D) • e^ = e^P{D + a)y.
Apply this to the following and check the results.
{a) (D»-SD + 2)e^-y = e2x(I)2 + D)y = e^^(^ + ^V
Vox-* ax/
(j8) {D»-SD-2)e'y, (7) (2)8 - 3 D + 2) e=^.
9. If y is a function of x and x = e* show that
Dxy = e-»D,y, Djy = c-a<A(D<- l)y, . • ., D^y = e-i"A(D«- 1) • • • (A -p + l)y.
10. Is the expression {hDx + kDy)'*, which occurs in Taylor's Formula (§54),
the nth power of the operator hDx + kD^ or is it merely a conventional symbol ?
The same question relative to (xD, + yD»)* occurring in Euler's Formula (§ 63) '•
COMPLEX NUMBERS AND VECTORS 153
71. Complex numbers. In the formal solution of the equation
ewe* -h Aa; 4- fl = 0, where ^ < 4 a«, numbers of the form m 4- n V— 1,
where m and n are real, arise. Such numbers are called complsx or
imaginary ; the part m is called the real part and n V— 1 the purs
imaginary part of the number. It is customary to write V— 1 = i and
to treat t as a literal (luantity subject to the relation i* = —1. Tlie defini-
tions for the enuality^ mlditlony and multipliratinn of complex num-
bers are ... . ,. r j i •* i .
a-{-bi = c -^ (h if and only if « = c, 6 = a,
[a + hi] -f [c + di-] = (a 4- c) + ib 4- d) i, (9)
la -f hi] [r 4- di] = (ffc _ Arf) -f- (cm/ -f- ^) t.
It readily follows that the commutative, associative, and distributive
laws hold in the domain of complex numJjers, namely,
« 4- ^ = i8 4- a, (a + ^) 4- y = a 4- (i8 + y),
«i8 = ^, («)8)y = «(i8y), (10)
a(P + y)=ap-h ay, (« 4- ;3) y = ay 4- )3y,
where Greek letters have been used to denote complex numbers.
Division is accomplished by the method of rationalization.
a -{■ bi _a -^ hi c — di _ (ac 4- bd) 4- (be — ad) i
c-^di~ c-^-dic — di" c^ -{- d^ ' ^ ^
This is always possible except when c* + (P = 0, that is, when both e
and d are 0. A complex number is defined as 0 when and only when
its real and pure imaginary parts are both zero. With this definition 0
has the ordinary properties that or 4- 0 = a and a 0 = 0 and that a/0 is
impossible. Furthermore if a product ap vanishes, either a or p vanishes.
For suppose
[« + bi] [c 4- di] = (ac — bd) + (ad 4- be) i = 0.
Then ac — bd = 0 and ad + bc^^O, (12)
from which it follows that either a = 6 = 0 or c = rf=0. From the
fact that a product cannot vanish unless one of its factors vanishes
follow the ordinary laws of cancellation. In brief, all the elementary
Uiws of real algebra hold also for the algebra of complex numbers.
By assuming a set of Cartesian codrdinates in the xy-plane and asso-
ciating the number a -^ bi to the point (a, /;), a graphical reprejtentation
is obtained which is the counterpart of the number scale for real num-
bers. The point (a, b) alone or the directed line from the origin to the
point (a, b) may be considered as representing the number a -^ bL
If OP and OQ are two directed lines representing the two numbers
' -f- bi and c -f- di, a reference to the figure shows that the line which
164 DIFFERENTIAL CALCITLUS
lepresents the sum of the numbers is OR, the diagonal of the parallelo-
gram of which OP and OQ are sides. Thus the geometric law for adding
complex numbers is the same as the law for compounding forces and is
known as the parallelogram law. A segment AB oi 2^ line possesses
magnitude, the length AB, and direction, the
direction of the line AB from A to B. A y] ^ j^^/a-^.^+^>
quantity which has magnitude and direction is
called a vector; and the parallelogram law is
called the law of vector addition. Complex num^
bers may therefore be regarded as vectors.
From tlie figiire it also appears that OQ and PR have the same mag-
nitude and direction, so that as vectors they are equal although they
start from different points. As OP + PR will be regarded as equal to
OP -f OQ, the definition of addition may be given as the triangle law
instead of as the parallelogram law ; namely, from the terminal end P
of the first vector lay off the second vector PR and close the triangle
by joining the initial end 0 of the first vector to the terminal end R of
the second. The absolute value of a complex number a -\-bi is the
magnitude of its vector OP and is equal to -waF+1?, the square root of
the smn of the squares of its real part and of the coefficient of its pure
imaginary part. The absolute value is denoted by |a -f Zii| as in the case
of reals. If a and p are two complex numbers, the rule |a|-j-|y8| = |a-|-)8|
is a consequence of the fact that one side of a triangle is less than the
sum of the other two. If the absolute value is given and the initial end
of the vector is fixed, the terminal end is thereby constrained to lie
upon a circle concentric with the initial end.
72. When the complex numbers are laid off from the origin, polar
coftrdinates may be used in place of Cartesian. Then
r = Va^ -f- Py <f> = ta.n-'^b/a*, a = r cos <^, b = r sin <f>
and a -]- ib = r (cos <f> -\- i sin <^). ■
The absolute value r is often called the modulus or magnitude of the
complex nmnber; the angle <f> is called the angle or argument of the
number and suffers a certain indetermination in that 2 mr, where n is
a positive or negative integer, may be added to <^ without affecting the
number. This polar representation is particularly useful in discussing
products and quotients. For if
cr = rj (cos <^j 4- i sin «^j), p == r^ (cos <t>^ + i sin t^^),
then «)8 = v,[co8(<^^-f-<^^+isin(<^, + <^.;)],
* Aa both coB^ and sin ^ are known, the quadrant of this angle is determined.
COMPLEX NUM15ERS AND VECTOUS 155
as may be seen by multiplication according to the rule. Hence the
magnitude of a product is the product of the tnagnitudes of the faetort^
ami the. angle, of a product w the sum of the angles of the factors; the
general rule lx?ing proved by induction.
The interpretation of multiplication by a complex number as an oper-
atum is illuminating. Let p l)e the multiplicand and a the multiplier.
As the product ap has a magnitude equal to the product of the magni«
tudes and an angle equal to the sum of the angles, the factor a used as
a multiplier may be interpreted as effecting the rotation of p through
the angle of « and the stretching of fi in the ratio |a|:l. From the
geometric viewpoint, therefore, multiplication by a complex number is
an operation of rotation and stretching in the plane. In the case of
a = cos <^ -\- i sin <^ with r = 1, the oj)eration is only of rotation and
hence the factor cos <^ -f i sin <^ is often called a cyclic factor or versor.
In particular the number i = V— 1 will effect a rotation through 90*
when used as a multiplier and is known as a quiulrantal versor. The
series of j)owers i, i* = — 1, t* = — i, i* = 1 give rotations through 90*,
180", 270°, 360°. This fact is often given as the reason for laying off
pure imaginary numbers hi along an axis at right angles to the axis
of reals.
As a particular product, the nth power of a complex niunber is
or" = (a -f iby = [r(cos <f> -f i sin <^)]'' = r"(cos n<f> -\- i sin m^) ; (15)
and (cos <^ + i sin <\>y = cos n<^ -f i sin 72 <^, (15^)
which is a special case, is known as De Moivre^s Theorem and is of use
in evaluating the functions of w^; for the binomial theorem may be
applied and the real and imaginary parts of the expansion may be
equated to cos n<f> and sin n€f>. Hence
cos n<^ = cos"<^ -^— — ^ cos''-^<t> sin^tfi
, n(n-l)(n~2)(n-3) , . ,
-h -^^ ^^^-71 — '^ ^ cos"-*<^ sinV
(16)
- 1. • . n(7i — 1)(» — 2) . ,. ... .
sni ni^ = n cos" *^ sin ^ ^^ -^ ^ cos""'^ sm*^ H .
As the nth root \fa of a must be a number which when raised to the
nth power gives a, the nth root may be written as
•Va = -Vr (cos <^/n -f i sin ^/n). (17)
The angle <^, however, may have any of the set of values
<^, </»4-l?7r, <^4-4 7r /> -f 2(»-l)ir,
166 DIFFERENTIAL CALCULUS
and the nth parts of these give the n different angles
n n n n n n n ^ ^
Hence there may be found just n different /ith roots of any given com-
plex number (including, of course, the reals).
The rooU of unity deserve mention. The equation x^ = l has in the real domain
one or two roots according as n is odd or even. But if 1 be regarded as a complex
number of which the pure imaginary part is zero, it may be represented by a i)oint
at a unit distance from the origin upon the axis of reals ; the magnitude of 1 is 1
and the angle of 1 is 0, 2 tt, • • • , 2 (n — 1) tt. The nth roots of 1 will therefore have
the magnitude 1 and one of the angles 0, 2 rr/n, • • • , 2 (n — 1) ir/n. The n nth roots
are therefore
2ir . . . 2ir „ 4ir , . . 4ir
1, a = cos Hism — , a'* = cos h *sm — , •••,
^ n n n n
2(n-l)ir , . . 2(n-l)^
^n-i — Q08— '— + ism— ^^ '—f
n n
and may be evaluated with a table of natural functions: Now a?" — 1 =0 is factor-
able as (z — l)(x'»-i + x"-^ H + X 4- 1) = 0, and it therefore follows that the
nth roots other than 1 must all satisfy the equation formed by setting the second
factor equal to 0. As a in particular satisfies this equation and the other roots are
a*, • • • , a"- 1, it follows that the sum of the n nth roots of unity is zero.
EXERCISES
1. Prove the distributive law of multiplication for complex numbers.
2. By definition the pair of imaginaries a -\-bi and a — hi are called covjugaie
imaginaries. Prove that (a) the sum and the product of two conjugate imaginaries
are real ; and conversely (/3) if the sum and the product of two imaginaries are both
real, the imaginaries are conjugate.
3. Show that if P{x, y) is a symmetric polynomial in x and y with real coeffi-
cients so that P(x, y) = P(y, x), then if conjugate imaginaries be substituted forx
and y, the value of the polynomial will be real.
4. Show that if a -j- 6t is a root of an algebraic equation P(x) = 0 with real
coefiScients, then a — 6i is also a root of the equation.
5. Carry out the indicated operations algebraically and make a graphical repre»
sentation for every number concerned and for the answer :
(a) (1 + i)», (^) (H- V3 0 (1 - 0, (7) (3 + V^) (4 + ^J^,
1-i l-iVS V2-iV3
6. Plot and find the modulus and angle in the following cases :
(a) - 2, 09) - 2 V:^!, (^) 3 + 4i, («) i - i V~8
COMPLEX NUMBERS AND VECTORS 157
7. Show that the modulus qfa quotient qftwo numbers ia the quotient t^f the moduk
and that the angle ia the angle of the numerator lesa that of the denominator.
8. Carry out the indicated operations trigonometricallj and plot:
(a) Tlie exampleu of Ex. 6, (fi) Vl-^iVl^i, (y) \/-2-f 2>/8i,
(«) (VrTi -¥ y/V^i)\ (0 \/V2 + V^, (0 V^2 + 2V8i,
(f,) -^10(008 200^+ t8in200°), (ff) V^, (i) Wi,
9. Find the equations of analytic geometry which represent the transforma*
tlon equivalent to multiplication by a = — 1 + V— 8.
10. Show that [z — a| = r, where z is a variable and a a fixed complex number,
is the e(iuation of the circle (x — a)* + {y — b)^ = r*.
11. Find cosSz and cos 8 x in terms of cosx, and sin Ox and sin 7 x In terms of
Minx.
12. Obtain to four decimal places the five roots "v^.
13. If « = X + iy and z' = x' + i/, show that x' = (cos 0 - i sin 0) z — a Is the
fonnula for 8hiftin«; the axes through the vector distance a = a 4- i6 to the new
origin {a, b) and turning them through the angle <f>. Deduce the ordinary equa-
tions of transfunuation.
14. Show^ that |z — a| = fc|z — jS], where k is real, is the equation of a circle ;
specify the position of the circle carefully. Use the theorem : The locus of points
whose distances to two fixed points are in a constant ratio is a circle the diameter
of which is divided internally and externally in the same ratio by the fixed points.
15. The transformation z' = , where a, 6, c, d are complex and ad^bcytO,
cz + d
is called the general linear tran^ormation of z into z'. Show that
Iz' — a'l = *;|z' — /S'l becomes |z — a| = ik
ca + d
.|z-/3|
c/S + d
Hence infer that the transformation carries circles into circles, and points which
divide a diameter internally and externally in the same ratio into points which
divide some diameter of the new circle similarly, but generally with a different ratio.
73. Functions of a complex variable. Let « = a- -f t> be a complex
variable represeiittible geometrically as a variable point in the a-y-plane,
which may be called the complex jAane. As z determines the two real
nmnbers x and y, any function F{xj y) which is the sum of two single
valued real functions in the form
F{x, y) = A' {x, y) + {Y{x, y) = /J (cos * + » sin ♦) (19)
will be completely determined in value if « is given. Such a function
is called a co7nplex function (and not a function of the complex variable,
for reasons tliat will appear later). The magnitude and angle of the
function are determined by
A . . r
« = VAM^T*, co6* = ^,8in* = -^. (20)
168 DIFFERENTIAL CALCtlLUS
The function F is continuous by definition when and only when both
X and Y are continuous functions of (x, y)\ /? is then continuous in
(x, y) and F can vanish only when -R = 0 ; the angle ^ regarded as a
function of (xy y) is also continuous and determinate (except for the
additive 2 ntr) unless 7? = 0, in which case X and Y also vanish and the
expression for * involves an indeterminate form in two variables and
is generally neither determinate nor continuous (§ 44).
If the derivative of F with respect to z were sought for the value
z = a -j- U), the procedure would be entirely analogous to that in the
case of a real function of a real variable. The increment A« = Aic + iAy
would be assumed for z and AF would be computed and the quotient
AF/Az would be formed. Thus by the Theorem of the Mean (§ 46),
AF_ AX + i^Y^ (X^-\.iY:)Ax + (X^ + ^QAy ^
Az Ax 4- iAy Ax + iAy ' ^ -^
where the derivatives are formed for (a, h) and where ^ is an infinitesi-
mal complex number. When Az approaches 0, both Ax and Ay must
approach 0 without any implied relation between them. In general the
limit of AF/Az is a double limit (§ 44) and may therefore depend on
the way in which Ax and Ay approach their limit 0.
Now if first Ay = 0 and then subsequently Ax == 0, the value of the
limit of AF/Az is X^ -{- iY^ taken at the point (a, h) ; whereas if first
Ax=zO and then Ay = 0, the value is — iXy + Yy. Hence if the limit
of AF/Az is to be independent of the way in which Az approaches 0, it
is surely necessary that
dx dx dy By
dx dy dy dx ^ ^
And orfnversely if these relations are satisfied, then
^dY _
and the limit is A'; -\-iY'^= Y^ — iX^ taken at the point (a, b), and is
independent of the way in which Az approaches zero. The desirability
of having at least the ordinary functions differentiable suggests the
definition: A complex function F(x, y) = X(x, y) -f iY{x, y) is con-
sidered as a function of the complex variable z = x -{- iy when and only
when X and Y are in general differentiable and satisfy the relations (22).
In this case the derivative is
AF (dx .dY\^. (.. ._. , ^
COMPLEX NUMBERS AND VECTORS 159
.^ , dF dX , .BY dV ,dX
'rii<s(> conditions may also be expressed in polar co5rdinate8 (Ex. 2).
A few words about the function «!>(/, y). This is a multiple valued function of
the variables (x, y), and the difference between two neighboring brancheii is the con-
KUint 2 V. The application of the discussion of § 45 to this caae shows at once that,
i!j any simply coiniected re«,Mon of the complex plane which contains no point (a, b)
such that Ii{a, b) = 0, the different branches of *(x, y) may be entirely separated
so that the value of 4> nuist return to its initial value when any closed curve is de-
scribed by the point (x, y). If, however, the region is multiply connected or contains
points for wliich li = 0 (which makes the region multiply connected because these
points nnist be cut out), it luivy happen that there will be circuits for which ♦,
although changing continuously, will not return to its initial value. Indeed if It can
1k' sh<»wn that * does not return to its initial value when changing continuously as
(/, y) describes the l)oun«lary of a region simply connected except for the excised
point,s, it may be inferred that there must be points in the region for which /? = 0.
An application of these results may be made to give a very simple demonstration
of the fundamental theorem of algebra that every equation of the nih degree has at least
one root. Consider the function
F(z) = z» 4- a,2«-^ + • • • + a^-iz + a„ = X(x, y) + iY^{x, y),
where A' and Y are found by writing z as x -\- iy and expanding and rearranging.
The functions X and Y will be polynomials in (x, y) and will therefore be every-
wliere finite and continuous in (x, y). Consider the angle ♦ of F. Then
♦ = ang. of F= ang. of «« (l + ^ + • • + -"-^ + -) = ang. of «" + ang. of (!+• • •)•
\ z z*-^ Z"/
Next draw about the origin a circle of radius r so large that
+ --
r r"-i r*
Then for all points z upon the circumference the angle of F is
* = ang. of F = n(ang. of z) + ang. of (1 + i?), |l|<«.
Now let the point (x, y) describe the circumference. The angle of z will change by
2ir for the complete circuit. Hence * must change by 2 nx and does not return to
it« initial value. Hence there is within the circle at least one point (a, b) for which
R{a,b) - 0 and consequently for which A' (a, b) = Oand Y{a^b) = Oand F(a, 6)=0.
Thus if a: = a + tZ>, then F{a) = 0 and the equation F{z) = 0 is seen to have at
least the one root a. It follows that z — n: is a factor of F{z) \ and hence by induc-
tion it may be seen that F(2) = 0 has just n roots.
74. The discussion of the algebra of complex numbers showed how
the sum, difference, j)r(){luct, quotient, real powers, and real roots of
such numbers could be found, and hence made it possible to compute
the value of any given algebraic expression or function of z for a given
value of z. It remains to show that any algebraic expression in « is
160 DIFFERENTIAL CALCULUS
really a functibn of z in the sense that it has a derivative with respect
to «, and to find the derivative. Now the differentiation of an algebraic
function of the variable x was made to depend upon the formulas of dif-
ferentiation, (6) and (7) of § 2. A glance at the methods of derivation
of these formulas shows that they were proved by ordinary algebraic
manipulations such as have been seen to be equally possible with imagi-
naries as with reals. It therefore may be concluded that an algebraic
expression in z has a derivative with respect to z and that derivative
may he found just as if z were a real variable.
The case of the elementary functions e'', log z, sin z, cos «, • • • other
than algebraic is different ; for these functions have not been defined
for complex variables. Now in seeking to define these functions when z
is complex, an effort should be made to define in such a way that : 1°
when z is real, the new and the old definitions become identical ; and
2** the rules of operation with the function shall be as nearly as possi-
ble the same for the complex domain as for the real. Thus it would be
desirable that De* = e' and «* + "' = e'e^, when z and w are complex.
With these ideas in mind one may proceed to define the elementary
functions for complex arguments. Let
^ = R(x, y) [cos $ (Xj y)-{- i sin ^ (x^ y)]. (24)
The derivative of this function is, by the first rule of (23),
= (i?^ cos * — iJ sin ^ • $^) 4- 1 (72^ sin ^ -f- 72 cos * • *^),
and if this is to be identical with ^ above, the equations
i?^ cos * — i?^^ sin * = ^ cos <E» K = ^
or
/2,8in * + i?*^ cos * = 7? sin ^ ^^ = 0
must hold, where the second pair is obtained by solving the first. If
the second form of the derivative in (23) had been used, the results
would have been R^ = 0, ^^ = 1. It therefore appears that if the
derivative of c*, however computed, is to be e*, then
72^ = 72, 7?; = 0, <I>^ = 0, *; = i
are four conditions imposed upon 72 and *. These conditions will be
satisfied if 72 = e* and * = y.* Hence define
e» = 6*+ •> = e*(cos y + i sin y). (25)
• The UM of the more general solutions R = Ge», ♦ = y + C would lead to expressions
which would not reduce to e* when y = 0 and 2 := a; or would not satisfy e» + ^ = e«c«».
COMPLEX NUMBERS AND VECTORS 161
With this definition />«* is surely e", and it is readily shown that the
exixjiiential law «?"■*•'* = e^e"^ holds.
Vi\Y tin' s|»'(i;il values \ iri^ iri, 2iri of x the value of c* is
ei" = i, 6'' = -l, ««•<=: 1.
Hence it appears that if 2 niri be added to «, e* is unchanged ;
«• + «"' = «•, period 2^1. (26)
Thus in the complex domain c" has the period 2 wiy just as cos x and
sin X have the real period 2 tt. This relation is inherent ; for
e^ = cos y + i sin y, e~»^ = cos y — i sin y,
and cos y = > sin y = — — — • (27)
The trigonometric functions of a real variable y may be expressed in
terms of the exponentials of yi and — yi. As the exponential has been
defined for all complex values of z, it is natural to use (27) to define
the trigonometric functions for complex values as
cos « = > sin « = — jp (21*)
With these definitions the ordinary formulas for cos {z + w)^ D sin «, • • •
may be obtained and be seen to hold for complex arguments, just as the
corresponding formulas were derived for the hyperbolic functions (§ 5).
As in the case of reals, the logarithm log z will be defined for com-
plex nimibers as the inverse of the exponential. Thus
if e* = w?, then log w = z -\-2 mri, (28)
where the periodicity of the function e' shows that the logarithm is not
unif/uely determined but admits the addition of 2n7ri to any one of iU
valuen, just as tan"* x admits the addition of rnr. If w is written as a
complex number u -f iv with modulus r = VV + t^ and with the angle
<^^ it follows that
w = u + iv = r (cos <l> -f i sin <t>) = re^^ = e****" + ♦• ; (29)
and log w = \ogr + <l>i = log Vu^ 4. v* -|- i tan"* (v/u)
is the expression for the logarithm of ta in terms of the modulus and
angle of w] the 2 7i7ri may l)e added if desired.
To this point the expression of a power «*, where the exponent b is
imaginary, has had no definition. The definition may now be given in
terms of exponentials and logarithms. Let
a* = «***«• or log a* = i log a.
162 DIFFERENTIAL CALCtJLUS
In this way the problem of computing a* is reduced to one already
solved. From the very definition it is seen that the logarithm of a
power is the product of the exponent by the logarithm of the base, as
in the case of reals. To indicate the path that has been followed in
defining functions, a sort of family tree may be made,
real numbers, x real angles, x
real powers and real trigonometric functions,
roots of reals, x" cosx, sin x, tan- ^x, « • •
I ' 1 I '
exponentials, logarithms real powers and roots
of reals, e*^, logx of imaginaries, 2«
exponentials of imaginaries, e«
I I
logarithms of imaginaries, log z trigonometric functions
I ^ of imaginaries
imaginary powers, 2*
EXERCISES
1. Show that the following complex functions satisfy the conditions (22) and
are therefore functions of the complex variable z. Find F'(2):
(a) x2 - 2/2 + 2ixy, (/3) x3 - 3(xi/2 + x2 - i/2) + i(3a.2y _ yS _ 6a.y)^
(^)^-^^' (5)logV^^T^ + itan-i|,
(c) e^cosy + ie»^siny, (f) sin x sinh y + i cos x cosh y.
2. Show that in polar coordinates the conditions for the existence of F'{z) are
-r- = --r— » -T- = r- with F'{z) = l \- 1 — )(cos0 — ism0).
dr r d<f> dr r d<t> \dr dr/^ '
3. Use the conditions of Ex. 2 to show from D log 2 = z- ^ that log z = log r + ipi.
4. From the definitions given above prove the formulas
(a) sin (x + iy) = sin x cosh ?/ + i cos x sinh y,
(/S) cos (x 4- iy) = cos x cosh y — i sin x sinh y,
(7) tan(x + iy) = "'"2» + JBinh^V.
cos 2 X + cosh 2 y
5. Find to three decimals the complex numbers which express the values of :
(^) «*", (P) c*, (7) ei + i^ __ (5) e-i-s
(«) sinjiri, (D cosi, ('?) sin (i + J V- 3), ((?) tan(- 1 - i),
(«)log(-l), («) logi, (X) log(i + iV38), (^) log(-l-i).
6. Owing to the fact that log a is multiple valued, a* is multiple valued in such
a manner that any one value may be multiplied by c2 »•''«'•■. Find one value of each
of the following and several values of one of them : ..
(a) 2*, 09) i^ (7) 4/<, (a) V^, (e) (i + i V^)^''"'.
COMPLEX NUMBERS AND VECTORS
168
7. Sliow that Da* = a» log a when a and t are complex.
8. Show that {c^Y = o^ ; and fill in such other iteps %m may be suggested by
the work in the text, which for the moet part has merely been iketched in a broad
way.
9. Show that if /(z) and g{z) are two functions of a complex variable, then
f{z) ± giz), ocfiz) with a a complex constant, /(r) g (z), f{t)/g{z) are also func-
tions of z.
10. Obtain logarithmic expressions for the inverse trigonometric functions.
Find sin-U*.
75. Vector sums and products. As stated in § 71, a vector is a quan-
tity which has mugnitudf and direction. If the magnitudes of two
vectors are equal and the directions of the two vectors are the same,
the vectors are said to lie equal irrespective of the
position which they occupy in space. The vector
— a is by definition a vector which has the same
magnitude as a but the opjwsite direction. The
vector ma is a vector which has the same direction
as a (or the opposite) and is m (or — w) times as
long. The law of vector or geometric addition is
the parallelogmm or triangle law (§ 71) and is still
applicable when the vectors do not lie in a plane
but have any directions in space ; for any two vec-
tors brought end to end determine a plane in which the construction
may be carried out. Vectors will be designated by Greek small letters
or by letters in heavy type. The relations of equality or similarity
between triangles establish the rules
(r-|-^ = ^4-cr, a-h()3-hy) = (a + i8)-|-y, m{a + P) = ina + mfi (30)
as true for vectors as well as for numbers whether real or complex. A
vector is said to be zero when its magnitude is zero, and it is writ-
ten 0. From the definition of addition it follows that
a -}- 0 = or. In fact as far as addition, subtract ion, and
multljjlivatiDn by numbers are concerned, vectors obey
the same formal laws as numbers.
A vector p may be resolved into components par-
allel to any three given vectors a, ft, y which are not
parallel to any one plane. For let a ptirallelepiped
be constructed with its edges parallel to the three
given vectors and with its diagonal equal to the vector whose compo-
nents are desired. The edges of the parallelepiped are then certain
164 DIFFERENTIAL CALCULUS
multiples aw, y)S, «y of a, p, y, and these are the desired components
of p. The vector p may be written as
p = xa + yP + zy* (31)
It is clear that two equal vectors would necessarily have the same
components along three given directions and that the components of a
zero vector would all be zero. Just as the equality of two complex
numbers involved the two equalities of the respective real and imagi-
nary parts, so the equality of two vectors as
p = xa-{-yl3-i-zy = x'a-]-y'/3-h z'y = p' (31')
involves the three equations x — x\y = y\z = z'.
As a problem in the use of vectors let there be given the three vectors or, /S, y
from an assumed origin O to three vertices of a parallelogram ; required the vector
to the other vertex, the vector expressions for the sides and diagonals of the paral-
lelogram, and the proof of the fact that the diagonals bisect
each other. Consider the figure. The side AB is, by the
triangle law, that vector which when added to OA = a
gives OB = /3, and hence it must be that AB = p— a.
In like manner AC = y— a. Now OD is the sum of OC
and CD, and CD = AB; hence OD = 7 -h /3 - «. The diag-
onal AD is the difference of the vectors OD and OA, and
is therefore 7 + /3 — 2 a. The diagonal -BC is 7 — /3. Now the vector from 0 to the
middle point of BC may be found by adding to OB one half of BC. Hence this
vector is /3 -I- i (7 — /3) or J (/3 + 7). In like manner the vector to the middle point of
AD is seen to be a + i (7 4- /3 — 2 a) or ^ (7 + /3), which is identical with the former.
The two middle points therefore coincide and the diagonals bisect each other.
Let a and ^ be any two vectors, \a\ and \p\ their respective lengths,
and Z (a, /?) the angle between them. For convenience the vectors may
be considered to be laid off from the same origin. The product of the
lengths of the vectors by the cosine of the angle between the vectors
is called the scalar product^
scalar product = a./8 = |a||)9| cos Z (a, )8), (32)
of the two vectors and is denoted by placing a dot between the letters.
This combination, called the scalar product, is a number, not a vector.
As 1/3 1 cos Z (a, )3) is the projection of p upon the direction of or, the
scalar product may be stated to be equal to the product of the length
of either vector by the length of the projection of the other upon it.
In particular if either vector were of unit length, the scalar product
would be the projection of the other upon it, with proper regard for
• The numberi aj, y, z are the oblique coordinates of the terminal end of p (if the
initial end be at the origin) referred to a set of axes which are parallel to a, /S, 7 and
upon which the unit lengths are taken as the lengths of a, /3, 7 respectively.
COMPLEX NUMBERS AND VECTORS 166
the sign ; and if both vectors are unit vectors, the product is the ooeine
of the angle between them.
Tlie scalar in'0<lu(rt, from its definition, is commutative bo that a* fist fi»a.
Moreover (ma)*p = a»(mP) = yn (a»fi)y thus allowing a numerical factor
w to 1)6 combined with either factor of the product. Furthermore tha
distributive Uiw
a*()3 -f- y) = a.p -h ify or (a -f- fi^y = a-y -f fi'y (33)
is satisfied as in the case of numbers. For if a be written as the product
aa^ of its length a by a vector a^ of unit length in the direction of a,
the first equation becomes
aa^>{fi + y) = aa^'fi 4- "a^-y or a^'{^ + y) = oc^^fi -f ffj-y.
And now a^»{fi -f- y) is the projection of the sum fi -\- y ui)on the direc-
tion of a, and a^'ft -f a,»y is the sum of the projections of fi and y upon
this direction ; by the law of projections these are equal and hence the
distributive law is proved.
The associative law does not hold for scalar products ; for (a^fi) y
means that the vector y is multiplied by the number a»^, whereas
a (j8«y) means that a is multiplied by (^'y), a very different matter.
The laws of cancellation cannot hold ; for if
tf.)3 = 0, then |a||^| cos Z (a, fi) = 0, (34)
and the vanishing of the scalar product a»fi implies either that one of
the factors is 0 or that the two vectors are perpendicular. In fact
a»p = 0 is called the corulltion of perpendicularity. It should be noted,
however, that if a vector p satisfies
p.a = 0, P'P = 0, p«y = 0, (35)
three conditions of perpendicularity with three vectors or, fi, y not
parallel to the same plane, the inference is that p = 0.
76. Another product of two vectors is the vector product^
vector product = a^p = vlaH^I sin Z (a, )3), (36)
where v represents a vector of unit length normal to the plane of a
and p upon that side on which rotation from a t-o
/3 through an angle of less than 180' appears posi- ax/3
tive or counterclockwise. Thus the vector product
is itself a vector of which the direction is perpen-
dicular to each factor, and of which the magni- «^*^____s^
tude is the product of the magnitudes into the
sine of the included angle. The magnitude is therefore equal to the
area of the parallelogram of which the vectors a and $ are the sides.
166 DIFFERENTIAL CALCULUS
The vector product will be represented by a cross inserted between the
letters.
As rotation from )8 to a is the opposite of that from a to ^, it follows
from the definition of the vector product that
Pxa = — a^p, not ax^ = ftxa, (37)
and the product is not commutative, the order of the factors must be
carefully observed. Furthermore the equation
arx/J = v\a\\p\ sin Z {a, ft) = 0 (38)
implies either that one of the factors vanishes or that the vectors a and
fi are parallel. Indeed the condition a^p = 0 is called the condition of
parallelism. The laws of cancellation do not hold. The associative law
also does not hold; for (arx/3)xy is a vector perpendicular to axp and y,
and since ax^ is perpendicular to the plane of a and fi, the vector (arxy3)xy
perpendicular to it must lie in the plane of a and ^ ; whereas the vec-
tor ax()3xy), by similar reasoning, must lie in the plane of p and y ; and
hence the two vectors cannot be equal except in the very special case
where each was parallel to p which is common to the two planes.
But the operation (ma)xj8 = a:x(m/8) = m(«;xj8), which consists in
allowing the transference of a numerical factor to any position in the
product, does hold ; and so does the distributive law
ax(/8 4-y) = ax^ + axy and (a + /3)xy = axy ^ fixy, (39)
the proof of which will be given below. In expanding according to
the distributive law care must be exercised to keep the order of the
factors in each vector product the same on both sides of the equation,
owing to the failure of the commutative law; an interchange of the
order of the factors changes the sign. It might seem as if any algebraic
operations where so many of the laws of elementary algebra fail as in
the case of vector products would be too restricted to be very useful ;
tliat this is not so is due to the astonishingly great number of problems
in which the analysis can be carried on with only the laws of addition
and the distributive law of multiplication combined with the possibility
of transferring a nmnerical factor from one position to another in a
product ; in addition to these laws, the scalar product a»/3 is commuta-
tive and the vector product axfi is commutative except for change of sign.
In addition to segments of lines, plane areas may be regarded as
vector quantities ; for a plane area has magnitude (the amount of the
area) and direction (the direction of the normal to its plane). To specify
on which side of the plane the normal lies, some convention must be
made. If the area is part of a surface inclosing a portion of space, the
COMPLEX NUMBERS AND VECTORS 167
normal is taken as the exterior normal. If the area lies in an isolated
plane, its positive side is determined only in connection with some
assi^Mied direction of description of its bounding curve ; the rule is : If
a person is a.ssuiued to walk along the boundary of an area in an
assigned direction and upon tliat side of the plane which
causes the inclosed area to lie upon his left, he is said -^^
to Ixi upon the positive side (for the iissigned direction
of description of the boundary), and the vector which
represents the area is tlie normal to that side. It has
bt*en mentioned that the vector product represented
an area.
That the projection of a plane area upon a given plane gives an area
which is the original area multiplied by the cosine of the angle between
the two planes is a fundamental fact of projection, following from the
simple ft'U't that lines parallel to the intersection of the two planes are
unchanged in length whereas lines perpendicular to the intersection
are multiplied by the cosine of the angle between the planes. As the
angle between the normals is the same as that l)etween the planes, the
projection of an area upon a plane ami the projeet'mn of the vector rei>-
resenting the area upon the normal to the plane are equivalent. The
projection of a closed area upon a plane is zero; for the area in the
projection is covered twice (or an even number of times) with opposite
signs and the total algebraic sum is therefore 0.
To prove the law ax(/i -|- y) = ax^ -|- axy and illustrate the use of
the vector interpretation of areas, construct a triangular prism with the
triangle on /8, y, and ^ -|- y as base and a as lateral edge. The total
vector expression for the surface of this prism is .^
P^a + yxor -|- arx(/3 + y) + J (P^y) -hM = 0,
and vanishes because the surface is closed. A cancel-
lation of the equal and opposite terms (the two
bases) and a simi)le transposition combined with the
rule fixa = — axfi gives the result
ax(p -f y) = — pxa — yxa = axp -f axy.
A system of vectors of reference which is particularly useful consists
of three vectors i, j, k of unit length directed along the axes JC, K, Z
drawn so that rotation from A' to Y appears positive from the side of
the j-y-plane upon which Z lies. The components of any vector r drawn
from the origin to the point (x, y, z) are
ri, »/j, z)l, and r = ri + i/j + «k.
168 DIFFERENTIAL CALCULUS
The products of i, j, k into each other are, from the definitions,
i.i = j.j = k.k = 1,
i.j = j.i = j.k = k.j = k.i = i.k = 0,
ixi = jxj = kxk = 0, ^ ^
ixj = - jxi = k, jxk = - kxj = i, kxi = _ ixk = j.
By means of these products and the distributive laws for scalar and
vector products, any given products may be expanded. Thus if
ar = aji + flr,j + agk and )8 = i^i + ^»J + igk,
then a.^ = aj)^ 4- (ij>^ + %K (^^)
ax/3 = (a^p^ - aj)^i -f (aj)^ - a^^j + (afi^ - a/j)k,
by direct multiplication. In this way a passage may be made from
vector formulas to Cartesian formulas whenever desired.
EXERCISES
1. Prove geometrically that a + (/3 + 7) = (« + /3) + 7 and m(a + /3) = mar + wi/3.
2. If a and /3 are the vectors from an assumed origin to A and B and if C
divides AB in the ratio m : n, show that the vector to C is 7 = {na + m^)/{m + n).
3. In the parallelogram ABCD show that the line BE connecting the vertex to
the middle point of the opposite side CD is trisected by the diagonal AD and
trisects it.
4. Show that the medians of a triangle meet in a point and are trisected.
5. If mj and m^ are two masses situated at Pj and Pg, the center of gravity or
center of mass of m^ and m^ is defined as that point G on the line P1P2 which
divides P^P^ inversely as the masses. Moreover if G^ is the center of mass of a
number of masses of which the total mass is M^ and if G^ is the center of mass of
a number of other masses whose total mass is M^, the same rule applied to Jf j and
M^ and G^ and G^ gives the center of gravity G of the total number of masses.
Show that
- _ m^r, 4- m^T^ ^^^ _ _ m^r^ + m^r^ + • • • + mnin _ Swr
wij + Trig mj + wig + . . . + win 2m '
where r denotes the vector to the center of gravity. Resolve into components to
^**^^ -_Smx __Smy _ _ Smz
2m 2m * ~ 2m
6. If a: and p are two fixed vectors and p a variable vector, all being laid off
from the same origin, show that (/> — /3).a = 0 is the equation of a plane through
the end of /3 perpendicular to a.
7. Let a, /3, 7 be the vectors to the vertices -4, B, C of a triangle. Write the
tliree equationg of tlie planes through the vertices perpendicular to the opposite
sides. Show that the third of these can be derived as a combination of the other
two ; and hence infer that the three planes have a line in common and that the
perpendiculars from the vertices of a triangle meet in a point.
COMPLEX NUMBERS AND VECTORS 169
8. Solve the problem analogous to Ex. 7 for the perpendicular biMctora of the
9. Not© that the length of a vector is y/ix»a. If a, /J, and 7 = /| — a are the
three sides of a triangle, expand 7.7 = (/S — a).(/3 — or) to obtain the law of cosines.
10. Show that the sum of the squares of the diagonals of a parallelogram equals
the 8um of the squares of the sides. What does the difference of the squares of the
diagonals equal ?
11. Show that — ^ a and - — — — are the components of B parallel and perpen-
dicular to a by showing 1° that these vectors have the right direction, and 2^ that
they have the right magnitude.
12. If cr, /3, 7 are the three edges of a parallelepiped which start from the same
vertex, show that (ax/9).7 is the volume of the parallelepiped, the volume being
considered positive if 7 lies on the same side of the plane of a and fi with the
vector ax/3.
13. Show by Ex. 12 that (ax^).7 = a*(/3x7) and (ax/3).7 = {pxy)»a ; and hence
infer that in a product of three vectors with cross and dot, the position of the cross
and dot may be interchanged and the order of the factors may be permuted cyc-
lically without altering the value. Show that the vanishing of (ax^).7 or any of
its equivalent expressions denotes that a, ^, 7 are parallel to the same plane ; the
condition ax/3*7 = 0 is called the condition of complanarity.
14. Assuming a = a^i + a„j + Ogk, /3 = 6,i + ftj + 6,k, 7 = Cjl + cj + c,k,
expand a»7, a«^, and ax(/jx7) in tenns of the coefficients to show
ax(^x7) = (a.7) /3 — (a./3) 7 ; and hence (ax/3)x7 = (a:.7) /3 — (7./S) a.
15. The formulas of Ex. 14 for expanding a product with two crosses and the
rule of Ex. 13 that a dot and a cross may be interchanged may be applied to expand
(ax/3)x(7x3) = (a.7x3)/3— 03.7x5) or = (ax/3.«)7— (ax/3.7) a
and (ax/S).(7x«) = (a.7)03.«) - {p-y){a.6).
16. If a and /3 arer two unit vectors in the xy-plane inclined at angles 9 and ^
to the z-axis, show that
a = icos^ + jsin^, /3 = icos0 + jsin0 ;
and from the fact that a.^ = cos(0 — 6) and ax^ = ksin(0 — ^ obtain by multi-
plication the trigonometric formulas for m\{<t> — B) and cos(0 — B).
17. If Z, m, n are direction cosines, the vector U + m j + nk is a vector of unit
length in the direction for which /, m, n are direction cosines. ShowHhat the
condition for perpendicularity of two directions (/, m, n) and (r, m', n*) is
If + mm' + nn' = 0.
18. With the same notations as in Ex. 14 show that
i J k
a»a = Oj* + a,* + a/ and ax^ =
ai a, a,
\ 6, 6g
and ax/5«7 =
«i «« ««
\\\
^ «, «•
170 DIFFERENTIAL CALCtTLUS
19. Compute the scalar and vector products of these pairs of vectors :
^ r6i + 0.3j-6k ri + 2j + 3k /^./i + ^
<^> to.l i- 4.2 j + 2.6k, (^)l-3i-2j + k, ^^Uj + i.
20. Find the areas of the parallelograms defined hy the pairs of vectors in
Ex. 10. Find also the sine and cosine of the angles between the vectors.
21. Prove ax\fix{yx8)] = {a.yxS) p - a»p yy-S = ^'S axy - p.y axS,
22. What is the area of the triangle (1, 1, 1), (0, 2, 3), (0, 0, - 1) ?
77. Vector differentiation. As the fundamental rules of differentiar
tion depend on the laws of subtraction, multiplication by a number,
the distributive law, and the rules permitting rearrangement, it follows
that the rules must be applicable to expressions containing vectors
without any changes except those implied by the fact that ax^ =^ pxa.
As an illustration consider the application of the definition of differen-
tiation to the vector product uxv of two vectors which are supposed
to be functions of a numerical variable, say x. Then
A (uxv) = (u 4- Au)x(v + Av) — UxV
= UxAv + AUxV -}- AUxAv,
A (uxv) Av AU , AUxAv
— i-j^ ^ = ux- h —-XV H y
Ax Ax Ax Ax
d(uxY) ,. A (uxv) dv . du
-^ — ^ = lim — ^^1^ ^ = ux—- + —-XV.
ax Axio Ax ax ax
Here the ordinary rule for a product is seen to hold, except that
the order of the factors must not be interchanged.
The interpretation of the derivative is important. Let the variable
vector r be regarded as a fimction of some variable, say x, and suppose
r is laid off from an assumed origin so that, as x varies,
the terminal point of r describes a curve. The incre-
ment Ar of r corresponding to Ax is a vector quantity
and in fact is the chord of the curve as indicated.
The derivative
dx ,. Ar dr .. Ar . ,._.
is therefore a vector tangent to the curve; in particular if
the variable x were the arc s, the derivative would have
the magnitude unity and would be a unit vector tangent to the curve.
The derivative or differential of a vector of constant length is per-
pendicular to the vector. This follows from the fact that the vector
COMPLEX NUMBERS AND VECTORS 171
then describes a circle concentric with the origin. It may ako U^ ftH<»n
analytically from the equation
rf(r.r) = rfr.r -f T»dT = 2 r.rfr = d conat. = 0. (43)
If the vector of constant length is of length unity, the increment Ar is
the chord in a unit circle and, apart from infinitesimaU of higher
order, it is equal in magnitude to the angle subtended at the center.
Consider then the derivative of the unit tangent t to a curve with
resiM'ct to the arc s. The magnitude of dt is the angle the tangent turns
through and the direction of dt is normal to t and hence to the cunre.
The vector quantity, ^^ ^
curvature C = t = -rs » (44)
ds dfT ^ ^
therefore has the magnitude of the curvature (by the definition in f 42)
and the direction of the interior normal to the curve.
This work holds ecjually for plane or space curves. In the case of a space curve
the plane which contains the tangent t and the curvature C is called the osculating
plane (§ 41). By definition (§ 42) the torsion of a space curve is the rate of turning
of the osculating plane with the arc, that is, d\p/ds. To find the torsion by vector
methods let c be a unit vector C/Vc«C along C. Then as t and c are perpendicular,
n = txc is a unit vector perpendicular to the osculating plane and dn will equal d^
in magnitude. Hence as a vector quantity the torsion is
_ dn d(txc) dt . . do ^ dc ,.^^
T = — = -^— ^ = -xc + tx- = tx_ , (46)
ds ds d» ds da
where (since dt/ds — C, and c is parallel to C) the first term ^e
drops out. Next note that dn is perpendicular to n because it
is the differential of a unit vector, and is perpendicular to t
becaust; dn = d(txc) = txdc and t.(tx(/c) = 0 since t, t, dc are
necessarily complanar (Ex. 12, p. 169). Hence T is parallel /
to c. It is convenient to consider the torsion as positive when u /
the osculating plane seems to turn in the positive direction when
viewed from the side of the normal plane upon which t lies. An inspection of the
figure shows that in this case dn has the direction — c and not + c. As C is a unit
vector, the numerical value of the torsion is therefore — C»T. Then
r=-c.T = -c.txi' = -c.t«l-^
ds ds VC.C
^ rd»r 1 ^ d 1 1 ^ d»r 1 ..^
Ld«* VC^ ds Vc.cJ *»• VC^
^ C d«r r'.r"xr"'
= t'— — -x-
Y.
C.C da» r".r"
where differentiation with respect to « is denoted by accents.
78. Another sort of relation between vectors and differentiation
comes to light in connection with the normal and directional deriva-
tives (§ 48). If F{xy i/y x) is a function which has a definite value at
172 DIFFERENTIAL CALCULUS
each point of space and if the two neighboring surfaces F= C and
F= C + dC are considered, the normal derivative of F is the rate of
change of F along the normal to the surfaces and
is written dF/dn. The rate of change of F along ^ -^"^^ ^^^
the normal to the surface F = C is more rapid than
along any other direction ; for the change in F be-
tween the two surfaces is dF = dC and is constant,
whereas the distance dn between the two surfaces is
least (apart from infinitesimals of higher order) along the normal. In
fact if dr denote the distance along any other direction, the relations
shown by the figure are
dr = sec ddn and -r- = -7- cos B. (46)
dV CLTh
If now n denote a vector of unit length normal to the surface, the
product ndF/dn will he a vector quantity which has both the magnitude
and the direction of most rapid increase of F. Let
dF
n— = VF=gradF (47)
be the symbolic expressions for this vector, where VF is read as "del JP"
and grad F is read as " the gradient of jP." If dr be the vector of which
dr is the length, the scalar product n.c?r is precisely cos Bdr, and hence
it follows that ,„
d F
dr>VF = dF and r^-VF = -7- » (48)
where r^ is a unit vector in the direction dr. The second of the equar
tions shows that the directional derivative in any direction is the com-
ponent or projection of the gradient in that direction.
From this fact the expression of the gradient may be found in terms
of its components along the axes. For the derivatives of F along the
axes are dF/dx^ ^F/dy^ dF/dzy and as these are the components of VF
along the directions i, j, k, the result is
Hence v = i| + jl + k|
may be regarded as a symbolic vector-differentiating operator which
when applied to F gives the gradient of F. The product
dr.VF = (<fa l + d,f^ + d.l^F= dF (60)
COMPLEX NUMBERS AND VECTORS 178
Is immediately seen to give the ordinary expression for dF. From this
form of grad F it does not appear that the gradient of a function is
independent of the choice of axes, but from the manner of derivation
of VF first given it does appear that grad F is a definite vector quan-
tity independent of the choice of axes.
In the case of any given function F the gradient may be found by
the application of the formula (49); but in many instances it may also
be found by means of the important relation (It*VF =» dF of (48). For
instance to prove the formula V(FG) = FVG -f GVF, the relation may
be applied as follows :
dT»V(FG) = d(FG) = FdG -f GdF
= Fdt.VG -f GdT.VF = dT.{FVG + GVF).
Now as these equations hold for any direction rfr, the di may be can-
celed by (36), p. 166, and the desired result is obtained.
The use of vector notations for treating assigned practical problems involving
computation is not great, but for handling the general theory of such pazts of
physics as are essentially concerned witli direct quantities, mechanics, hydro-
mechanics, electromagnetic theories, etc., the actual use of the vector algorisms
considerably shortens the formulas and has the added advantage of operating
directly upon the magnitudes involvfed. At this point some of the elements of
mechanics will be developed.
79. According to Newton's Second Law, when a force acts upon a
particle of mass m, the rate of change of momentum is equal to the
force acting^ and takes place in the direction of the force. It therefore
appears that the rate of change of momentum and momentum itself
are to be regarded as vector or directed magnitudes in the application
of the Second Law. Now if the vector r, laid off from a fixed origin
to the point at which the moving mass m is situated at any instant of
time ty be differentiated with respect to the time ^, the derivative di/dt
is a vector, tangent to the curve in which the particle is moving and of
magnitude equal to ds/dt or i?, the velocity of motion. As vectors*,
then, the velocity v and the momentum and the force may be written as
IT -, rfv eft- . .^ . rfv rf«r ^ ^
Hence F = ^- = ..^ = .,f if i -^t ^ 1?'
From the equations it appears that the force F is the jirtxluct of the
mass m by a vector f which is the rate of change of the velocity regartied
* In applications, it Is usual to denote vectors by heavy type and to denot« the magni-
tudes of those vectors by corresponding italic letters.
174 DIFFERENTIAL CALCULUS
as a vector. The vector f is called the acceleration; it must not be con-
fused with the rate of change dv/dt or d^s/d^ of the speed or magnitude
of the velocity. The components /p, /y, /^ of the acceleration along the
axes are the projections of f along the directions i, j, k and may be
written as f»i, f -j, f 'k. Then by the laws of differentiation it follows
-^^ dt dt dt
, _dh . ^(P(T.i) _fe
^^ •'^ df'^ df ^ df
Hence /- = -^' •^^='^' '^^^d^'
and it is seen that the components of the acceleration are the acceler-
ations of the components. If X, Y, Z are the components of the force,
the equations of motion in rectangular coordinates are
^^ = ^> -^=1-. -^ = ^- (52)
Instead of resolving the acceleration, force, and displacement along
the axes, it may be convenient to resolve them along the tangent and
normal to the curve. The velocity v may be written as vt, where v is
the magnitude of the velocity and t is a unit vector tangent to the
curve. Then . ■,/ xn ^ -.^
, dY d(vt) dv ^ dt
I = = — ^^ — := t 4- V •
dt dt dt dt
■n i. dt dtds ^ V ,^^^
where R is the radius of curvature and n is a unit normal. Hence
It therefore is seen that the component of the acceleration along the
tangent is d^s/d^, or the rate of change of the velocity regarded as a
number, and the component normal to the curve is v^/R. If T and N
are the components of the force along the tangent and normal to the
curve of motion, the equations are
T = mft = m-^y N = mf^ = ^ p *
It is noteworthy that the force must lie in the osculating plane.
If r and r -f Ar are two positions of the radius vector, the area of
the sector included by them is (except for infinitesimals of higher order)
COMPLEX NUMBERS AND VECTORS 175
aA = ^rx(r + Ar) = }rxAr, and is a vector quantity of which the
direction is normal to the plane of r and r -f- Ar, that is, to the phuie
through the origin tangent to the curve. The rate of description of area,
or the ureal velfjcUy^ is therefore
- = hmirx-=irx- = lr.T. (64)
The projections of the areal velocities on the coordinate planes, which
are the same a.s the areal velocities of the projection of the motion on
those planes, are (Ex. 11 below)
1/ dz dy\ 1/ dx dz\ 1/ dy dx\ ,,,^
2(y^-^i)' H^^-^^)' 2\^drydt)' (^')
If the force F acting on the mass m passes through the origin, then
r and F lie along the same direction and rxF = 0. The equation of
motion may then be integrated at sight.
dy ^ dy -^ ^
m-T- = F, ^'irx— - = rxF = 0,
at at
dy d . . ^ ,
rx— = — (rxv) = 0, rxv = const.
ac az
It is seen that in this case the rate of description of area is a constant
vector, which means that the rate is not only constant in magnitude
but is constant in direction, that is, the path of the particle m must lie
in a plane through the origin. When the force passes through a fixed
point, as in this case, the force is said to be central. Therefore when a
particle moves under the action of a central force, the motion takes place
in a plane passing through the center and the rate of description of
areas, or the areal velocity, is constant.
80. If there are several particles, say n, in motion, each has its own equation
of motion. These equations may be combined by addition and subsequent reduction.
d^i, - dPT„ „ (Pr„ _
(Pt (Pt (Pt <P
But m,-^ + '^^=«-^ + •• + ^ d^" = ^aK'i + '^r. + •• • + '^•)-
Let m^ij + rn^Fj + • • • + m,r„ = (mj + m, + • • • + m,) f = if f
or - _ m^T^ + 1141, + • . . -f m,r, _ Zmi _ Zmr
^ S = ^1 + ''s + • • • + F. = S''- ^^^
Then
176 DIFFERENTIAL CALCULUS
Now the vector r which has been here introduced is the vector of the center of
mass or center of gravity of the particles (Ex. 5, p. 168). The result (66) states, on
comparison with (61), that the center of gravity of the n masses moves as if all the
mass M were concentrated at it and all the forces applied there.
The force F,- acting on the ith mass may be wholly or partly due to attractions,
repulsions, pressures, or other actions exerted on that mass by one or more of the
other mafises of the system of n particles. In fact let F,- be written as
F< = F,o + Fa + F,-2 + • • • + Fi„,
where F^- is the force exerted on m,- by tyij and F.o is the force due to some agency
external to the n masses which form the system. Now by Newton's Third Law,
when one particle acts upon a second, the second reacts upon the first with a
force which is equal in magnitude and opposite in direction. Hence to Fy above
there will correspond a force F^-,- = — F.y exerted by rm on mj. In the sum 2F,- all
these equal and opposite actions and reactions will drop out and SF,- may be re-
placed by SFio, the sum of the external forces. Hence the important theorem that :
The motion of the center of mass of a set of particles is as if all the mass were concen-
trated there and all the external forces were applied there (the internal forces, that is,
the forces of mutual action and reaction between the particles being entirely
neglected).
The moment of a force about a given point is defined as the product of the force
by the perpendicular distance of the force from the point. If r is the vector from
the point as origin to any point in the line of the force, the moment is therefore
rxF when considered as a vector quantity, and is perpendicular to the plane of the
line of the force and the origin. The equations of n moving masses may now be
combined in a different way and reduced. Multiply the equations by r^, ij, • • •, r„
and add. Then
dv dv dv
etc Cui at
or mi - iiXYi + wig - igXYg + • • • + '^ ^ 'nXVn = h'^'^i + ^^'''^i + ' * ' + r„xF,
or — (mjT^xY^ + m2r2xv2 + • • • + w„r„xYn) = SrxF. (66)
This equation shows that if the areal velocities of the different masses are multiplied
by those masses, and all added together, the derivative of the sum obtained is equal
to the moment of all the forces about the origin, the moments of the different forces
being added as vector quantities.
This result may be simplified and put in a different form. Consider again the
resolution of F,- into the sum F,o 4- F,i + • • • + F,„, and in particular consider the
action F^/ and the reaction Fj, = — F^- between two particles. Let it be assumed
that the action and reaction are not only equal and opposite, but lie along the line
connecting the two particles. Then the perpendicular distances from the origin to
the action and reaction are equal and the moments of the action and reaction are
equal and opposite, and may be dropped from the sum 2r,xF,-, which then reduces
to Sr^xF^o. On the other hand a term like m,r,xv,- may be written as r,x(/n,v,). This
product is formed from the momentum in exactly the same way that the moment
Is formed from the force, and it is called the moment of momentum. Hence the
equation (66) becomes
COMPLEX NUMBERS AND VECTORS 177
- (total moment of momentum) = moment of eztenml foroet.
dt
Hence the re8ult that, as vector quantities : JTte rale qf change of the momaU ^
monumtum of a system of particles is equal to the moment qf the external /orem (the
forct'H between tlie masseA being entirely neglected under the aflumpUon that Acdoo
and reaction lie along the line connecting the mimnon).
EXERCISES
1. Apply the definition of differentiation to prove
{a) d(u.y) = u.dv + v^u, (fi) d [u.(yxw)] = du.(vxw) + u.(dTxw) + Q.(Txdw).
2. Differentiate under the assumption that vectors denoted by early letters of
the alphabet are constant and those designated by the later letters are variable :
(a) ux(vxw), (fi) a cost + b sine, (7) (u.u)u,
du
dz
(«) nx — , (e)u.(^-x— j, (rt c(*.ii).
3. Apply the rules for change of variable to-show that -— = , where
djr *'•
accents denote differentiation with respect to x. In case r = arl + yj show that
1/ VC»C takes the usual form for the radius of curvature of a plane curve.
4 . The equation of the helix is r = ia cos 0 + ja sin 0 + k60 with s = Va* + 6* ^ ;
show that the radius of curvature is (o^ + h^)/a.
5. Find the torsion of the helix. It is 6/ (a* + 6^).
6. Change the variable from s to some other variable i in the formula for torsion.
7. In the following cases find the gradient either by applying the formula which
contains the partial derivatives, or by using the relation dr«VF = dF, or both :
(a) r.r = x^ 4. ^a + z\ (/S) logr, (7) r = \^,
(a) log(x3 + y2) = log [r.r - (k.r)^], (e) (rx«).(rxb).
8. Prove these laws of operation with the symbol V :
(a) V(F + (?) = VF+ VG, (/3) G^{F/0) = GVF- FVO,
9. If r, 0 are polar coordinates in a plane and fj is a unit vector along the radius
vector, show that dr,/df = nd<p/dt where n is a unit vector perpendicular to the
ulius. Thus differentiate r = rr^ twice and separate the result into components
.liong the radius vector and perpendicular to it so that
10. l*rove conversely to the text that if the vector rate of description of area la
onsumt, the force must be central, that is, rxF = 0.
11. Note that rxY*i, rxy.j, rxyk are the projections of the areal yelooities upon
the planes x = 0, y = 0, 2 = 0. Hence derive (64^) of the text.
178 DIFFERENTIAL CALCULUS
12. Show that the Cartesian expressions for the magnitude of the velocity and
of the acceleration and for the rate of change of the speed dv/dt are
Vx'2 + 2^2 4. ^2
where accents denote differentiation with respect to the time.
13. Suppose that a body which is rigid is rotating about an axis with the
angular velocity u = d<f>/dt. Represent the angular velocity by a vector a drawn
along the axis and of magnitude equal to w. Show that the velocity of any point
in space is y = axr, where r is the vector drawn to that point from any point of
the axis as origin. Show that the acceleration of the point determined by r is in a
plane through the point and perpendicular to the axis, and that the components are
ax(axr) = (a«r)a — ta^i toward the axis, {d&/dt)xi perpendicular to the axis,
under the assumption that the axis of rotation is invariable.
14. Let f denote the center of gravity of a system of particles and r/ denote the
vector drawn from the center of gravity to the ith particle so that r,- = f + i- and
V, = V + v/. The kinetic energy of the ith particle is by definition
Jmft)? = iwiiVi.v,- = ^mf(v + ▼/)•(▼ + v.O-
Sum up for all particles and simplify by using the fact S7M,r^ = 0, which is due to
the assumption that the origin for the vectors r/ is at the center of gravity. Hence
prove the important theorem : The total kinetic energy of a system is equal to the
kinetic energy which the total mass would have if moving with the center of gravity
plus the energy computed from the motix)n relative to the center of gravity as origin^
that is,
T = \ ^rmvf = ^ Mv^ + i SwifV^^.
15. Consider a rigid body moving in a plane, which may be taken as the xy-
plane. Let any point r^ 01 the body be marked and other points be denoted rela-
tive to it by r'. The motion of any point r' is compounded from the motion of Tq
and from the angular velocity a = kw of the body about the point r^. In fact the
velocity v of any point is v = Vq + axr'. Show that the velocity of the point denoted
by r' = kxVo/w is zero. This point is known as the instantaneous center of rotation
(§ 39). Show that the coordinates of the instantaneous center referred to axes at
the origin of the vectors r are
1 dyn . 1 dXn
X = r.i = Xo - - -^«, y = r.] = 2/0 + - -f .
ia dt u> dt
16. If several forces Pj, Fg, • • -, P„ act on a body, the sum R = SF,- is called
the resultant and the sum 2r,xF,-, where r,- is drawn from an origin O to a point
in the line of the force F,-, is called the resultant moment about 0. Show that the
resultant moments Mo and Mo' about two points are connected by the relation
M(y = Mo + TA(y(Ro)j where Mo'(Ro) means the moment about 0' of the resultant
R considered as applied at O. Infer that moments about all points of any line
parallel to the resultant are eijual. Show that in any plane perpendicular to R
there in a point (/ given by r = RxMo/R-R, where 0 is any point of the plane,
such that Mo/ is parallel to R.
PART II. DIFFERENTIAL EQUATIONS
CHAPTER VII
GENERAL INTRODUCTION TO DIFFERENTIAL EQUATIONS
81. Some geometric problems. The application of the differential
calculus to j)liinc curves hius given a means of determining some
geometric jiroperties of the curves. For instance, the length of the
subnormal of a curve (§ 7) is ydy/dx, which in the case of the parabola
i/ = A:px is 2 JO, that is, the subnormal is constant. Suppose now it
were desired conversely to find all curves for which the subnormal is
a given constant m. The statement of this problem is evidently con-
tained in the equation
dy , , ,
y — = VI or yy' = w or ydy = mdx.
Again, the radius of curvature of the lemniscate r* = a* cos 2 ^ is found
to be It = ays r, that is, the radius of cui'vature varies inversely as the
radius. If conversely it were desired to find all cui'ves for which the
radius of curvature varies inversely as the radius of the curve, the state-
ment of the problem would be the equation
\l
k
Nil
where k is a constant called a factor of proportionality.*
Equations like these are unlike ordinary algebraic equations because,
in addition to the variables a, y or r, <^ and certain constants m or A-,
they contain also derivatives, as dy/ilx or dr/di^t and d^r/d<ft% of one of
the variables with respect to the other. An equation which contains
* Many problems in geometry, mechanics, and physics are stated in terms of Taria-
tion. For purposes of analysis the statement x varies as y, or z ac v> ^ written as a; = Jry,
Intrmluiiu^ a constant k calle.l a faft«»r of proportionality to convert the variation into
an equation. In like manner the staUMuent x varies inversely as y, or x x l/y, beoomea
Z = k/y, and x varies jointly with y and z heoomes r = kyz.
17U
180 DIFFERENTIAL EQUATIONS
derivatives is called a differential equation. The order of the differential
equation is the order of the highest derivative it contains. The equar
tions above are respectively of the first and second orders. A differen-
tial equation of the first order may be symbolized as * («, y, y') = 0,
and one of the second order as ^(a;, y, y', y") = 0. A function y =f(x)
given explicitly or defined implicitly by the relation F(x, y) = 0 is
said to be a solution of a given differential equation if the equation is
true for all values of the independent variable x when the expressions
for y and its derivatives are substituted in the equation.
Thus to show that (no matter what the value of a is) the relation
4ay - x^ + 2 a^logx = 0
gives a solution of the differential equation of the second order
it is merely necessary to form the derivatives
dx X dx^ x^
and substitute them in the given equation together with y to see that
\dx/ \dxV 4an ^ xV 4a2\ ^ x^ ^ xV
is clearly satisfied for all values of x. It appears therefore that the given relation
for y is a solution of the given equation.
To integrate or solve a differential equation is to find all the functions
which satisfy the equation. Geometrically speaking, it is to find all the
curves which have the property expressed by the equation. In mechan-
ics it is to find all possible motions arising from the given forces. The
method of integrating or solving a differential equation depends largely
upon the ingenuity of the solver. In many cases, however, some method
is immediately obvious. For instance if it be possible to separate the
variables, so that the differential dy is multiplied by a function of y
alone and dxhy o. function of x alone, as in the equation
i> (y) dy = ^ (x) dx, then f*^ (y) dy = Cijf (x) dx -\- C (1)
will clearly be the integral or solution of the differential equation.
As an example, let the curves of constant subnormal be determined. Here
ydy = mdx and y^ = 2 »nx + C.
The variables are already separated and the integration is immediate. The curves
are parabolas with semi-latus rectum equal to the constant and with the axis
GENERAL INTRODUCTION 181
coincident witli the axis of x. If in particular it were dednd to determine that
curve whose eubnormal waa m and which paased through the origin, it would
merely be n<«€e88ary to substitute (0, 0) In the equation y* = 2 mx + C to aaoertain
what particular value must be aaaigned to C in order that the curre paM through
(0, 0). The value is C = 0.
Another example might be to determine the curves for which the a^interoept
varies a^ the abscissa of the point of tangency. As the expreision (f 7) for the
x-intercept is x — ydx/dy, the statement is
x-y—^kx or (l-l:)« = y--.
dy dy
Hence (1-4)^ = — and (1 - ifc)logy = logx + C.
V -*
If demred, this expression may I>e changed to anothf^r form by using each aide of
the eijuality as an exponent with the base e. Then
e(i-«:)iocir = ^x^c or yi-* = e^'x = C'x.
As C la an arbitrary constant, the constant C = c^is also arbitrary and the solution
may simply be written as y^-* = Cz, where the accent has been omitted from the
constant. If it were desired to pick out that particular curve which passed through
the point (1, 1), it would merely be necessary to determine C from the equation
P-* = C 1, and hence V — \.
As a third example let the curves whose tangent is constant and equal to a be
detenniiied. The length of the tangent is y Vl + y^V/ a"d hence the equation is
y ^— = a or y* -f— = a or 1 = y
/ y^ y
The variables are therefore separable and the results are
dx = ^L^^dy and x + C = V^TT^i _ « log ^Jl^^I^ .
y V
If it be desired that the tangent at the origin be vertical so that the curve
through (0, a), the constant C is 0. The curve is the tractrix or "curve of pursuit"
as described by a calf dragged at the end of a rope by a person walking along
a straight line.
82. Problems which involve the radius of curvature will lead to differ-
ential equations of the second order. The method of solving such
problems is to reduce the equation^ if posslblef to one of the first order.
For the second derivative may be written as
dx ^ dy
182 DIFFERENTIAL EQUATIONS
is the expression for the radius of curvature. If it be given that the
radius of curvature is of the form /(a:) 4* (y') ovf{y) ^ (y'),
^4jp^=/(-)<^(y) or (L!^=/(2,)<^(y), (3)
rfx ^ dy
the variables x and y' or y and y' are immediately separable, and an
integration may be performed. This will lead to an equation of the
first order ; and if the variables are again separable, the solution may
be completed by the methods of the above examples.
In the first place consider curves whose radius of curvature is constant. Then
(1 -I- y"^\h dy" dx , y' x-C
i-^l-i-L. = a or — r = — and —-^=z = ,
dy^ (1 + /2)| a Vl + 2/'2 a
dx
where the constant of integration has been written as — C/a for future conven-
ience. The equation may now be solved for y' and the variables become separated
with the results
"' — or dy = — ^ dx.
Va2 - (X - C)2 Va2 - (X - C)2
Hence y-C'^- Va^ - (x - Cf or (x - Cf + {y - C'f = a^.
The curves, as should be anticipated, are circles of radius a and with any arbi-
trary point (C, C") as center. It should be noted that, as the solution has required
two successive integrations, there are two arbitrary constants C and O' of integra-
tion in the result.
As a second example consider the curves whose radius of curvature is double
the normal. As the length of the normal is y Vl -|- y"^, the equation becomes
lL±|^ = 2,VITr^ or l±f; = ±^y,
dy dy
where the double sign has been introduced when the radical is removed by cancel-
lation. This is necessary ; for before the cancellation the signs were ambiguous
and there is no reason to assume that the ambiguity disappears. In fact, if the
curve is concave up, the second derivative is positive and the radius of curvature
is reckoned as positive, whereas the normal is positive or negative according as
the curve is above or below the axis of x ; similarly, if the curve is concave down.
Let the negative sign be chosen. This corresponds to a curve above the axis and
concave down or below the axis and concave up, that is, the normal and the radius
of curvature have the same direction. Then
dy 2 v'dy'
— = - r-T-^ and logy = - log(l + y"^) -|- log 2 C,
where the constant has been given the form log 2 C for convenience. This expres-
sion may be thrown into algebraic form by exponentiation, solved for y', and then
GENERAL INTRODUCTION 18S
ydy
V2 Cy - ^
j/(l + l^ = 2C or i^ = ^5 — I? or — =^?= = dr,
Hence x - C = C vew- » ^ - V2 Cy - y«.
The curves are cycloids of which the generating circle hiui an arbitmry radiiu C
and of which tito cusps are upon the x-axis at the points C ± 2kwC. If the pod-
live sign lia<l been taken in the equation, the curves would have been entirely
different ; see Kx. 6 (a).
The number of arbitrary constants of integration which enter into
the solution of a differential equation depends on the number of inte-
grations which are performed and is equal to the order of the equation.
This results in giving a family of curves, deijendent on one or more
parameters, as the solution of the equation. To pick out any particular
member of the family, additional conditions must be given. Thus, if
there is only one constant of integration, the curve may be required
to pass through a given point ; if there are two constants, the curve
may be required to pass through a given point and have a given slope
at that point, or to pass through two given points. These additional
conditions are called initial conditions. In mechanics the initial condi-
tions are very important ; for the point reached by a particle describing
a curve under the action of assigned forces depends not only on the
forces, but on the point at which the particle started and the velocity
with which it started. In all cases the distinction between the constants
of integration and the given constants of the problem (in the foregoing
examples, the distinction between C, C and 7/t, A*, a) should be kept
clearly in mind
EXERCISES
1. Verify the solutions of the differential equations :
(7) (l + x2)y^ = l,'2x=Ce»'-C-Je-^ («) y -{■ xi/ = x*y^, xy = C^x + C,
(e) V" + y'/x = 0, 1/ = Clogx + Ci, (0 y = Ce* + C^^\ k" + 2y = 8/,
(v) y^'-y^x^ j/ = Ce« + e-i'f CiC08^^+ C,sin^^j-x«.
2. Determine the curves whicli have tlie following properties:
(or) The subtangent is constant ; jr* = Ce*. If through (2, 2), y" = 2"e«-«.
(/3) Tlie right triangle funned by the tangent, subtangent, and ordinate has the
constant area k/2 ; the hyperbolas xy + C}/ + A; = 0. Sliow that if the curve pa«es
through (1, 2) and (2, 1), tl»e arbitrary constant C is 0 and the given I; is — 2.
(7) The normal is constant in lengtli ; the circles (x — C)* + y* = 4*.
(d) The nonnal varies as the square of the ordinate ; catenaries l:y=coeh lf(x— C).
If in particular the curve is perpendicular to the y-axis, (7 = 0.
(e) The area of the right triangle formed by tlie tAiigent, normal, and x-axis It
inversely proportional to the slope ; the circles (x — C)* + y* = *•
184 DIFFERENTIAL EQUATIONS
3. Determine the curves which have the following properties:
(a) The angle between the radius vector and tangent is constant; spirals
(/9) The angle between the radius vector and tangent is half that between the
radius and initial line ; cardioids r = C(l — cos 0).
(y) The perpendicular from the pole to a tangent is constant ; r cos (0 — C) = k.
(S) The tangent is equally inclined to the radius vector and to the initial line ;
the two sets of parabolas r = C/(l ± cos 0).
(e) The radius is equally inclined to the normal and to the initial line ; circles
r = C cos 0 or lines r cos 0 = C.
4. The arc « of a curve is proportional to the area J., where in rectangular
coordinates A is the area under the curve and in polar coordinates it is the area
included by the curve and the radius vectors. From the equation ds = dA show
that the curves which satisfy the condition are catenaries for rectangular coordi-
nates and lines for polar coordinates.
5. Determine the curves for which the radius of curvature
(a) is twice the normal and oppositely directed ; parabolas {x— C)^ = C\2y — C).
(/S) is equal to the normal and in same direction ; circles (x — C)^ + y^ = C'^.
(7) is equal to the normal and in opposite direction ; catenaries.
(5 ) varies as the cube of the normal ; conies kCy^ — C^ (x + C')^ = k,
( e ) projected on the x-axis equals the abscissa ; catenaries.
( f ) projected on the x-axis is the negative of the abscissa ; circles.
(ri) projected on the x-axis is twice the abscissa.
(^) is proportional to the slope of the tangent or of the normal.
83. Problems in mechanics and physics. In many physical problems
the statement involves an equation between the rate of change of some
quantity and the value of that quantity. In this way the solution of
the problem is made to depend on the integration of a differential equa^
tion of the first order. If x denotes any quantity, the rate of increase
in X is dx/dt and the rate of decrease in jc is — dx/dt ; and consequently
when the rate of change of ic is a function of cc, the variables are
immediately separated and the integration may be performed. The
constant of integration has to be determined from the initial conditions ;
the constants inherent in the problem may be given in advance or their
values may be determined by comparing x and t at some subsequent
time. The exercises offered below will exemplify the treatment of
such problems.
In other physical problems the statement of the question as a differ-
ential equation is not so direct and is carried out by an examination of
the problem with a view to stating a relation between the increments
or differentials of the dependent and independent variables, as in some
geometric relations already discussed (§ 40), and in the problem of the
tension in a rope wrapped around a cylindrical post discussed below.
GENERAL INTRODUCTION
186
r+Ar
The method may be further illustrated by the derivation of the differ-
ential equations of the curve of equilibrium of a flexible string or
chain. Let p be the density of the chain so that pA« is the mass of
the length Ax; let X and Y be the components
of the force (estimated per unit mass) acting on
the elements of the chain. Let T denote the
tension in the chain, and t the inclination of
the element of chain. From the figure it then
appears that the components of all the forces
acting on A« are
(r 4- Ar) cos (r + At) - jT cos t + Xp^H = 0,
(r-h Ar) sin (t + At) - r sin r + 1>A« = 0;
for tliese must be zero if the element is to be in a jxwition of equi'
lihriuni. The equations may be written in the form
A(rcosT)-f XpA« = 0, A(rsinT)-h 1>A« = 0;
and if they now be divided by A« and if As be allowed to approach
zero, the result is the two equations of equilibrium
where cos r and sin t are replaced by their values dx/ds and dy/d».
If the string is acted on only by forces parallel to a given directioa, let the
y-axis be taken as parallel to that direction. Then the component X will be lero
and the first equation may be integrated. The result is
d8\ dsj
T--C
T=C
da
dx'
This value of T may be substituted in the second equation. There is thua obtained
a differential equation of the second order
mh^^-^
r
+ pr = o.
(*-)
Vi + i^
If this equation can be integrated, the form of the curve
of equilibrium may be found.
Another problem of a different nature in strings is to
consider the variation of the tension in a rope wound around
a cylinder without overlapping. The forces acting on the
element As of the rope are the tensions T and r+ AT, the
normal pressure or reaction li of the cylinder, and tlie force
of friction which is proixirtional to the pressure. It will
be assumed that the normal reaction lies in the angle A^ and that the ooeiBcient
of friction is /« so that the force of f rictiun \& uli. The compouenLs along the ndiu*
and along the tangent are
n
186 DIFFERENTIAL EQUATIONS
( r + AT) sin A0 - R cos (^A0) - fiR sin (tfA0) = 0, 0 < ^ < 1,
(r + AjT) cos A0 + /J sin (tfA0) - fiR cos (^A0) - T = 0.
Now discard all infinitesimals except those of the first order. It must be borne in
mind that the pressure R is the reaction on the infinitesimal arc As and hence is
itself infinitesimal. The substitutions are therefore rd0 for (T + AT) sin A^, R for
R cos ^A0, 0 for R sin 0A4>, and T + dT for (T + AT) cos A(p. The equations there-
fore reduce to two simple equations
Td<f>-R = 0, dT-fiR = 0,
from which the unknown R may be eliminated with the result
dT = iiTd<t> or T = CeH> or T = T^ef^i*^
where T^ is the tension when <f> is 0. The tension therefore runs up exponentially
and affords ample explanation of why a man, by winding a rope about a post, can
readily hold a ship or other object exerting a great force at the other end of the
rope. If II is 1/3, three turns about the post will hold a force 535 Tq, or over 25
tons, if the man exerts a force of a hundredweight.
84. If a constant mass m is moving along a line under the influence
of a force F acting along the line, Newton's Second Law of Motion (p. 13)
states the problem of the motion as the differential equation
mf=F ov m-^ = F (5)
of the second order ; and it therefore appears that the complete solution
of a problem in rectilinear motion requires the integration of this equa^
tion. The acceleration may be written as
_^ dv _ dv dx __ dv
^ " dt~ dxdt~^ dx^
and hence the equation of motion takes either of the forms
m- = F or mv- = F. (6')
It now appears that there are several cases in which the first integration
may be performed. For if the force is a function of the velocity or of
the time or a product of two such functions, the variables are separated
in the first form of the equation ; whereas if the force is a function of
the velocity or of the coordinate a; or a product of two such functions,
the variables are separated in the second form of the equation.
When the first integration is performed according to either of these
methods, there will arise an equation between the velocity and either
the time t or the coordinate x. In this equation will be contained a
constant of integration which may be determined by the initial condi-
tions, that is, by the knowledge of the velocity at the start, whether in
GENERAL INTRODUCTION 187
time or in position. Finally it will be possible (at least theoretically)
to solve the equation and express the velocity as a function of the time
t or of the ]K)sition x, .is the case may be, and integrate a second time.
The currying through in practice of this sketch of the work will be
exempli tied in the following two examples.
Suppose a particle of mass m is projected vertically upward with the velocity V.
Solve the problem of the motion under the assumption that the resistance of the
air varies as tlie velocity of the particle. Let the distance be measured vertically
upward. The forces acting on the particle are two, — the force of gravity which is
the weight W = ni^, and the resistance of the air which is ho. Both these forces
are negative because they are directed toward diminishing values of x. Hence
n^= — mg — ho or m— = — mg — kv,
at
where the first form of the equation of motion has been chosen, although in this
case the second form would be equally available. Then integrate.
dv
m
-dt and log/gr +-»\ = - -< + C.
\ m / m
As by the initial conditions v = V when t = 0, the constant C is found from
log/y + ^F\ = --0+C; hence ^ = c""'
\ m ) m ^^k^
m
is the relation between v and t found by substituting the value of C. The solution
for V gives
Hence * ~ ~ Tvik ^ "*" )^ "* ~ T^ "^
If the particle starts from the origin x = 0, the constant C is found to be
Hence the position of the particle is expressed in terms of the time and the prob-
lem is solved. If it be desired to find the time which elapses before the particle
comes to rest and starts to drop back, it is merely necessary to substitute v = 0 in
the relation connecting the velocity and the time, and solve for the time <= T;
and if this value of t be substituted in the expression for x, the total distance JC
covered in the ascent will be found. The results are
As a second example consider the motion of a particle vibrating up and down
at the end of an elastic string held in the field of gravity. By Hookers Law for
188 DIFFERENTIAL EQUATIONS
elastic strings the force exerted by the string is proportional to the extension of
the string over its natural length, that is, F = fcAf . Let I be the length of the string,
A J, the extension of the string just sufficient to hold the weight W = mg &t rest so
that kA^l = mg^ and let x measured downward be the additional extension of the
string at any instant of the motion. The force of gravity mg is positive and the
force of elasticity — fc(Aoi + x) is negative. The second form of the equation of
motion is to be chosen. Hence
mv — = mg — k {A^l + a;) or mv — = — fee, since mg = kAJ.
dx dx
Then mvdv = — kxdx or mv^ = — kx^ -{■ C.
Suppose that x = a is the amplitude of the motion, so that when x = a the velocity
D = 0 and the particle stops and starts back. Then C = ka^. Hence
dx
^/?V^2ZT2 or ^ = -^/idt,
\ m -y/oi _ a;2 \ m
and 8in-i- = -v/— i+ C or x = asini \ —t + c]-
a Mm \ \m /
Now let the time be measured from the instant when the particle passes through
the position x = 0. Then C satisfies the equation 0 = asm C and may be taken as
zero. The motion is therefore given by the equation x = asmVk/mt and is
periodic. While i changes by 2 tt Vm/fc the particle completes an entire oscilla-
tion. The time T = 2 tt y/m/k is called the periodic time. The motion considered
in this example is characterized by the fact that the total force — /ex is propor-
tional to the displacement from a certain origin and is directed toward the origin.
Motion of this sort is called simple harmonic motion (briefly S. H. M.) and is of
great importance in mechanics and physics.
EXERCISES
1. The sum of $100 is put at interest at 4 per cent per annum under the condition
that the interest shall be compounded at each instant. Show that the sum will
amount to $200 in 17 yr. 4 mo., and to $1000 in 57^ yr.
2. Given that the rate of decomposition of an amount x of a given substance is
proportional to the amount of the substance remaining undecomposed. Solve the
problem of the decomposition and determine the constant of integration and the
physical constant of proportionality if x = 5.11 when « = 0 and x = 1.48 when
t = 40 min. Ana. k = .0309.
8. A substance is undei^oing transformation into another at a rate which is
a«8umed to be proportional to the amount of the substance still remaining untrans-
formed. If that amount is 85.6 when < = 1 hr. and 13.8 when t = 4 hr., determine
the amount at the start when t = 0 and the constant of proportionality and find
how many hours will elapse before only one-thousandth of the original amount
will remain.
4. If the activity A ot a, radioactive deposit is proportional to its rate of
diminution and is found to decrease to ^ its initial value in 4 days, show that A
■atlsflea the equation .4 /il^ = c-o"8r.
0.284S
0.1864
GENERAL INTRODUCTION 189
5. Suppose that amounts a and b respectively of two labiUiioet are ioTolved in
a reaction in whicli the velocity of transformation dx/di is proportional to the prod-
uct (a — x)(b — x) of the amounts remaining untranaformed. Integrate on tiie
supposition tliat a^^b.
hi - \ -i- ^""^ *""*
log ^^^— ^' = (a - 6) « ; and if 898 0.4866
^yP"^) 1266 0.8879
determine the product *(o — 6).
6. Integrate the equation of Ex. 6 if a = 6, and determine a and I; if x = 9.87
when ( = 15 and x = 13.69 wiien i = 56.
7. If the velocity of a chemical reaction in which three substances are inToived
is proportional to the continued product of the amounts of the sabsCanoes remaining,
show that the equation between x and the time is
(a- 6)(6-c)(c-a) * \t=0.
8. Solve Ex. 7 if a = 6 ^i c ; also when a = 6 = c. Note the very different
forms of the solution in the three cases.
9. The rate at which water runs out of a tank through a small pipe issuing
horizontally near the bottom of the tank is proportional to the square root of the
height of the surface of the water above the pipe. If the tank is cylindrical and
half empties in 30 min., show that it will completely empty in about 100 min.
10. Discuss Ex. 0 in case the tank were a right cone or frustum of a cone.
11. Consider a vertical column of air and assume that the pressure at any levei
is due to the weight of the air above. Show that p = PqC-^ gives the preMore at
any height A, if Boyle's Law that the density of a gas varies as the pressure be used.
12. Work Ex. 11 under the assumption that the adiabatic law pocp*-* repre-
sents the conditions in the atmosphere. Show that in this case the pressure would
become zero at a finite height. (If the proper numerical data are inserted, the
height tunis out to be about 20 miles. The adiabatic law seems to correspond
better to the facts than Boyle's Law.)
13. Let I be the natural length of an elastic string, let Ai be the exuMision, and
assume Hooke's Law that the force is proportional to the extension in the form
A7 = klF. Let the string be held in a vertical position so as to elongate under its
own weight W. Show that the elongation is \k\Vl.
14. The density of water under a pressure of p atmospheres is /> = 1 -f 0.00004 p.
Show that the surface of an ocean six miles deep is about 600 ft. below the position
it would have if water were incompressible.
15. Show that the equations of the curve of equilibrium of a string or chain are
in polar co-ordinates, where R and 4 are the components of the force along the
radius vector and perpendicular to it.
190 DIFFERENTIAL EQUATIONS
16. Show that dT-\- pSds = 0 and T + pRN = 0 are the equations of equilib-
rium of a string if R is the radius of curvature and S and N are the tangential and
normal components of the forces.
17.* Show that when a uniform chain is supported at two points and hangs down
between the points under its own weight, the curve of equilibrium is the catenary.
18. Suppose the mass dm of the element ds of a chain is proportional to the pro-
jection dx of ds on the x-axis, and that the chain hangs in the field of gravity.
Show that the curve is a parabola. (This is essentially the problem of the shape
of the cables in a suspension bridge when the roadbed is of uniform linear density ;
for the weight of the cables is negligible compared to that of the roadbed.)
19. It is desired to string upon a cord a great many uniform heavy rods of
varying lengths so that when the cord is hung up with the rods dangling from it
the rods will be equally spaced along the horizontal and have their lower ends on
the same level. Required the shape the cord will take. (It should be noted that
the shape must be known before the rods can be cut in the proper lengths to hang
as desired.) The weight of the cord may be neglected.
20. A masonry arch carries a horizontal roadbed. On the assumption that the
material between the arch and the roadbed is of uniform density and that each
element of the arch supports the weight of the material above it, find the shape of
the arch.
21. In equations (4') the integration may be carried through in terms of quadra-
tures if pF is a function of y alone ; and similarly in Ex. 15 the integration may be
carried through if * = 0 and pR is a function of r alone so that the field is central.
Show that the results of thus carrying through the integration are the formulas
J -\/(fpYdyy-C^ J ■y/{JpRdrf-C^
22. A particle falls from rest through the air, which is assumed to offer a resist-
ance proportional to the velocity. Solve the problem with the initial conditions
c = 0, z = 0, i = 0. Show that as the particle falls, the velocity does not increase
indefinitely, but approaches a definite limit V = mg/k.
23. Solve Ex. 22 with the initial conditions v = Vq, x = 0, t = 0, where Vq is
greater than the limiting velocity V. Show that the particle slows down as it falls.
24. A particle rises through the air, which is assumed to resist proportionally to
the square of the velocity. Solve the motion.
25. Solve the problem analogous to Ex. 24 for a falling particle. Show that
there is a limiting velocity V = Vmg/k. If the particle were projected down with
an initial velocity greater than F, it would slow down as in Ex. 23.
26. A particle falls towards a point which attracts it inversely as the square of the
distance and directly as its mass. Find the relation between x and t and determine
the total time T taken to reach the center. Initial conditions t) = 0, x = a, t = 0.
^, a _i2x-a / • _ -l/a\^
— t = -C08 + Vox - x'^ T = 7rk *(-) .
a 2 a \2/
• Ezflroises 17-20 should be worked ah initio by the method by which (4) were derived,
not by applying (4) directly.
GENERAL INTRODUCTION 191
27. A particle starts from the origin with a velocity V and moTes in a mediam
whicti rcHifiUi proportionally to the velocity. Find the reUUona between Telodtj
and dlKtance, velocity and time, and disUnce and time ; alao the limitinir dlntancfl
traversed.
-tt -*i
» = r- to/m, 0 = Fe * , te = mr(l - « - ), mV/k,
28. Solve Ex. 27 under the assumption that the resistance varies is V«.
29. A particle falls t^)ward a point which attracts inversely as the cube of the
diKtiince and directly as the mass. The initial conditions are z = a, v = 0, t = 0.
Show that x*^ = a!^ — W«/a* and the total time of descent is T = o*/ Vl.
30. A cylindrical spar buoy stands vertically in the water. The buoy is prcMed
down a little and released. Show that, if the resistance of the water and air be
neglected, the motion is simple harmonic. Integrate and determine the constants
from the initial conditions x = 0, c = V, < = 0, where x measures the displacement
from the position of equilibrium.
31. A particle slides down a rough inclined plane. Determine the motion. Note
that of the force of gravity only the component ing sin i acts down the plane,
whereas the component rng cos i acts perpendicularly to the plane and develops the
force nmg cos i of friction. Here t is the inclination of the plane and /« is the
coefficient of friction.
32. A bead is free to move upon a f rictionless wire in the form of an inverted
cycloid (vertex down). Show that the component of the weight along the tangent
to the cycloid is proportional to the distance of the particle from the vertex. Hence
detennine the motion as simple harmonic and fix the constants of integration by
the initial conditions that the particle starts from rest at the top of the cycloid.
33. Two equal weights are hanging at the end of an elastic string. One drops
off. Detennine completely the motion of the particle remaining.
34. One end of an elastic spring (such as is used in a spring balance) is attached
rigidly to a point on a horizontal table. To the other end a particle is attached.
H the particle be held at such a point that the spring is elongated by the amount
a and then released, detennine the motion on the assumption that the coefficient
of friction between the particle and the table is /t ; and discuss the possibility of
different cases according as the force of friction is small or large relative to the
force exerted by the spring.
85. Lineal element and differential equation. The idea of a curve
as m(ule up of the ])i)ints upon it is familiar. Points, however, have no
extension and therefore must be regarded not as i)ieces of a curve but
merely as i)ositions on it. Strictly speaking, the pieces of a curve are
the elements A.s- of arc ; but for many purposes it is convenient to re-
place the complicated element A,s by a piece of the tangent to the curve
at some point of the arc A-^, and from this point of view a curve is made
up of an infinite numU'r of infinitesimal elements tangent to it. This
is analogous to the })oint of view by which a curve is regarded as
192 rjFFERENTIAL EQUATIONS
up of an infinite number of infinitesimal chords and is intimately related
to the conception of the curve as the envelope of its tangents (§65).
A point on a curve taken with an infinitesimal portion of the tangent
to the curve at that point is called a lineal element of the curve. These
concepts and definitions are clearly equally available in two or three
dimensions. For the present the curves under dis-
cussion will be plane curves and the lineal elements
will therefore all lie in a plane. "^^ Aa:,y,p)
To specify any particular lineal element three
coordinates x, y, p will be used, of which the two (x, y) determine the
point through which the element passes and of which the third p is
the slope of the element. If a curve f{x^ y) — ^ is given, the slope at
any point may be found by differentiation,
dx dxf dy ^ ^
and hence the third coordinate p of the lineal elements of this particular
curve is expressed in terms of the other two. If in place of one curve
f(Xy y) = 0 the whole family of curves f(x, y) = C, where C is an
arbitrary constant, had been given, the slope p would still be found
from (6), and it therefore appears that the third coordinate of the lineal
elements of such a family of curves is expressible in terms of x and y.
In the more general case where the family of curves is given in the
unsolved form F(x, ?/, C) = 0, the slope p is found by the same formula
but it now depends apparently on C in addition to on x and y. If, how-
ever, the constant C be eliminated from the two equations
F(x,y,C) = 0 and £ + |^i> = 0, (7)
there will arise an equation $ (x, y, p) = 0 which connects the slope p
of any curve of the family with the coordinates (a;, y) of any point
through which a curve of the family passes and at which the slope of
that curve is^. Hence it appears that the three coordinates (x, y^p) of
the lineal elements of all the curves of a family are connected by an equa-
tion *(x, y, p) = 0, just as the coordinates (x, y, z) of the points of a
surface are connected by an equation ^{x, y, z) = 0. As the equation
*(^> y> «) = 0 is called the equation of the surface, so the equation
♦(iC) y> V) = 0 is called the equation of the family of curves ; it is, how-
ever, not the finite equation F(a;, y, C) = 0 but the differential equation
of the family, because it involves the derivative p — dy/dx of y by a?
instead of the parameter C.
GENERAL INTRODUCTION 198
Ak an example of the elimination of a constant, conaide^r tll^ ca«e of the parmbolM
y« = Cz or j^/x = C.
ThQ differentiation of the equation in the second form gives at onoe
- yVx* + 2 vp/z = 0 or y = 2xp
aa the differential equation of the family. In the unsolved form the work la
2vp = 0, y« = 2vpa5, v = 2xp.
The result is, of course, the same in either case. For the family here treated It
makes little difference which method is followed. As a general rule it la perhaps
beKt to solve for the constant if the solution is simple and leads to a simple form
of the function /(z, y) ; whereas if the solution is not simple or the form of the
function is complicated, it is best to differentiate first because the differentiated
e(iuation may be simpler to solve for the constant than the original equation, or
because the elimination of the constant between the two equations can be con-
ducted advantageously.
If an equation * {xy y,p) = 0 connecting the three coordinates of the
lineal element be given, the elements which satisfy the equation may
be plotted much as a surface is plotted ; that is, a pair of values (x, y)
may l)e assumed and substituted in the equation, the equation may then
bo solved for one or more values of j), and lineal elements with these
values of JO may be drawn through the point (x, y). In this manner the
elements through as many points as desired may be found. The de-
tached elements are of interest and significance chiefly from the fact
that they can be assembled into curves f — in fact, into the curves of a
family F(x, y, C) = 0 of which the equation ^(x, y, y>) = 0 is the differ-
ential equation. This is the converse of the problem treated above and
requires the integration of the differential equation * (x, y, jp) = 0 for its
solution. In some simple cases the assembling may be accomplished
intuitively from the geometric properties implied in the equation, in
other cases it follows from the integration of the equation by analytic
means, in other cases it can be done only approximately and by methods
of computation.
As an example of intuitively assembling the lineal elements into curves, take
♦(X, l/,p) = l/V + I/^-r2 = 0 or p=±— — !^'
The quantity Vr* — y* may be interpreted as one leg of a right triangle of which
y is the other leg and r the hypotenuse. The slope of the hypotenuse Is then
± y/ Vr^ — y'^ according to the position of the figure, and the differential eqioatlon
^(<^« 2/« P) = 0 states that the coordinate p of the lineal element which MtiifleH it
is the negative reciprocal of this slope. Hence the lineal element Is perpendicular
to the hypotenuse. It therefore appears that the lineal elements are tangent to cir-
cles of radius r described about points of the x-azis. The equation of these circles la
194 DIFFERENTIAL EQUATIONS
(X — C)2 + J/* = r*, and this is therefore the integral of the differential equation.
The correctness of this integral may be checked by direct integration. For
dx y Vr* — y^
86. In geometric problems which relate the slope of the tangent of a
curve to other lines in the figure, it is clear that not the tangent but
the lineal element is the vital thing. Among such problems that of the
orthogonal trajectories (or trajectories under any angle) of a given family
of curves is of especial importance. If two families of curves are so
related that the angle at which any curve of one of the families cuts
any curve of the other family is a right angle, then the curves of either
family are said to be the orthogonal trajectories of the curves of the
other family. Hence at any point (x, y) at which two curves belonging
to the different families intersect, there are two lineal elements, wie
belonging to each curve, which are perpendicular. As the slopes of two
perpendicular lines are the negative reciprocals of each other, it follows
that if the coordinates of one lineal element are (x, y, p) the coordinates
of the other are (ic, y, — 1/p) ; and if the coordinates of the lineal ele-
ment (x, y, p) satisfy the equation $ (cc, y, p) — 0, the coordinates of the
orthogonal lineal element must satisfy ^ (xj y, — 1/p) = 0. Therefore
the rule for finding the orthogonal trajectories of the curves F(x, y, C)= 0
is to find first the differential equation ^(x, y,p) = ^ of the family, then
to replace phy — 1/p to find the differential equation of the orthogonal
family, and finally to integrate this equation to find the fa.mily. It may
be noted that if F(%) = X (x, y) -\- iY(xj y) is a function ot z = x -{- iy
(§ 73), the families X(xj y) = C and Y(x, y) = K are orthogonal.
As a problem in orthogonal trajectories find the trajectories of the semicuibical
parabolas (x — C)' = y^. The differential equation of this family is found as
3(x-C)2 = 2j^, x-C = (|yp)i, {lyp)^ = y^ or |p = 2/i
This is the differential equation of the given family. Replace jp by — 1/p and
integrate :
— — = y ' or 1 + -pyi = 0 or dx -{■ - y^ dy = 0, and x-\- -y^ = C.
op 2 2 8
Thus the differential equation and finite equation of the orthogonal family are found.
The curves look something like parabolas with axis horizontal and vertex toward
the right.
Given a differential equation * (a*, y, p)= 0 or, in solved form,
p = <t> (x, y) ; the lineal element affords a means for obtaining graphically
and num^^rically an approximation to the solution which passes through
GENERAL INTRODUCTION
195
an assigned point P^ix^t t/o)' ^^^ ^^® value p^ of p at this point may be
computtHl from tlie equation and a lineal element P^P^ may be draim,
the length being taken small. As the lineal element is tangent to the
curve, its end point will not lie upon the curve but will depart from it
by an infinit^isimal of higher order. Next the slope />, of the
element which uatisties the equation and passes
through Pj may be found and the element P^P^
may be drawn. This element will not be tangent
to the desired solution but to a solution lying near
that one. Next the element P^, may be drawn,
and so on. The broken line PJ*^P.J\ -is clearly
an approximation to the solution and will be a better approximation
the shorter the elements P<Pf+i are taken. If the radius of curvature
of the solution at P^ is not great, the curve will be bending rapidly and
the elements must be taken fairly short in order to get a fair approx-
imation ; but if the radius of curvature is great, the elements need not
be tiiken so small. (This method of approximate graphical solution
indicates a method which is of value in proving by the method of
limits that the equation j9 = ^ (ar, y) actually has a solution ; but that
matter will not be treated here.)
rP.(x«iyotP*>
Let it be required to plot approximately that solution of j^p -f z = 0 which
through (0, 1) and thus to find the ordinate for x = 0.5, and the area under
the curve and the length of the curve to this point. Instead of aasuming the lengths
of the successive lineal elements, let the
lengths of successive increments Jx of
X be taken as Jx = 0.1. At the start
Xq = 0, ^0 = 1, and from p = — x/y it
follows that Pq = 0. The increment Sy
of y acquired in moving along the tan-
gent is iy = pSx = 0. Hence the new
point of departure (/,, y^) is (0.1, 1) and
the new slope is p^ = — x^/y^ = — 0.1.
The results of the work, as it is contin-
ued, may be grouped in the table. Hence it appears that the final ordinate is
y = 0.90. By adding up the trapezoids the area is computed as O.iS, and by find-
ing the elements 5s = Vax'-* -|- 3y* the length is found as 0.61. Now the particular
equation here treated can be integrated.
i
Sx
8y
Xi
M
Pi
0
...
0.
1.00
0.
1
0.1
0.
0.1
1.00
-0.1
2
0.1
-0.01
0.2
0.09
-0.2
3
0.1
-0.02
0.3
0.07
-0.81
4
0.1
-0.08
0.4
0.94
-0.48
5
0.1
-0.04
0.6
0.90
...
1/p + X = 0, ydy + xdx
+ y« = C, and hence x* + y* = 1
is the solution which passes through (0, 1). The ordinate, area, and length found
from the curve are therefore 0.87, 0.48, 0.52 respectively. The erroni In the
approximate results to two places are therefore respectively 3, 0, 2 percent. If iz
had been chosen as 0.01 and four places had been kept in the computations, the
errors would have been smaller.
196 DIFFERENTIAL EQUATIONS
EXERCISES
1. In the following cases eliminate the constant C to find the differential equa-
tion of the family given :
(a) x2 = 2 Cy + C72, (iS) y = Ox + Vl - C\
(7) «* - y* -Cx, (3) V = x tan {x + O),
^•'a«-C fta^c ' \dxl xy dx
2. Plot the lineal elements and intuitively assemble them into the solution :
(a) yp + X = 0, 03) xp - y = 0, (7) r^ = 1.
Check the results by direct integration of the differential equations.
3. Lines drawn from the points (± c, 0) to the lineal element are equally in-
clined to it. Show that the differential equation is that of Ex. 1 (e). What are the
curves?
4. The trapezoidal area under the lineal element equals the sectorial area formed
by joining the origin to the extremities of the element (disregarding infinitesimals
of higher order), (a) Find the differential equation and integrate. (^) Solve the
same problem where the areas are equal in magnitude but opposite in sign. What
are the curves ?
5. Find the orthogonal trajectories of the following families. Sketch the curves.
{a) parabolas y^ = 2, Cx, Arts, ellipses 2 x^ ■{■ y^ z=z C .
(/3) exponentials y = Ce*^, Ans. parabolas \ ky^ + x = C.
(7) circles (x — C)* + 2/^ = a^, Ans. tractrices.
(a) x2 - 2/2 =, (72^ (,) cy^ ^ a.8^ (f) a-f + y| = cl,
6. Show from the answer to Ex. 1 (e) that the family is self -orthogonal and
illustrate with a sketch. From the fact that the lineal element of a parabola makes
equal angles with the axis and with the line drawn to the focus, derive the differ-
ential equation of all coaxial confocal parabolas and show that the family is self-
orthogonal.
7. If * (x, y, p) = 0 is the differential equation of a family, show
*(x,y,ff^) = 0 and *(x,!,,f+il)
\ 1 -f- mpj \ 1 - mp)
mp)
are the differential equations of the family whose curves cut those of the given
family at tan-i m. What is the difference between these two cases ?
8. Show that the differential equations
*(|,r,«) = 0 and *(- r^g, r, ^) = 0
define orthogonal families in polar coordinates, and write the equation of the family
which cuts the first of these at the constant angle tan-i m.
9. Find the orthogonal trajectories of the following families. Sketch.
(a) r = cc«, OS) r = C(l - cos^), (7) r = C<t>, («) r^ = C^ cos20.
GENERAL INTRODUCTION 197
10. Recompute the approximate solution of vp + 2 = 0 under th« oonditloni of
the text but with Sx = 0.05, and carry the work to three decimala.
11. Plot the approximate solution otp — xy between (1, 1) and the y-AzU. Take
ax = — 0.2. Find the ordinate, area, and length. Check bj intflfrmdon and
comparison.
12. Plot the approximate solution of p = ~ x throogh (1, 1), takiog te = 0.1 and
following the curve to its intersection with the OB-axlB. Find a]«o the area and the
length.
13. Plot the solution of p = Vx« + y^ from the point (0, 1) to ita Intersection
with the z-axis. Take to = — 0.2 and find the area and length.
14. Plot the solution of p = 8 which starts from the origin into the first quad-
rant (a is the length of the arc). Take «x = 0.1 and carry the work for five steps
tu find the final ordinate, the area, and the length. Compare with the true integral.
87. The higher derivatives ; analytic approximations. Although a
dittereiitial equation <t>(j-, ?/, i/') = 0 does not determine the relation
between x and y without the application of some process equivalent to
intt^gration, it does afford a means of computing the higher derivatives
simply by differentiation. Thus
flfo d^ d^ d^ ,
is an equation which may be solved for y" as a function of ar, y, y';
and y" may therefore be expressed in terms of x and y by means of
♦ (a:, y, y') = 0. A further differentiation gives the equation
which may be solved for y'" in terms of ar, y, y', y"; and hence, by the
preceding results, y'" is expressible as a function of x and y ; and so
on to all the higher derivatives. In this way any property of the inte-
grals of <j>(a;, y, y') = 0 which, like the radius of curvature, is expressi-
ble in terms of the derivatives, may be found as a function of x and y.
As the differential equation *(ic, y, y') = 0 defines y' and all the
higher derivatives as functions of x, y, it is clear that the values of the
derivatives may be found as y^, yo', y^", -at any given point (x^, yj.
Hence it is possible to write the series
y = yo + yi (« - *o) + i y^ (^ - ^0)' + i yJ" (« - * J* + • • • • i^)
If this power series in a; — x^ converges, it defines y as a function of
X for values of x near x^ ; it is indeed the Taylor development of the
198 DIFFERENTIAL EQUATIONS
function y (§ 167). The convergence is assumed. Then
It may be shown that the function y defined by the series actually
satisfies the differential equation ^(x, y, y') = 0, that is, that
for all values of x near x^. To prove this accurately, however, is beyond
the scope of the present discussion ; the fact may be taken for granted.
Hence an analytic expansion for the integral of a differential equa-
tion has been found.
As an example of computation with higher derivatives let it be required to deter-
mine the radius of curvature of that solution of y' = tan {y/x) which passes through
(1, 1). Here the slope y\^^ ^ at (1, 1) is tan 1 = 1.667. The second derivative is
y" = — - = — tan - = sec
ax ax X X x^
From these data the radius of curvature is found to be
_ (1-1- 7/2^f 2/ x2 1
B=^ ^ ^ ' = sec ^ , i?(i,i) = secl = = 3.250.
/ xxy' — y tan 1 — 1
The equation of the circle of curvature may also be found. For as yf^^ j. is positive,
the curve is concave up. Hence (1 — 3.260 sin 1,14- 3.250 cos 1) is the center of
curvature ; and the circle is
(x -I- 1.735)2 4- (y _ 2.757)2 = (3.250)2.
As a second example let four terms of the expansion of that integral of
X tan y' = y which passes through (2, 1) be found. The differential equation may-
be solved ; then
dx \x/ dx^ x2 + 2/2
cPy _ (a;2 + y'i){x -~ l)y^^ + (3y2 - x^)y' - 2a;yy^2 + 20^
dx« (a;2 4. y2)2
Now it must be noted that the problem is not wholly determinate ; for y' is multi-
ple valued and any one of the values for tan-i ^ may be taken as the slope of a
solution through (2, 1). Suppose that the angle be taken in the first quadrant ; then
tan-i i = 0.462. Substituting this in y", we find 2/J2. i) = — 0.0152 ; and hence may
be found i/J^' i) = 0.110. The series for y to four terms is therefore
y = 1 + 0.462 (X - 2) - 0.0076 (x - 2)2 + 0.018 (x - 2)8.
It may be noted that it is generally simpler not to express the higher derivatives in
terms of x and y, but to compute each one successively from the preceding ones.
88. Picard has given a method for the integration of the equation
y' = ^(Xj y) by successive approximations which, although of the highest
theoretic value and importance, is not particularly suitable to analytic
GENERAL INTRODUCTION 199
uses in finding an approximate solution. The method ia this. Let the
ecjuatiou y' = ^(j*, y) be given in solved form, and suppose (a?^, yj is
the point through which the solution is to pass. To find the first
approximation let y be held constant and equal to y^, and integrate the
equation y' = <t»(x, y^. Thus
dy = 4»(x, y^dx'y y=y^+ f <^(a?, y^dx =/j(a;), (9)
where it will be noticed that the constant of integration has been chosen
so that tlie curve i)a88e8 through (x^, y^. For the second approximation
let y have the value just found, substitute this in ^(^, y), and integrate
again. Then
With this new value for y continue as before. The successive deter-
minations of y as a function of x actually converge toward a limiting
function which is a solution of the equation and which passes through
(•*"o» ^0)- ^^ "^^^y ^^ noted that at each step of the work an integration
is required. The difficulty of actually performing this integration in
formal i)i*actice limits the usefulness of the method in such erases. It is
clear, however, that with an integrating machine such as the integraph
the method could be applied as i-apidly as the curves <t>(x,fi(x)) could
he plotted.
To see how the method works, consider the integration of y' = x + y to find the
integral through (1, 1). For the first approximation y = 1. Then
dy = {x+l)dx, y = \x^ + x-{-C, y = \x^ -^ x - \ =f^{x).
From this value of y the next approximation may be found, and then still another :
dy = [X + (ix'a + X- i)]da;, y = ix« + x«- Jx + i =/,(«),
dy = [X +/,(2;)]dx, y = ,»ix* + ix» + Jx^ + Jx + ^.
In this case there are no diflBculties which would prevent any number of appli-
cations of the method. In fact it is evident that if y' is a polynomial in x and y, the
result of any number of applications of the method will be a polynomial in x.
The method of vndetemiined coefficients may often be employed to
advantage to develop the solution of a differential equation into a
series. The result is of course identical with that obtained by the
application of successive differentiation and Taylor's series as ubove ;
the work is sometimes shorter. Let the equation be in the form
y' = <t> (xj y) and assume an integral in the form
y = y,-\- a^{x - x^ + a^(x - x^* -h a,(x -arj« + .- -. (10)
200 DIFFERENTIAL EQUATIONS
Then ^ (a;, y) may also be expanded into a series, say,
^(x, y) = ^0 + ^i(^ - ^o) + ^a(^ - ^o)' + ^8 (^ - ^o)' H- -^
But by differentiating the assumed form for y we have
y* = a^ + 2 a, (X - a:,) + 3 ^3 (a; - x^f + 4 a, (a; - a^,)« + • • • .
Thus there arise two different expressions as series in a; — a;^j for the
function y\ and therefore the corresponding coeflB.cients must be equal.
The resulting set of equations
<^i = \y 2a2 = ^j, ^a^ = A^, 4a^ = ^3, ... (11)
may be solved successively for the undetermined coefficients aj, a^^ a^,
a^f . • . which enter into the assumed expansion. This method is partic-
ularly useful when the form of the differential equation is such that
some of the terms may be omitted from the assumed expansion (see
Ex. 14).
As an example in the use of undetermined coefficients consider that solution of
the equation y" = Vx^ + 3 y^ which passes through (1, 1). The expansion will pro-
ceed according to powers of x — 1, and for convenience the variable may be changed
to t = X — 1 so that
are the equation and the assumed expansion. One expression for ]/ is
/ = aj + 2 ttgi + 3a8«2 + 4a^t8 + . . ..
To find the other it is necessary to expand into a series in t the expression
y' = V(l + 0' + 3 (1 + a^i + a^t^ + a^t*f.
If this had to be done by Maclaurin's series, nothing would be gained over the
method of § 87 ; but in this and many other cases algebraic methods and known
expansions may be applied (§ 32). First square y and retain only terms up to the
third power. Hence
y' = 2 Vi + i(i + Sa{)t + i(l + 6a^ + 3 af ) t^ + f (a^a^ + a,) i^.
Now let the quantity under the radical be called 1 + A and expand so that
y' = 2 VrM = 2(1 + i/i - I A2 + ^^^8).
Finally raise k to the indicated powers and collect in powers of t. Then
1^ = 2+1(1 + 80,)
«2
+ i(l + 6a2 + 8ai^)
-A(l + 3ai)'»
+ i {a^a^ + 03)
-^(l + 3a0(l+6a, + 8ai2)
GENERAL INTRODUCTION 201
Hence the succeasive equationj} for determining the ooeffidenta an a| = 2 ^^
2a, = i(l + 8ai)ora, = |,
8a, = i(l + 6a, + 8a?)- ^(1 + 8a,)« or a, = H.
4a, = I {a^a^ + a,) - x»j(l + 8aj)(l + 6a, + 8a«) + ,«|(1 + «a,)» or a, s HI-
Therefore to five terms the expansion desired is
1/ = 1 + 2 (X - 1) + Hx - i)« + H («-!)■ + Hi (»- 1)*.
The methcxls of developing a solution by Taylor's series or by un-
determined coefficients apply equally well to equations of higher order.
For example consider an equation of the second order in solved form
y" = ^ (xy y, y ') and its derivatives
Evidently the higher derivatives of 1/ inay be obtained in terms of x,
y, y' ; and y itself may be written in the expanded form
y = % + y;(« - ««) + i yo (* -*«)' + 4 yn'' - 'd* n -i\
+ ^iyi-(^-a;,)*+ ••-, ^"^
where any desired values may be attributed to the ordinate y^ at which
the curve cuts the line x = x^^ and to the slope j/q of the curve at that
point. Moreover the coefficients y^', yj", • • • are determined in such a way
that they depend on the assumed values of y^ and yj. It therefore is
seen that the solution (12) of the differential equation of the second
order really involves two arbitrary constants, and the justification of
writing it as F(x, y, C^, C,^ = 0 is clear.
In following out the method of undetermined coefficients a solution
of the equation would be assumed in the form
y = ^0+ ^o(-^ - ^o) + «2(^ - ^o)'+ «.(^ - ^o)*+ «4(^- ^o)*+ •• •» (13)
from which y' and y" would be obtained by differentiation. Then if the
series for y and y' be substituted in y" = <^ (j-, y, y') and the result
arranged as a series, a second expression for y" is obtained and the
comparison of the coefficients in the two series will afford a set of equa-
tions from which the successive coefficients may be found in terms of
y^ and yj by solution. These results may clearly be generalized to the
case of differential equations of the nth order, whereof the solutions
will de})end on n arbitrary constants, namely, the values assumed for
y and its first n — l derivatives when x = x^.
202 DIFFERENTIAL EQUATIONS
EXERCISES
1. Find the radii and circles of curvature of the solutions of the following equa-
tions at the points indicated :
(a) y' = Vx2 + y-^ at (0, 1), (/S) yy' + x = 0 at (Xq, y^).
2. Find yj;; ^ = (5 V2 - 2)/4 if y' = Vx^ + y^.
3. Given the equation y^y"^ + xyy"^ - yy" ■¥ x^ = 0 of the third degree in y' so
that there will be three solutions with different slopes through any ordinary point
(X, y). Find the radii of curvature of the three solutions through (0, 1).
4. Find three terms in the expansion of the solution of y' =3 e^ about (2, |).
5. Find four terms in the expansion of the solution of y =log sin xy about (^ ir, 1).
6. Expand the solution of y' = xy about (1, y^) to five terms.
7. Expand the solution of y' = tan (y/x) about (1, 0) to four terms. Note that
here x should be expanded in terms of y, not y in terms of x.
8. Expand two of the solutions of y^y"^ + xyy'^ — yy' + x^ = 0 about (— 2, 1)
to four terms.
9. Obtain four successive approximations to the integral of y'=xy through (1, 1).
10. Find four successive approximations to the integral of y' = x + y through
(0, Vo)'
11. Show by successive approximations that the integral of y' = y through (0, y^)
is the well-known y = yoC*.
12. Carry the approximations to the solution of y' = — x/y through (0, 1) as
far as you can integrate, and plot each approximation on the same figure with the
exact integral.
13. Find by the method of undetermined coefficients the number of terms indi-
cated in the expansions of the solutions of these differential equations about the
points given :
{a) y" = Vx + y, five terms, (0, 1), (/3) y' = Vx + y, four terms, (1, 3),
(7) 2/' = aJ + y, n terms, (0, y^), («) y' = Vx^ + y2, four terms, (f, \).
14. If the solution of an equation is to be expanded about (0, y^) and if the
change of x into — x and y' into — y' does not alter the equation, the solution is
necessarily symmetric with respect to the y-axis and the expansion may be assumed
to contain only even powers of x. If the solution is to be expanded about (0, 0)
and a change of x into — x and y into — y does not alter the equation, the solution
is symmetric with respect to the origin and the expansion may be assumed in odd
powers. Obtain the expansions to four terms in the following cases and compare
the labor involved in the method of undetermined coefficients with that which
would be involved in performing the requisite six or seven differentiations for the
application of Maclaurin's series :
{a) V = , about (0, 2), (/3) y' = sin xy about (0, 1),
Vx2 + y2
(7) y' = ew about (0, 0), (5) y' = x»y -f- xy» about (0, 0).
15. Expand to and including the term x^ :
(a) y" = y^ ^-xy about Xq = 0, y^ = a^,, y^ = ay^ (by both methods),
(/S) xy" + y' + y = 0 about x^ = 0, y^ = a^, y^= -a^ (by und. coeffs.).
CHAPTER Vm
THE COMMONER ORDINARY DIFFERENTIAL EQUATIONS
89. Integration by separating the variables. If a differential equa^
tion of the first order may be solved for y' so tliat
l/' = <t>(x,y) or M(x,y)dx-irN(x,y)dy = 0 (1)
(where the functions ^, M, N are single valued or where only one spe-
cific branch of each function is selected in case the solution leads to
multiple valued functions), the differential equation involves only the
first power of the derivative and is said to be of the first degree. If,
furthermore, it so happens that the functions <^, 3/, N are products of
functions of x and functions of y so that the equation (1) takes the form
y = <^i(^)«^.(y) or M^(x)MJiy)dx^N^{x)Njiy)dy = 0, (2)
it is clear that the variables may be separated in the manner
and the integration is then immediately performed by integrating each
side of the equation. It was in this way that the numerous problems
considered in Chap. VII were solved.
As an example consider the equation yy' + xy^ = x. Here
ydy-{-x{y^^\)dx = Q or -i?z?L + xdx = 0,
and \ log (2/2 _ 1) ^. J a;2 _ (; or {y^ - l)^ = C.
The second form of the solution is found by taking the exponential of both sidM
of the first form after multiplying by 2.
In some differential equations (1) in which the variables are not
immediately separable as above, the introduction of some change of
variable, whether of the dei)endent or independent variable or both,
may lead to a differential equation in which the new variables are sepa-
rated and the integration may be accomplished. The selection of the
proper change of variable is in general a matter for the exercise of
ingenuity ; succeeding jxiragraphs, however, will point out some special
204 DIFFERENTIAL EQUATIONS
types of equations for which a definite type of substitution is known
to accomplish the separation.
As an example consider the equation xdy — ydx = x Vx'^ + y^ dx, where the varia-
bles are clearly not separable without substitution. The presence of Vx^ + y^
suggests a change to polar coordinates. The work of finding the solution is :
X = r cos $^ y = r sin ^, dx = cos 6dr — r sin Odff, dy = sin ddr + r cos OdO ;
then xdy — ydx = r^dO, x Vx^ + y^ dz = r^ cos $d (r cos 6) .
Hence the differential equation may be written in the form
r^dO = r^ cos Od (r cos 0) or sec OdO = d (r cos ff),
and log tan a tf + iir) = r cos ^ + C7 or log "^ ^^^ =x+ G.
cos 6
■y/x^ _L 7/2 I y
Hence *^ ^ = Cef^ (on substitution for ^).
X
Another change of variable which works, is to let y = vx. Then the work is :
x{vdx + xdv) — vxdx = x^Vl + vl^dx or du = Vl + r^dx.
dt)
Then > = dx, sinh-it) = x + C, y = x sinh (x + C).
Vl + ^
This solution turns out to be shorter and the answer appears in neater form than
before obtained. The great difference of form that may arise in the answer when
different methods of integration are employed, is a noteworthy fact, and renders a
set of answers practically worthless ; two solvers may frequently waste more time
in trying to get their answers reduced to a common form than each would spend in
solving the problem in two ways.
90. If in the equation y' = <l> (», y) the function <^ turns out to be
<l>(y/x)f a function of y/x alone, that is, if the functions M and N are
homogeneous functions of Xj y and of the same order (§ 53), the differ-
ential equation is said to be homogeneous and the change of variable
y = vx OT X = vy will always result in separating the variables. The
statement may be tabulated as :
if ^ = Jy\ c,,w,-^-„^. f y = '^^
dx
(Ay substitute I ^"""^ (3)
\x/' \oTx = vy. ^
A sort of corollary case is given in Ex. 6 below.
As an example take y(l + e^jdx + ^(y - x)dy = 0, of which the homogeneity
Is perhaps somewhat disguised. Here it is better to choose x = vy. Then
0 and dx = vdy + ydv.
0 or *? + L±i:d„ = o.
y tj + c*'
at
C or x-\-y0i =C,
(l + C)dx + c«'(l-t))dy
Hence
(t> + c«')dy + y(l + c^)dtJ
Hence
logy + log(t» + c«')
COMMONER ORDINARY EQUATIONS 206
If the differential equation may be arranged bo that
% + W)y = U^)r or g+r,(y)>!=r.(y)^, (4)
where the second form differs from the first only through the inter-
change of x and y and where X^ and A'^ are functions of x alone and
Kj and Y^ functions of y^ the equation is called a Bernoulli equation; and
in particular if n = 0, so that the dei)endent variable does not occur on
the riglit-hand side, the equation is called linear. The substitution
which separates the variables in the respective cases is
y = ve-A<'>*'' or x = ve-'f^^^^'^. (5)
To show that the separation is really accomplished and to find a general
formula for the solution of any Bernoulli or linear equation, the sub-
stitution may be carried out formally. For
The substitution of this value in the equation gives
dv r,^_ „ _ _r,^_ dv
Hence
dx ^ if *
or
t;i- - = (1 - 7i) Ix/^- "> A'*^«te, when n ^ 1,*
yi- " = (1 - n) ef- -i)/'^«'''r f X/^- »> A*" dx\ . (6)
There is an analogous form for the second form of the equation.
The equation (x^y^ + xy) dy = dx may be treated by this method by writing it m
dx
4/x = y^x^ 80 that F. = — y. y, = y*. n = 2.
dy
Then let x = veS- *^''' = wr* "^.
-^ dx do ly* , Ay< iy* do iv*
Tljen l/x = — e' + vye^ — yve^ = -r «
dy dy dy
and ^e^»^ = l/«o^e^ or ^ = y»e^%,
dy 0*
and - - =(y« - 2)c^*^ + C or 1 = 2 - y« + Ce"^*'.
0 X
This result could have been obtained by direct substitution in the formula
xi- = (1 - n)c<-»^/ »•>''»'[ Jr/^->/''«*"dy].
but actually to carry the method through is far more instructive.
* If n= 1, the variables are separated in the original equation.
206 DIFFERENTIAL EQUATIONS
EXERCISES
1. Solve the equations (variables immediately separable) :
(a) (1 + X) y + (1 - 2/)a;/ = Oi Ans^_xy = Cev-
(/3) a{xdv + 2ydx) = xydy, (7) Vl- x^dy +Vl- y^dx = 0,
(5) (1 + y*^) dx - (2/ + Vl + y)(l + «)^ dy = 0.
2. By various ingenious changes of variable, solve :
(a) (x + y) V = a^» ^'^^ X + y = a tan (jz/a + C).
{P) (X - 2/2) dx + 2 xydy = 0, (7) xdy - ydx = {x^ + y^) dx,
(3) y' = x-y, (e) yy' + y2 + a; + 1 = 0.
3. Solve these homogeneous equations :
(or) (2Vxy-x)y' + y = 0, ^ns. Vx/y + log y = (7.
y
(/S) X€* + y — aJ/ = 0, ^ns. y + x log log C/x = 0.
(7) («^ + V^) dy = a;ydx, (8) xy'-y= Vx^+V^.
4. Solve these Bernoulli or linear equations :
(cr) y' + y/x = y2, " ^ns. xy log Cx + 1 = 0.
(/S) y' — y CSC X = cosx — 1, Ans. y = sin x + C tan ^ x.
(7) xy" ■\-y = y^ log x, ^ns. y-i = log x + 1 + Cx.
(3) (1 + y^) dx = (tan- 1 y - X) dy, (e) ydx + (axV - 2 x) dy = 0,
(f) xy' - ay = X + 1, (77) yy' + ^ y^ = cosx.
5. Show that the substitution y = vx alv^ays separates the variables in the
homogeneous equation y^ = <f> (y/x) and derive the general formula for the integral.
6. Let a differential equation be reducible to the form
dy _ /g^x + h^y + cA a^b^ — a„\ ^0,
dx ~ VttgX + h^y + Cg/ * or a^\ - a^\ = 0.
In case a^^ — a^b^ -^ 0, the two lines a^x + 6j2/ + Cj = 0 and a^ ■\-bc^y ^ c^ — ^
will meet in a point. Show that a transformation to this point as origin makes
the new equation homogeneous and hence soluble. In case a^ftg — ^^\ — 0? t.he
two lines are parallel and the substitution z = a^x + b^y or z = a^x + b^y will
separate the variables.
7. By the method of Ex. 6 solve the equations :
(a) (32/-7x + 7)dx + (72/-3x + 3)d2/ = 0, Ans. (y - x + l)2(?y + x- 1)6 = C.
(/3) (2x + 3j/-6)/ + (3x + 2y-5)=0, (7) (4x+32/+l)dx+(x + y+l)dy=0,
(«) (2x + y) = y'(4x + 22/-l), (') T = L'"'' l~ 1 S '
dx \2 X — 2 y + 1/
8. Show that if the equation may be written as jt/'(xy) dx + xg (xy) dy = 0,
where /and g are functions of the product xy, the substitution v = xy will sepa-
rate the variables.
9. By virtue of Ex. 8 integrate the equations :
{a) {V + 2xy2 _ xV)dx + 2x^ydy = 0, Ans. x + x^y = C(l- xy).
(/S) (y + xy«) dx + (X - x^y) dy = 0, (7) (1 + xy) xy^dx + {xy - 1) xdy = 0.
COMMONER ORDINARY EQUATIONS 207
10. By any method that is applicable solve the following. If more than one
Kthfxl is applicable, state what methods, and any apparent r«a«ont for cbooa-
,i.K one:
(a) y' + ycoiix = ir«in2», (/J) (2z*y + 8y*)<ix =:(x'-|' Ssy*)d^,
(y) (4x + 2y-l)/ + 2x + y+l = 0, («) vv^_±jn^ = x,
(t) y' fiin'y + fiin a; cos y = sin x, (f) Va« + x*(l — /) = X + y,
(,) (x»y» + x'T/* + atj^ + l)y + (a^ - xV - «y + l)«y', (^ y' = alnix -y),
V
(() xydy - y*dx = (x + y)«e"'dx, («) (1 - y«)dx = axy(x + \)dy.
91. Integrating factors. If the equation Mdx -h ^^y = 0 by a suitar
ble rearrangriiuiit of the terms can be put in the form of a sum of total
differentials of certain functions w, v, • • • , say
du + dv -{ = 0, then w -f r H = C (7)
is surely the solution of the equation. In this case the equation is called
an exact differential equation. It frequently happens that although the
equation cannot itself be so arranged, yet the equation obtained from
it by multiplying through with a certain factor ft (a:", y) may be so
arranged. The factor /x (a;, y) is then called an integrating factor of the
given equation. Thus in the case of variables separable, an integrating
factor is 1/M^N^ ; for
^[A/.M,<^ + ^.iV,rf,] = ^</.+M,, = 0; (8)
and the integration is immediate. Again, the linear equation may be
treated by an integrating factor. Let
dy H- X^ydx = X^dx and fi = e/^**^ ; (9)
then «'/-^>''' dy -f A'^e/-^'''' ydx = ef^^''' X^dx (10)
d[yeS''^^^'\±=eS'^^^X^dx, and yeS^^'^= jeS^^'^X^x. (11)
In the case of variables separable the use of an integrating factor is
therefore implied in the process of separating the variables. In the
case of the linear equation the use of the integrating factor is somewhat
shorter than the use of the substitution for separating the variables.
In general it is not possible to hit upon an integrating factor by inspec-
tion and not practicable to obtain an integrating factor by analysis, but
the integration of an equation is so simple when the factor is known,
and the equations which arise in practice so frequently do have simple
integrating factors, that it is worth while to examine the equation to
see if the factor cannot be determined by inspection and trial. To aid
in the work, the differentials of the simpler functions such as
or
208 DIFFERENTIAL EQUATIONS
da^^xdy-^- ydx, ic^(a^ -\-f)=xd^ -\- ydy,
y^xdy-ydx ^ ^^.^^ ^ yj^-xdy
X x^ y x^-\-f ' ^ ^
should be borne in mind-
Consider the equation (x^e"^ — 2 mxy^) dx + 2 mx^ydy = 0. Here the first term
jc*e=^ will be a differential of a function of x no matter what function of x may be
assumed as a trial /x. With /u = 1/x* the equation takes the form
^ + 2mm-m = d^ + mdt 0.
\ X2 X* / X2
The integral is therefore seen to be e=^ + my^/x^ = C without more ado. It may-
be noticed that this equation is of the Bernoulli type and that an integration by
that method would be considerably longer and more tedious than this use of an
integrating factor.
Again, consider {x -\- y)dx ^ {x — y)dy = 0 and let it be written as
xdx + ydy + ydx — xdy = 0 ; try fi .= l/{x^ -^ y^) -,
xdx + ydy ydx-xdy^ ^^ 1 d log (x^ + 2/^) + d tan-i ? = 0,
x2 + 2/2 x2 + y2 2 ^^ ^^ f^ y
and the integral is log Vx* + y2 ^ tan-i {x/y) = C, Here the terms xdx + ydy
strongly suggested x^ + y^ and the known form of the differential of tan-i {x/y)
corroborated the idea. This equation comes under the homogeneous type, but the
use of the integrating factor considerably shortens the work of integration.
92. The attempt has been to write Mdx -\- Ndy or fi (Mdx + Ndy)
as the sum of total differentials du -\- dv -] , that is, as the differential
dF of the function u -\- v -\ , so that the solution of the equation
Mdx + Ndy = 0 coidd be obtained as F = C When the expressions
are complicated, the attempt may fail in practice even where it theoreti-
cally should succeed. It is therefore of importance to establish condi-
tions under which a differential expression like Pdx -f Qdy shall be the
total differential dF of some function, and to find a means of obtaining
F when the conditions are satisfied. This will now be done.
dF dF
Suppose Pdx -f Qdy = dF = -^ dx -\- -k- dy -j (13)
then p = ^ 0 = — gP gQ _ d^F
dx dy dy dx dxdy
Hence if Pdx -f Qdy is a total differential c?F, it follows (as in § 52) that
the relation /^ = Qi must hold. Now conversely if this relation does
hold, it may be shown that Pdx -f Qdy is the total differential of a
function, and that this function is
COMMONER ORDINARY EQUATIONS 209
r c ^*>
or F= j Q(x,y)dy-\- j P(x, y;)dx,
where the fixed value x^ or y^ will naturally be so chosen as to simplify
the integrations as much as possible.
To show that these expressions may be taken as F it is merely neces-
sary to compute their derivatives for identification with P and Q. Now
These differentiations, applied to the first form of F, require only the
fact that the derivative of an integral is the integrand. The first turns
out satisfactorily. The second must be simplified by interchanging the
order of differentiation by // and integration by x (Leibniz's Rule,
S 119) and by use of the fundamental hypothesis that i^ = Q^.
^j Pdx+Q(x^,y)=j^ —dx-^Q(x^,y)
=J ^dx^Q{x^,y)=Q{x,y)
-\-Q(x,,y)=Q(x,y).
The identity of P and Q with the derivatives of F is therefore estab-
lished. The second form of F would be treated similarly.
Show that (x* + log y)dx-\- x/ydy = 0 is an exact differential equation and obtain
the solution. Here it is first necessary to apply the test P^ = ^ . Now
— (x* + log y) = - and = -•
ty^ * ' y bxy y
Hence the test is satisfied and the integral is obtained by applying the formula ;
j^ V + logy)dx + J ?dy = lx» + xlogy = C
j^' - dy + J(x« + log l)dx = X log y + ix» = C.
» or
It should be noticed that the choice of x^ = 0 simplifies the integration in the flrrt
case because the substitution of the lower limit 0 Is easy and becaoae the •eoond
f integral vanishes. The choice of y^ = 1 introduces oorreeponding almpUfloationi ii
I the second case.
210
DIFFERENTIAL EQUJlTIONS
Derive the partial differential equation which any integrating factor of the differ-
ential equation Mdx + Ndy = 0 must saiisfy. If n is an integrating factor, then
d/iM_d/jLN
dy ~ dz '
tuMdx -V fiNdy = dF and 1^=^ =
Hence
dy dx \dx dy J
(15)
is the desired equation. To determine the integrating factor by solving this equa-
tion would in general be as difficult as solving the original equation; in some
iSpecial cases, however, this equation is useful in determining fi.
93. It is now convenient to tabulate a list of different types of dif-
ferential equations for which, an integrating factor of a standard form
can be given. With the knowledge of the factor, the equations may
then be integrated by (14) or by inspection.
Equation Mdx -f Ndy = 0 :
I. Homogeneous Mdx -j- Ndy = 0,
II. Bernoulli dy ■+■ X^ydx = X^dx,
III. M=yf(xy)y N = xg(xy)y
dM dN
Factor fi :
1
Mx-{- Ny
y-n^a-
n)JXidx^
1
IV. If
V. If
dy
dx
N
dN dM
dx dy
M
=/(»>),
f(y),
VL Type xfy^(mydx + nxdy) = 0,
VII. afy^(mydx + nxdy) -f- x"i\^{jpydx -f qxdy)
0,
Mx — Ny
eSfWy.
j^km-l-aykn-l-li^
\k arbitrary.
\k determined.
The use of the integrating factor often is simpler than the substitu-
tion y = vx in the homogeneous equation. It is practically identical
with the substitution in the Bernoulli type. In the third type it is
often shorter than the substitution. The remaining types have had no
substitution indicated for them. The proofs that the assigned forms
of the factor are right are given in the examples below or are left as
exercises.
To show tliat n = {Mz + Ny)-^ is an integrating factor for the homogeneous
CMe, it is poMible simply to substitute in the equation (16), which fi must satisfy,
and show that the equation actually holds by virtue of the fact that M and N are
COMMONEK ORDINAEY EQUATIONS 211
inogeneous of the same degree, — tbl« fact being uaed to simplify the remit by
.UiT'H Formula (80) of { 68. But It is easier to proceed directly to show
= 1(^!L\ or 1/1_L_UA/1_^V where * = ^
) or 1/1_L_)=A(1-^V where ^ =
/ d|/Vcl + 0/ 2B5\yl + W
V/x -^ Ny dx \Mz + Ny/ dy\tl-\-^J etr \y 1 + ^/ Jtfx
( hving to the homogeneity, ^ is a function of y/z alone. Differentiate.
a /I 1 \ 1 ii>' 1^1 »^ -y^^l^ » V
ai/ Vc 1 + 0/ X (1 + 0)» X y (1 + ^)« ' x« 5x \v 1 + ^/
this is an evident identity, the theorem is proved.
I'o find the condition tliat the integrating factor may be a function of x only
I to find the factor when the condition is satisfied, the equation (15) which m
li. s may be put in the more compact form by dividing by /i.
j,l?f_jvl?e = ?^-?^ or Ml^2§Jt-N'J2^ = ^Ji-'Ji. (16-)
fidy fidx dx ty dy dx ^ dy
w if M (&nd hence log /a) Is a function of x alone, the first term vanishes and
l^ = ^^9^=/(x) or logM= r/(x)dx.
dx N J
I hi> < >t;ibli8he8 the rule of type IV above and further shows that in no other csm
1 M ^f a function of x alone. The treatment of type V is clearly analogous.
Integrate the equation x*y{iiydx + '2xdy) + x^(4ydx + 3xdy) = 0. This U of
;»* VII ; an integrating factor of the form /* = xPy will be assumed and the ex-
lumts p, <r will be determined so as to satisfy the condition that the equation be
txact differential. Here
P = M3f = 3xP + V + * + 4xP + V+^ Q = /i^' = 2xP + »i^+» + 8x<» + »y».
Then P; = 3(<r + 2)xP + *i/^+i + 4(<r + 1)xP + V
= 2(p + 5)xP + V+^ + 3(p + 3)xP + V= Q;.
Hence if 3(«r + 2) = 2(p + 6) and 4(«r + 1) = 3(/) + 8),
the relation P'^= Qx will hold. This gives <r = 2, p = 1. Hence /& = xy*,
tnd f'iSx^y* + 4x^y»)dx + J* Ody = J x»y* + x«y« = C
Is the solution. The work might be shortened a trifle by dividing through in the
first place by x^. Moreover the integration can be performed at sight without the
use of (14).
94. Several of the most important facts relative to integrating factors
and solutions of Mdx -f- AVy = 0 will now be stated as theorems and
the proofs will be indicated below.
1. If an integrating factor is known, the corresponding solution may
be found ; and conversely if the solution is known, the corresponding
integrating factor may be found. Hence the existence of either implies
the existence of the other.
2. U F = C and c; = C are two solutions of the equation, either most
be a function of the other, as G = ^{F)] and any function of either is
212 DIFFERENTIAL EQUATIONS
a solution. If /a and v are two integrating factors of the equation, the
ratio /i/v is either constant or a solution of the equation ; and the jjrod-
uct of II by any function of a solution, as fi^(F)y is an integrating fax^
tor of the equation.
3. The normal derivative dF/dn of a solution obtained from the
factor fi is the product fi -y/W+W^ (see § 48).
It has already been seen that if an integrating factor >x is known, the corre-
sponding soUition F = C may be found by (14). Now if the solution is known, the
equation
dF = F'^dx + F'ydy = /* {Mdx + Ndy) gives F^ = fiM, F^ = tiN;
and hence /* may be found from either of these equations as the quotient of a
derivative of F by a coefficient of the differential equation. The statement 1 is
therefore proved. It may be remarked that the discussion of approximate solutions
to differential equations (§§ 86-88), combined with the theory of limits (beyond the
scope of this text), affords a demonstration that any equation Mdx + Ndy = 0,
where M and N satisfy certain restrictive conditions, has a solution ; and hence it
may be inferred that such an equation has an integrating factor.
If A* be eliminated from the relations F^ = fxM, Fy = ^jlN found above, it is seen
that
MF^-NF^ = 0, and similarly, 3f G; - iV^G^ = 0, (16)
are the conditions that F and G should be solutions of the differential equation.
Now these are two simultaneous homogeneous equations of the first degree in M
and N. If M and N are eliminated from them, there results the equation
^X-^X = 0
= J{F, G) = 0, (160
which shows (§ 62) that F and G are functionally related as required. To show
that any function * (F) is a solution, consider the equation
3f*; - N^;, = (MF'y -NF'^) *'.
As F is a solution, the expression ifF^—iV^F^ vanishes by (16), and hence M^y—N^'^
also vanishes, and 4> is a solution of the equation as is desired. The first half of 2
is proved.
Next, if /A and v are two integrating factors, equation {!(/) gives
j^aiogM j^giogM^j^aiogv j^aiogv ^^ ^aiog^A ^b\ogn/^^^
dy dx dy dx dy dx '
On comparing with (16) it then appears that log (n/v) must be a solution of the
equation and hence fi/v itself must be a solution. The inference, however, would
not hold if /i/y reduced to a constant. Finally if /* is an integrating factor leading
to the solution F = C^ then
dF = M {Mdx + Ndy), and hence m* {F) {Mdx + Ndy) = d f* (F) dF.
It therefore appears that the factor m* {F) makes the equation an exact differen-
tial and must be an integrating factor. Statement 2 is therefore wholly proved.
COMMONER ORDINARY EQUATIONS 218
The third proposition i« proved simply by dlflerenti&tion and wihrtttutfon. Foi
dF_5F&5e£dy__ dz ^.djf
dn" dx dn dy dn dn dn
And if r denotes the inclination of the curve F = C, it follows that
dy M , dv N _ dx M
tanT = -^ = » Bin T = -^ = —COST— —
dx N dn VJf' + N* <*» VJf* f J^
Hence dF/dn = n VM'^ + JV^^ and the proposition is proved.
EXERCISES
1. Find the integrating factor by inspection and integrate:
(a) xdy ^ydx = (x* + V^) dx, (fi) {y^ -xy)dx-\- x^y = 0,
(7) ydx — xdy + logxdx = 0, (S) y(2xy + C)dx — fFdy = 0,
(») (1 + xy)i/dx + (1 - x|/)xdy = 0, (0 (x - y2)dx + 2x^1/ = 0,
(n) (ly" + y) dx - xdy = 0, (^) a (xdy + 2 ydx) = xj/dy,
(« ) {x^ + 1/^) (a^ + ydy) + Vl + (x« + y^) {ydx - xdy) = 0,
(k) x^ydx - (x« + i/») dy = 0, (X) xdy - ydx = zVx*-y«dy.
2. Integrate these linear equations with an integrating factor :
(a) / + ay = sin fcx, (/3) y' + y cot x = secx,
(7) (x+l)y'-2y = (x+l)\ («) (H-x^)y' + y :^ c«^»',
and 05), (a), (f) of Ex. 4, p. 20«.
3. Show that the expression given under II, p. 210, is an integrating factor foi
the Bernoulli equation, and integrate the following equations by that method :
(a) y' — y tan X = y* sec x, 03) 3 y*/ + y» = x — 1,
(7) y' -^-y cosx = y«sin 2x, (3) dx + 2xydy = 2ax*y»dy,
and (a), (7), (e), (1?) of Ex. 4, p. 206.
4. Show the following are exact differential equations and integrate :
{a) (3 x'* + 6 xy2) (Lc + (8 x'^y + 4 y^) dy = 0, (/3) sin x cos ydx + cos x sin ydy = 0,
(7) (6x-2y + l)dx4-(2y-2x-3)di/ = 0, («) (x» + 3 xy^) dx + (y« + 8 x*y) dy = 0.
2xyjMdx + t::i^dy = 0, (f) (l + c^dx + c^(l - -\dy = ©.
y y^ \ vJ
(if) e»(x« + y2 + 2x)dx+ 2yef*dy = 0, (^) (ysinx - l)dx + (y- cosx)dy = 0.
5. Show that {Mx— Ny)-^ is an integrating factor for type III. Determine
the integrating factors of the following equations, thus render them exact, and
Integrate:
(a) (y + x)dx + xdy = 0, ifi) (y» ^ xy)dx -^ x»dy = 0,
(7) (x« ±y^) dx - 2 xydy = 0, (3) (x«j/* + xy) ydx + (xV - l)«'y = ^
{*) (Vxy-l)xdy-(Vxy+l)ydx = 0, (f) x»dx + (8x«y + 2y»)dy = 0,
and £x8. 3 and 0, p. 206.
6. Show that the factor given for type VI is right, and that the form irfven fo»
type VII is right if k satisfies k{qm — pn) = q{a - y) - p{fi — 8).
214 DIFFERENTIAL EQUATIONS
7. Integrate the following equation^ of types IV- VII :
(a) (y* + 2j/)dx + (x/ + 22/*-4x)dy=±0, (/S) (a;2 + 2/2 + 1) dx - 2 asydy = 0,
(7) (3x2+6x2/ + 3y2)dx+(2x2 + 3a;2/)dy = 0, (5) {2 x^y^ + y) - {x^y - S x) y" = 0,
(e) (2x2y-3y*)dx + (3x3 + 2x2/3)di/ = 0,
16 (2 - 2/0 sin (3x - 2 2/) + y' sin (x - 2 ?/) = 0.
8. By virtue of proposition 2 above, it follows that if an equation is exact and
homogeneous, or exact and has the variables separable, or homogeneous and under
types IV-VII, so that two different integrating factors may be obtained, the solu-
tion of the equation may be obtained without integration. Apply this to finding
the solutions of Ex. 4 (^3), (5), (7) ; Ex. 5 (a), (7).
9. Discuss the apparent exceptions to the rules for types I, III, VII, that is,
when Mx -}- Ny = 0 or Mx — Ny = 0 or qm — pn = 0.
10. Consider this rule for integrating Mdx + Ndy=0 when the equation is known
to be exact : Integrate Mdx regarding y as constant, differentiate the result regard-
ing y as variable, and subtract from N ; then integrate the difference with respect
to y. In symbols,
C = f{Mdx + Ndy) = fMdx + fl^- — f Mdx\dy.
Apply this instead of (14) to Ex. 4. Observe that in no case should either this
formula or (14) be applied when the integral is obtainable by inspection.
95. Linear equations with constant coefficients. The type
«og + «.£! + - + «.-.2 + «». = ^(-) (17)
of differential equation of the nth. order which is of the first degree in
y and its derivatives is called a linear equation. For the present only
the case where the coefficients a^, a^, •••, a„_i, a^ are constant will be
treated, and for convenience it will be assumed that the equation has
been divided through by a^ so that the coefficient of the highest deriva-
tive is 1. Then if differentiation be denoted by D, the equation may be
written symbolically as
(D" -f a^ir-' -f . . . + «_^D 4. a^) y = X, V (17')
where the symbol D combined with constants follows many of the laws
of ordinary algebraic quantities (see § 70).
The simplest equation would be of the first order. Here
■^-a^y = X and y = e°i^ Ce-''^='Xdx, (18)
as may be seen by reference to (11) or (6). Now it D — a^ be treated
as an algebraic symbol, the solution may be indicated as
(X>-a^y = .Y and y = ^^x, (18')
COMMONER ORDINARY EQUATIONS 215
where the operator (/> — ftj)"* is the inverse of D — a^. The lolatioii
which h'dH just been obtained shows that the interpretiUion which must
be assigned to the inverae operator is
-J—(,) = ef^Je-^(,)dx. (19)
where («) denotes the function of x upon which it operates. That the
integrating operator is the inverse of D — a^ may be proved by direct
differentiation (see Ex. 7, p. 152).
This operational method may at once be extended to obtain the solu-
tion of equations of higher order. For consider
^ + aj^-h«^ = ^ or (D^^a^D-^a;)y = X. (20)
Let a, and a^ be the roots of the equation L^ -f a^D -f o, = 0 so that
the differential equation may be written in the form
liy'-(a, + a;)D-\-a^a^y==X or (Z)- a^)(/) - a^y = X (20')
The solution may now be evaluated by a succession of steps as
(I)-a;)y = -^^ X = e^'^'Je-'^'Xdx,
or y = e'^ Ce^<'^-''*^A Ce-'^^'Xdx \dx, (20")
The solution of the equation is thus reduced to quadratures.
The extension of the method to an equation of any order is immediate.
The first step in the solution is to solve the equation
D" 4- a^ir-^ H h a,_i/) + a, = 0
80 that the differential equation may be written in the form
(D-a,)(Z)-a^...(Z)-(r,_0(Z>-«,)y = ^; aH
whereupon the solution is comprised in the formula
y = e'^ re(»«-»-««)' C ... r<;(«.— .)' Ce-'^'X{dxy, (17"^
where the successive integrations are to be performed by beginning
upon the extreme right and working toward the left Moreover, it
appears that if the operators Z) — or,, Z) — tf,_i, •••,/> — a^t» ^ — fl^i '^f^^
successively applied to this value of y, they would undo the work here
216 DIFFERENTIAL EQUATIONS
done and lead back to the original equation. As n integrations are
required, there will occur n arbitrary constants of integration in the-
answer for y.
As an example consider the equation (D^ _ 4 D) y = x^. Here the roots of the
algebraic equation D« — 4D = 0 are 0, 2, — 2, and the solution for y is
y = - — i— x2 = r e2^ r e- 2xe- 2x r e^'^^idxf.
The successive integrations are very simple by means of a table. Then
Ce^=^^dx = i x2e2^ - J xe^'^ + J e2a: + C^,
Ce-*' f^'^Hdxf = r(ix2e-2^- ^xe-2^+ ^6-^^+ C^e-^=')dx
= _|x2e-2^_^e-2a:+ 0^6-*=^+ Cg,
y = fe*^ fe-*^ re2aa2(da;)8 == /"(_ ^a;2 - | + Cie-2«+ C^t^^y^^
= -i^»*- ia; + Cie-2=«'+ C3e2x+ Og.
This is the solution. It may be noted that in integrating a term like C^e-*« thft
result may be written as C^e-*^, for the reason that C^ is arbitrary anyhow ; and,;
moreover, if the integration had introduced any terms such 2^^e-^^^\e^^^ 5, these
could be combined with the terms Cyer^^^ G^e^^^ Cg to simplify the form ol
the results.
In case the roots are imaginary the procedure is the same. Consider
—fL + y = sin X or (1)2 + 1) y = sin x or {D + i){D— i)y = sin x.
dx2
Then y = sin x = e^ Ce-^^ fe^ sin x (dx)2, t = V^.
D-iD + i J J ^ ' '
The formula for j e^ sin &xdx, as given in the tables, is not applicable when
a2 4- 62 33 0, as is the case here, because the denominator vanishes. It therefore be^
comes expedient to write sin x in terms of exponentials. Then
e- 2 ixj Qxx (cto)2 ; for sin x
2i
Now i-e**re-2<* r(e2fx_i)((ix)2==i-e»^ re-2terj_e2ix_a;+ cjdx
X gfx 4. g- to
Now C.e-i- + C,e- = (C, + C,) ^ + (C, - C,) i' ^.' .
Hence this expression may be written as Cj cosx + C^sinx, and then
y = — i X cosx + C^ cosx + Cj sin x.
The Bolution of such equations as these gives excellent opportunity to cultivate the
art of manipulating trigonometric functions through exponentials (§ 74).
COMMONER ORDINARY EQUATIONS 217
96. The general method of solution given above may be oonsiderably
simplified in case the function X(x) has certain special forms. In the
jBrst phuje suppose JC = 0, and let the equation be P{D)y r= 0, where
P{D) denotes the symbolic polynomial of the nth degree in D, Suppose
the roots of P{D) = 0 are a,, ff,, • • • , a^t and their respective multiplicities
are //ij, m.^, • • • , w^, so that
(Z) - ar^)-* • . (Z) - a^^{D - a,)-Hy = 0
is the form of tlu* difTeiential erjuation. Now, as above, if
(D-«j""y = U, then y = ^^ } ^^^ 0 = e"- j^-Jo(dx)^,
Hence y = e'^'(C^ ^ c^ -\- C^ + - ■ -{- C^x'^ ">)
is annihilated by the application of the operator (D — a,)*», and there-
fore by the application of the whole operator P(D), and must be a solu-
tion of the equation. As the factors in P(D) may be written so that
any one of them, as (D — a,)"", comes last, it follows that to each factor
(Z> — a,)"*"' will correspond a solution
y. = e-i-CCfl + C«a; + . ■ . 4- Cim,<c^-'), P(D) y, = 0,
of the equation. Moreover the sum of all these solutions,
y=^ e'.-(Ca + Q^ + • • -f C.«..2--s-»), (21)
will be a solution of the equation; for in applying P(D) to y,
P(D) y = P(/>) y^ + P(Z)) y, -h . . . + P(Z>) y* = 0.
Hence the general rule may be stated that: The solution of the dif-
ferential equation P(D)y = 0 of the nth order may be found by multiply-
ing each e"' by a polynomial of(m — l)st degree in x {where a is a root of
Vie equation P (D) = 0 of multiplicity 7n and where the coefficients of the
polynomial are arbitrary) and adding the results. Two observations
may be made. First, the solution thus found contains n arbitrary con-
stants and may therefore be considered as the general solution ; and
second, if there are imaginary roots for P (D) = 0, the exponentials aris-
ing from the pure imaginary parts of the roots may he converted into
trigonometric functions.
As an example Uke (i>* - 2 D* + D*) y = 0. The roots are 1, 1, 0, 0. Hence the
"•""">■"« V = «-(C, + C^)+(C. + C.x).
Agahi If (Z>* + 4) 2/ = 0, the rootA ofl>« + 4 = 0are±l±< and the solution Ic
218 DIFFERENTIAL EQUlTIOKS
or y = e'(Ciete+ C^e-^) + €-'{0^6^^+ C,e-«)
= e'(Ci cosx + Cj sinx) + e-'{C^ cosx + C^ sinx),
where the new Cs are not identical with the old Cs. Another form is
y=ze'A cos (x + 7) + e-^ B cos (x + 5),
where 7 and «, -4 and B, are arbitrary constants. For
and if 7 = tan-i(-^V then C^ cos x + Cg sin x = Vcf + C| cos (x + 7).
Next if X is not zero but if any one solution I can be found so that
P(D) I = A', then a solution containing^ n arbitrary constants tnay be
found by adding to I the solution of P{D)y = 0. For if
P(D)I=X and P(D)y = 0, then P(D) (I -{- y) = X.
It therefore remains to devise means for finding one solution /. Thisi
solution I may be found by the long method of (17'"), where the inte-
gration may be shortened by omitting the constants of integration sinces
only one, and not the general, value of the solution is needed. In th©
most important cases which arise in practice there are, however, som©
very short cuts to the solution I. The solution I of P(D)y = A ia
called the particular integral of the equation and the general solu-,
tion of P(D) y = 0 is called the complementary function for the equari
tion P(D) y = X.
Suppose that X is a polynomial in x. Solve S3anbolically, arrange
P (D) in ascending powers of D, and divide out to powers of D equal to
the order of the polynomial A. Then
P(D)I=X, I = ^^X = [q(D)+^^X, (22)
where the remainder R (D) is of higher order in D than A in x. Then
P(D)I = P(D)Q(D)X + R(D)X, iJ(Z))A = 0.
Hence Q (D) x may be taken as /, since P (D) Q(D)X = P(D)I = X. £■
this method the solution / may be found, when A is a polynomial, 9
rapidly as P (D) can be divided into 1 ; the solution of P (D) y = 0 ma)
be written down by (21) ; and the sum of / and this will be the requiret
solution of P{p)y = A containing n constants.
As an example consider (D» + 4 D^ + 3 D) y = x^*. The work is as follows :
8i) + 4Da + D»
L^Ji-I^ D8 + 4D+i)3 dLs 9 ^27 P(2>)J
COMMONER ORWNAEY EQUATIONS 219
Hence /= «(«)x' = ^(i-^B + if i^)«' = ix*-^*-* |?«.
For D> + 4I>' + 3D = 0 the roota are 0, • 1, — 8 &nd the complemenury f imcUoB
or solution of P(D)y = 0 would be Ci + C,e-* + C,«-"*. Hence the aolution of
the equation P(D)j/ = x* is
y = Ci + C,r- + C,e-«« + Jx» - |x« + |f x.
It should be noted that in this example D is a factor of P{D) and ha« been taken oat
before dividing? ; this shortens the work. Furthermore note that, in interpreting
1/D as integration, the constant may be omitted because any one value of / will do.
97. Next suppose that X = Ctf, Now De^ = ae", D^e" = o*««,
and P(Z))e« = P(a)e-*; hence ^ W | ^^ «"1 = ^«"-
But P(D)r= Ce", and hence / = —— e" (23)
is clearly a solution of the equation, provided a is not a root of P(D) = 0.
If P(a) = 0, the division by P{a) is impossible and the quest for / has
to be directed more carefully. Let a be a root of multiplicity m so that
P (Z>) = (D - a)"*Pj(Z>). Then
P^(D) (D - a)-I = Ce«, (Z> - a)-/ = ^^^
and I^^e'-f---f(dx)- = ^f^' (23')
For in the integration the constants may be omitted. It follows that
when A' = Ce*"^, the solution / may be found bi/ direct substitution.
Now if X broke up into the smn of terms A' = A'j + A'^ H and if
solutions /j, I.jy-' were determined for each of the equations P(D)I^= Xj,
P{D) /j = A'jj, • • •, the solution / corresponding to A would be the sum
/j -H /j H . Thus it is seen that the above short methods apply to
equations in which A is a smn of terms of the form Cx* or C«**.
As an example consider (D* — 2 2>» + l)y = c*. The roots are 1, 1, — V - V
and or = 1. Hence the solution for / is written as
{D + 1)2(D_ 1)«I = e« (D- 1)«J = Jc*. 7 = Jcz^.
Then y = e'{C^ + C^) + e-'(C, + C^x) + } cx^.
Again consider (IX*— 5D + 6)y = x + c*«. To find the I^^ corresponding to ^
' 6-6D + 7>» \6^86 ^ / 6 86
Ko find the 7, corresponding to e*^, substitute. There are three cases,
h tnr — 6 m + o
220 DIFFERENTIAL EQUATIONS
according as m is neither 2 nor 8, or is 3, or is 2. Hence for the complete solution,
when m is neither 2 nor 8 ; but in these special cases the results are
y = C^^' + C^e^^ + jx + ^V - a^e^"', y = C^^^ + Cgca^ + ^x + /^ + xe»^.
The next case to consider is where X is of the form cos ^x or sin ^.
If these trigonometric functions be expressed in terms of exponentials,
the solution may be conducted by the method above ; and this is per-
haps the best method when ± ^i are roots of the equation P (D) = 0.
It may be noted that this method would apply also to the case where
X might be of the form e''^ cos /Sx or e"^ sin px. Instead of splitting the
trigonometric functions into two exponentials, it is possible to combine
two trigonometric functions into an exponential. Thus, consider the
equations
p (D) y = e«=^ cos ^x, P(p)y= e*^ sin px,
and P(D)y = e"'' (cos /3x + i sin fix) = e^'' + ^•>. (24)
The solution I of this last equation may be found and split into its
real and imaginary parts, of which the real part is the solution of the
equation involving the cosine, and the imaginary part the sine.
When X has the form cos ftx or sin /3x and ± pi are not roots of the
equation P{D) = 0, there is a very short method of finding /. For
L^cos px= — pF cos px and D^ sin px — — p^ sin px.
Hence if P(D) be written as P^(D^ + DP^{D^) by collecting the even
terms and the odd terms so that P^ and P^ are both even in 2), the
solution may be carried out symbolically as
/ = :^COS . = ^^^^ l^^^^^ cos X = ^^^_ ^ Ij^^^^_ ^ COS X,
p,(-^-DP,(-^
By this device of substitution and of rationalization as if Z> were a surd,
the differentiation is transferred to the numerator and can be performed.
This method of procedure may be justified directly, or it may be made
to depend upon that of the paragraph above.
Consider the example (D* + l)y = cosic. Here /3i = i is a root of D^ + 1 = 0.
As an operator D» is equivalent to — 1, and the rationalization method will not
work. If the first solution be followed, the method of solution is
- 1 €<» , 1 e-*^ 1 e^ 1 e-^ 1 , . .^ 1 .
If the second suggestion be followed, the solution may be found as follows :
COMMONER ORDINARY EQUATIONS 221
X 11
Now / = — (C08X + iainx) = -islnx iz cosx.
Hence is^xsinx for (Z)* + 1) J = coex,
and I = ~)xcoex for (!>" + l)I = «ln«.
The complete solution is V = C^ cosx + C, sin x + } x sin x,
and for (D* + l)y = sinx, y = C, cosx + CjSinx— ^xcosx.
As another example take (/>• — 8 D + 2)i/ = cosx. The roots are 1, 2, neitbex
is equal to ± pi = ± i^ and the method of rationalization is practicable. Then
.1 1 1 + 3Z) 1 , ^ , ^
The complete solution is y = C^e-* + C,c-«» + x^(co8x — Ssinx). Tbe extreme
simplicity of this substitution-rationalization method is noteworthy.
EXERCISES
1. By the general method solve the equations :
(7) (I>a-4D+2)y = x, (3) (2)3 + 2>»_ 4D - 4)y = x,
(e) (Z>»4-5X>a + 6/))y = x, (f) (7)2 + D + l)y = xeT,
(t,) (7/J+D+ l)y = 8in2x, {$) (D^ - 4)y = x -^ e^^,
(0 (/)2+3D + 2)y = x + co8X, (<«) (D* - 4i)2)y = 1 - sinx,
(X) (Z>2 + 1)1/ = cosx, (m) (D2 + l)y = 8ecx, {p) (Z)^ + l)y = tanz.
2. By the rule write the solutions of these equations :
(a) (7>»+32)+2)y = 0, (/3) {D» + SIfi + D ^ 5)y = 0,
(7) (/)-l)»y = 0, (5) (2>« + 2D2 + l)|/ = 0,
(e) (Z)3-3D2 + 4)2/ = 0, (f) (!>♦- I)^ - 9Z>a- 11 i)- 4)y = 0,
(i,) (/)'-6Z)2 + 92))y = 0, (^) (2)*-4D8 + 8Z>2_8D + 4)y = 0,
(() (D5~2 2)* + i>»)y = 0, (k) (1>»-D2+D)y = 0,
(X) (Z><-l)«y = 0, (m) (i)6_13D» + 262)2 + 82D+ I04)y = C.
3. By the short method solve (7), (3), (e) of Ex. 1, and also :
(a) (2)*-l)y = x*, (/9) (2)» - 61)2 + 11 2)- 6)y = x,
(7) (2>» + 32)a+22))y = x2, («) (2)« - 32>» - 62) + 8) j/ = x,
(e) (2)« + 8)y = x« + 2x + l, (f) (2)8 - 3 2)^ - 2) + 3) y = x«,
(1,) (2)<-22)« + 2)2)y = x, ((9) (2)* + 22)» + 32)2 + 22)+ l)y = l + x + x«
(i) (2)»-l)i/ = x2, {k) (2)*-22)» + 2)a)j/ = x«.
4. By the short method solve (a), (/3), (ff) of Ex. 1, and also :
(a) (2)2-32)+2)y = c', (^) (2)« - 2)» - 3 D» + 62)- 2)y = «»«
(7) (7)2- 22)4- l)y = e', (3) (2)» - 3 2)» + 4)y = c«',
(e) (7)2 + i)y = 2e' + x»-x, (0 (2)» + l)y = 3 + e-'+6c«',
(tj) (7>*4-2 7)a+ l)y = c'+4, (^) (7)« + 3 2)a + 3 2) + l)y = 2e-«
(i) (2)a-2 2))i/ = e2'+l, (r) (2)» + 2 2)« + 2))y = ««' + x« + X,
(X) (2)a_a2)y = e« + e*», 0*) (D«- 2aD+ o«)y =« e*+ 1.
222 DIFFERENTIAL EQUATIONS
5. SolT* by the short method (i,), (*), (k) of Ex. 1, and also :
.^wx,i_D-2)y=8in*, (^) (I>« + 2D + l)y = Se^x- cosx,
> (/)■ + 4)y = x« + coex, (3) (D' + i;»- D- 1)2/ = co82x,
.)(Di+l)«ir = co8X. (f) (D»-i>» + D-l)y = cosx,
(,J (D»-6D+6)y = co8X-e2« (<?) (2)8 - 2 2)2- 32))y =3x2 + sinx,
(.)(Di-l)«y = 8inx, (*) (D" + 3 2) + 2)y = e^xsinx,
cosx.
(X) (D*-l)y = Cco8X, (m) (D»-32>2 + 42)-2)y = e« +
(,) (I>«-2D+4)y = c'8inx, (o) (2>a + 4)y = sin3x + e^ + x^,
(») (D« + l)y = 8inix8inJx, (p) (2)8 + l)y = e2^sinx + e2sin^,
(O (JDi+4)y = 8in«x, (t) (2)* + 322) + 48)y = xe-2x + e2xcos2tx
6. If X has the form e^X^, show that I = ^— - e^'^X^
gax.
(2)) ' P{D+a) '
This enables the solution of equations where Xj is a polynomial to be obtained by
a short method ; it also gives a way of treating equations where X is e«^ cos/Sx or
««» sin /te, but is not an improvement on (24) ; finally, combined with the second
suggestion of (24), it covers the case where X is the product of a sine or cosine by
a polynomial. Solve by this method, or partly by this method, (f) of Ex. 1 ; (/c), (X),
(»)» (/>)» ir) ^^ ^^- ^ 5 ^^^ ^^^
{a) (D»-2D + l)y = x2e«% (/S) (2)3 + 3i)2 + 3D+ 1)2/ = (2 - x2)e-^
(7) {IP + n2)y = x*e^ («) (D*-22)3-32)2 + 42)+4)y = x2e^
(c) (2)« - 7 2)- 6)2/ = e2'>=(l + x), (f) (D- 1)2?/ = e^ + cosx + x2e%
(ir) (D - 1)«2/ = X - x«e^ (^) {IP + 2)y = x^e^=^ + e^ cos 2 x,
(») (D» - l)y = xc' + co82x, {k) (2)2 - l)y = X sin X + (1 + x2) e^,
(X) (I>» + 4)y = X sin x, (a*) (D* + 22)2 + l)i/ = 3^2 cos ax,_
(r) (D« + 4) y = (X sin x)2, (o ) (2)2 - 2 2) + 4)2y = xe^ cos V3 x.
7. Show that the substitution x = e', Ex. 9, p. 152, changes equations of the type
x"2)»y + aiX«-i2)'»-iy + • • • + On-ixDy + a„2/ = X(x) (26)
Into equations with constant coefficients ; also that ox + 6 = e< would make a simi-
lar simplification for equations whose coefficients were powers of ox + 6. Hence
integrate :
(a) (x«D«-xD+2)y = xlogx, (/S) (x»2)8 - x22)2 + 2 x2) - 2) y = x' + 3 x,
(7) [(2x-l)»2)»+(2x-l)2)-2]y=0, {S) {x^IP + SxD + l)y = {1- x)-^,
(t) (x»D» + xD-l)y = xlogx, (f) [(x + 1)22)2 _ 4(x + l)i) + 6]y = x,
(if) {x*IP + 4 xD + 2) y = C, {$) (x»2)2- 3 x^D + x) y = log x sin log x + 1,
(i) (x*2>* + 6x«2)» + 4x22)2 - 2x2)- 4)y = x2 + 2coslogx.
8. If L be self-induction, R resistance, C capacity, i current, 5' charge upon the
plates of a condenser, and f{t) the electromotive force, then the differential equa-
tions for the circuit are
Solve (a) when/(() = e- •* sin « and {fi) when/(<) = sin bt. Reduce the trigonometric
part of the particular solution to the form K sin {bt + 7). Show that if R is small
and 6 Is nearly equal to 1/ VlC, the amplitude K is large.
COMMONER ORDINARY EQUATIONS 223
98. Simultaneous linear equations with constant coefRcient^. If
there Ui given two (or in genenil n) linear equiitions with constant
coefficients in two (or in general 7i) dependent variableis and one inde-
pendent variable t, the symbolic method of solution may still be used
to advantage. Let the equations be
when there are two variables and where D denotes differentiation by t.
The equations may also be written more briefly as
P,{D)x-\'Q^{D)y = R and P^(D) x -h Q^(D) y = S.
The ordinary algebraic process of solution for x and y may be employed
because it depends only on such laws as are satisfied equally by the
symbols /), P^(D), Qi(D)y and so on.
Hence the solution for x and y is found by multiplying by the ap-
propriate coefficients and adding the equations.
P,(D)x-\-Q^(D)y = R,
P,(D)x-^Q^(D)y = S.
Then IP^(D) Q^(D) - P^D) Q,(Z))] x = Q./D) R - Q,{D) S,
lP,(D)Q(D)-P,(D)Q^(D)]y = P^(D)S-P^(D)R. ^ >
It will be noticed that the coefficients by which the equations are multi-
])lied (written on the left) are so chosen as to make the coefficients of
X and y in the solved form the same in sign as in other respects. It may
also be noted that the order of P and Q in the symbolic products is im-
material. By expanding the operator P^{D) Q^J^D) — P^(D) Q^(D) a certain
polynomial in D is obtained and by applying the operators to R and A'
as indicated certain functions of t are obtained. Each equation, whether
in a; or in y, is quite of the form that has been treated in §§ 95-97.
As an example consider the solution for x and y in the case of
2 1^ + (4 D - 3) y = 0.
(2D»-4)a;-Dy = 2t
2Dx-\-{iD-S)y = 0.
Then [(4D- 3) (22)2 - 4) + 21>2]x = (4D- 3)2t,
[2 2>« + (2I>»-4)(4D-3)]y=-(2D)2t,
or 4(2D«-l>»-4D + 3)x:^8-6«, 4(27>'' - i>« - 4D + 3)y = - 4.
The roots of the polynomial in D are 1, 1, — IJ ; and the particular solution 7, fot
/ is — J (, and J, f or y is — \. Hence the solutions have the form
cPx dy
dt^ dt
-4x =
or
{2Lf^-4)x-I>y = 2t,
Solve
4D-3
-2D
D
2i>»-4
2S4 DIFFERENTIAL EQUATIONS
The arbitrary oonatants which are introduced into the solutions for x
and y are not independent nor are they identical. The solutwns must
be tJbttUutsd into one of the equations to establish the necessary relations
h$iwem the constants. It will be noticed that in general the order of the
equation in D for i and for y is the sum of the orders of the highest
derivatives which occur in the two equations, — in this case, 3 = 2+1.
The order may be diminished by cancellations which occur in the formal
algebraic solutions for x and y. In fact it is conceivable that the coeffi-
cient P (L—P Q. of X and y in the solved equations should vanish and
the solution become illusory. This case is of so little consequence in
practice that it may be dismissed with the statement that the solution
is then either impossible or indeterminate ; that is, either there are no
functions x and y of / which satisfy the two given differential equations,
or there are an infinite number in each of which other things than the
constants of integration are arbitrary.
To finish the example above and determine one set of arbitrary constants in
temM of the other, subetitute in the second differential equation. Then
-S(K^e^ + K^te* + K^e- I ' _ ^) = 0,
or e'{2 Cj + 2 C, + iTj + K^) + te*(2 C^-{- K^) - Se-^\C^ -{■SK^) = 0.
Ai the terms €*, te*, e~i* are independent, the linear relation between them can
bold only if each of the coeflScients vanishes. Hence
and C, = -8ir,, 2C^ = -K2, 2C^ = -K^.
Hence » = (C, + C,0C-3^,e-^-i<, y = -2{C^-\- 0^1)^ + K^e-^'- ^
are the finished solutions, where C^, C^, K^ are three arbitrary constants of inte-
gration and might equally well be denoted by Cj, Cj, Cg, or K^, K^, K^.
99, One of the most important applications of the theory of simultaneous equa-
NM with constant coefficients is to the theory of small vibrations about a state of
igwIfAi-liiM in a eonaeroative* dynamical system. If gj , Q'g » •••»<?« are n coordinates
(Me Bn. ld-40, p. 112) which specify the position of the system measured relatively
•The potential energy V is defined as - dr= dTr= Q^dq^ + Q^dq^ + • • • + Qndqn,
TWs Is the immediate exteDsion of Q^ as given in Ex. 19, p. 112. Here dW denotes the
differential of work and d H' = 2P,.rfr, = 2 {Xidxi + Yidyi + Z.t?2.). To find 0, it is
gMeraUy qtilekeM to compute d W from this relation with dxi , dt/,- , dzi expressed in terms
M lae differentials tf^i . • • . , c/?,. The generalized forces Q,- are then the coefficients of
^*' I ^ ^ !f! ? potential V, the differential d IT must be exact. It is frequently
auy to and K dIrecUy In terms of ^i. .... ^^ rather than through the mediation of
dBr'' dT ^ °*** ***' *' '* """""y ^«^^'' ^ leave the equations in the form
^ jT ~ C" ■ ^ "*****' ^***" ^ Introduce Kand L.
COMMONER ORDINARY EQUATIONS 225
to a position of stable equilibrium in which all the q'B Tanish, the deTelopmani of
the potential energy by Maclaurin*B Formula gives
ViQi , 9a, •••,<?-)= ^0 + ^i(<?i . 9i, • • • , 9.) + V,(q^, 7j, • • • , 7.) + • • • .
where the first term is constant, the second is linear, and the third la quadratic, and
where the supposition that the 9's take on only small values, owing to the reatricdoii
to small vibrations, shows that each term is infinitesimal with respect to the
ing. Now the constant term may be neglected in any ezpreasion of potential
As the position when all the q's are 0 ia assumed to be one of equilibrium, the foroet
must ail vanish when the ^'s are 0. This shows that the coefficients, {dV/dq,)o = 0,
of tlic linear expression are all zero. Hence the first term in the expansion is the
(juadratic term, and relative to it the higher terms may be disregarded. As the
jjosition of equilibrium is stable, the system will tend to return to the position
where all the q's are 0 when it is slightly displaced from that position. It follows
that the quadratic expression must be definitely positive.
The kinetic energy is always a quadratic function of the velocities 9i, 9s,* • •, ^a
with coefficients which may be functions of the q's. If each coefficient be expanded
by the Maclaurin Fonnula and only the first or constant term be retained, the
kinetic energy becomes a quadratic function with constant coefficients. Hence the
Lagrangian function (cf . § 160)
when substituted in the formulas for the motion of the system, gives
dtdq^ ^q^~ ' dldq^ dq^~ ' "' dtdqn dg, ~ '
a set of equations of the second order with constant coefficients. The equations
moreover involve the operator D only through its square, and the roots of the equa-
tion in D must be either real or pure imaginary. The pure imaginary roots intro-
duce trigonometric functions in the solution and represent vibrations. If there were
real roots, which would have to occur in pairs, the positive root would represent
a term of exponential fonn which would increase indefinitely with the time, — a
result which is at variance both with the assumption of stable equilibrium and
with the fact that the energy of the system is constant.
When there is friction in the system, the forces of friction are supposed to vary
v^ith the velocities for small vibrations. In this case there exists a dissipative func-
tion F(^i , 92, • • ' , 9m) which is quadratic in the velocities and may be assumed to
have constant coefficients. The equations of motion of the system then become
dt dq^ ^q^ dq^ ' ' dt d^^ ^^ ?^,
which are still linear with constant coefficients but involve first powers of the
operator D. It is jThysically obvious that the r(x>ts of the equation in D must be
negative if real, and nmst have their real parts negative if the roots are complex ;
for otherwise the energy of the motion would increase indefinitely with the time,
whereas it is known to be steadily dissipating its initial energy. It may be added
that if, in addition to the internal forces arising from the potential V and the
226 DIFFERENTIAL EQUATIONS
fricUoiuU foroM arising from the dissipative function F, there are other forces
imprvMed on the system, these forces would remain to be inserted upon the right-
hand side of the equations of motion just given.
The fact that the equations for small vibrations lead to equations with constant
ooeAdentB by neglecting the higher powers of the variables gives the important
phyirical theorem of the superposition of small vibrations. The theorem is : If with
a certain set of initial conditions, a system executes a certain motion ; and if with
a diflennt set of initial conditions taken at the same initial time, the system
executes a second motion ; then the system may execute the motion which consists
of merely adding or superposing these motions at each instant of time ; and in
particular this combined motion will be that which the system would execute under
initial conditions which are found by simply adding the corresponding values in
the two sets of initial conditions. This theorem is of course a mere corollary of the
linearity of the equations.
EXERCISES
1. Integrate the following systems of equations :
(a) Ite - Dy + aj = cos t, mx, - Dy + Bx — y = e^*,
Ifi) SDx-\-3x-\-2y = e*, 4x- 31>y + Sy.= Zt,
(7) D»x - 3x - 4y= 0, jyzy ^ x + y = 0,
y— 7x 2x + 6y 3x + 42/ 2x-\-6y
(0 t2>t + 2(x-y) = l, tDy + x + 5y = «,
{^) Dx = ny — mz, I>y = lz — nx, Bz =mx— ly,
(tf) I>»x-8x-4y + 3 = 0, DUy + x - 8y + 5 = 0,
(t) D«x-42)»y + 41>»i;-x = 0, 2)*y - 4 D^x + 4 .D2?/ - y = 0.
2. A particle vibrates without friction upon the inner surface of an ellipsoid.
Discuss the motion. Take the ellipsoid as
g + ^ + ^ = l; then x = Csin(:^t+C,), , = irsi„(^< + jr.).
S. Same as Ex. 2 when friction varies with the velocity.
4. Two heavy particles of equal mass are attached to a light string, one at the
middle, one at one end, and are suspended by attaching the other end of the string
to a axed point. If the particles are slightly displaced and the oscillations take
place without friction in a vertical plane containing the fixed point, discuss the
notion.
5. If there be given two electric circuits without capacity, the equations are
where i, . i, are the currents in the circuits, Zj , i, are the coefficients of self-
induction, H,, H, are the resistances, and M is the coefficient of mutual induction.
(a) Integrate the equations when the impressed electromotive forces E^^ , E^ are
«ero in both circulu.. {p) Also when £, = 0 but E^ = sinpf is a periodic force.
(>) Discuss the cases of loose coupling, that is, where MyL^L^ is small ; and the
**T!L!!T *^'*^*"«' ^^^ ^ ^^®'® MyL^L^ is neariy unity. What values forp
are aspedally noteworthy when the damping is small ?
COMMONER ORDINARY EQUATIONS 227
6. If the two circuitJi of Ex. 6 have c&paciUM C|, C, and If 9,, g, an tbe
charges on the condenflera ao that i^ = dq^/dt, if = dq^dt are the cuirenta, the
equations are
' dt^ ^ df ^ ' dt^ C^ *' * dfi^ dl^^ * dt
Integrate when the resistances are negligible and JFi= J?,= 0. If T, a twVCj^^
and r, = 2 x y/C^L^ are the periods of the individual separate circuito and
e = 2 irAf VC^, and if T^ = T,, show that VT« + 0« and Vr« - 0» an the
independent periods in the coupled circuits.
7. A uniform beam of weight 6 lb. and length 2 ft. is placed orthogonallj
across a rough horizontal cylinder 1 ft. in diameter. To each end of the beam is
suspended a weight of 1 lb. upon a string 1 ft. long. Solve the motion produced
by giving one of the weights a slight horizontal velocity. Note that in finding the
kinetic energy of the beam, the beam may be considered as rotating about its
middle point (§ 39).
CHAPTER IX
ADDITIONAL TYPES OF ORDINARY EQUATIONS
100. Equations of the first order and higher degree. The degree of
a diflferential equation is defined as the degree of the derivative of
highest order which enters In the equation. In the case of the equation
♦(at, y, y')= 0 of the first order, the degree will be the degree of the
equation in y'. From the idea of the lineal element (§ 85) it appears
that if the degree of ♦ in y' is w, there will be n lineal elements through
each point (x, y). Hence it is seen that there are n curves, which are
oompounded of these elements, passing through each point. It may be
pointed out that equations such as y' = a;Vl + y^, which are apparently
of the first degree in y', are really of higher degree if the multiple value
of the functions, such as Vl H-y**, which enter in the equation, is taken
into consideration ; the equation above is replaceable by y'^ = ar^ + xhf^
which is of the second degree and without any multiple valued function.*
First suppose that the differential equation
♦ (^, y, y") = y - U^y y)l x [y' - U^, 2/)] • • • = o (i)
may be solved for y'. It then becomes equivalent to the set
of equations each of the first order, and each of these may be treated
by the methods of Chap. VIII. Thus a set of integrals t
^i(^,y, C) = 0, F,(^,y, C) = 0, ... (2)
may be obtained, and the product of these separate integrals
F{x, y, C) = F^(x, y, C) • F^x, y, C) . • . = 0 (2')
k the complete solution of the original equation. Geometrically speak-
ing, each integral F,(a;, y, C) = 0 represents a family of curves and the
product represents all the families simultaneously.
*»U b tktwiofi spiiarent that the idea of degree as applied in practice is somewhat
tTki IMil* MMUnt C or any desired function of C may be used in the different
■MUtlOM bMMM C la an arbitrary constant and no specialization is introduced by its
rvpaatod nM to this way.
S88
ADDITIONAL ORDINAEY TYPES 229
As an ex&mple consider /* + 2 ^y ootx = y*. Solve.
1^+ 2y'i/cotx + y«cot«x = y*(l + cot^«) = i^cMx,
and (/ + ycotx— ycscx)^^ + ycotx + ycscjf) =0.
These equations both come under the type of variables separable. Integrate
dy 1 — coex. dcosx
— = : dx= —
V sinx 1 + cosx
y(l + ooex) = C,
, dy l + cosx. dcoBX ,. ^ _
and — = ; dx = , y(l — cosx)=C
y sinx 1 — cosx
Henco [y(l + cosx) + C][y(l — cosx) + C] = 0
i8 the solution. It may be put in a different form bj multiplying out. Then
y«sin«x + 2Cy+ C = 0.
If the equation cannot be solved for y' or if the equations resulting
from the solution cannot be integrated, this first method fails. In that
case it may he possible to solve for y or for x and treat the equation by
differentiation. Lety'=^. Then if
y-JK^,Ph dx ^ dx^dpdx W
The equation thus found by differentiation is a differential eqiiation of
the first order in dp/dx and it may be solved by the methods of Chap.
VIII to find F{p, X, C) = 0. The two equations
y=f(^>P) and F(p,x,C) = 0 (S*)
may be regarded as defining x and y parametrically in terms of p, or p
may be eliminated between them to determine the solution in the form
O (x, y, C) = 0 if this is more convenient. If the given differential equar
tion had been solved for a*, then
.=/(..) and |4 = |.|g- W
The resulting equation on the right is an equation of the first order in
dp/dy and may be treated in the same way.
As an example take xp^ — 2yp-{-ax = 0 and solve for y. Then
p dx dx p^ dx p
b-lYi'd-'h'' - '-^-^='>-
X
or -
P
The solution of this equation is x = Cp. The solution of the given equation is
2y = app + — , x = Cp
P
when expressed parametrically in terms of p. If p be eliminated, then
xt
2 y = — + aO parabolas.
C
280 DIFFERENTIAL EQUATIONS
Am another example take p*y + 2 j>x = y and solve for x. Then
^ l+p + y/l + l^^zzO, or ydp+l>dy = 0.
Th« aolution of thla i« py = C and the solution of the given equation is
2x = y(--p), py = C, or y^ = 2Cx+C^.
Two special types of equation may be mentioned in addition, although
their method of solution is a mere corollary of the methods already
given in general. They are the equation homogeneous in (x, y) and
Clairau^t equation. The general form of the homogeneous equation is
♦CPi yA) = ^' T^ equation may be solved as
p^^iA or as 1=Ap), y = ^f(p)) (5)
and in the first case is treated by the methods of Chap. VIII, and in
the second by the methods of this article. Which method is chosen
rests with the solver. The Clairaut type of equation is
y=px-\-f(jp) (6)
and comes directly under the methods of this article. It is especially
noteworthy, however, that on differentiating with respect to x the result-
ing equation is j j
[x+/(^)]| = 0 or 1 = 0. (6-)
Hence the solution for ^ is ^ = C, and thus y = Cx -{- f(C) is the solu-
tion for the Clairaut equation and represents a family of straight lines.
The rule is merely to substitute C in place of p. This type occurs very
frequently in geometric applications either directly or in a disguised
form requiring a preliminary change of variable.
101. To this point the only solution of the differential equation
♦ (x, y, p)=0 which has been considered is the general solution
F(x, y, C)=0 containing an arbitrary constant. If a special value,
•aj 2, is given to C, the solution F(Xy y, 2) = 0 is called a particular
solution. It may happen that the arbitrary constant C enters into the
expression F(x, y, C) = 0 in such a way that when C becomes positively
infinite (or negatively infinite) the curve F(xy y, C) = 0 approaches a
definite limiting position which is a solution of the differential equation ;
•unh iMilutionH are called infinite solutions. In addition to these types
of solution which naturally group themselves in connection with the
generml solution, there is often a solution of a different kind which is
ADDITIONAL ORDLNAKY TYPES 281
known as the singular solution. There are several different definitiom
for the singuhir solution. That which will be adopted here is : A »ing%^
lar solution is the envelope of the family of curves defined by the
general solution.
The consideration of the lineal elements (§ 85) will show how it is
that the envelope (§ 65) of the family of particular solutions which
constitute the general solution is itself a solution of the equation. For
consider the figure, which represents the particular solutions broken up
into their lineal elements. Note that the envelope is made up of those
lineal elements, one taken from each particular so-
lution, which are at the points of contact of the envelope
envelope with the curves of the family. It is seen '^^y^^\*^^
that the envelope is a curve all of whose lineal X •'"""'v^
elements satisfy the equation ♦(x, y,p)= 0 for the
reason that they lie upon solutions of the equation. Now any curve
whose lineal elements satisfy the equation is by definition a solution
of the equation; and so the envelope must be a solution. It might
conceivably happen that the family F(xy y, C)= 0 was so constituted
as to envelope one of its own curves. In that case that curve would
be both a particular and a singular solution.
If the geneiul solution F(a;, y, C) = 0 of a given differential equation
is known, the singular solution may be found according to the rule for
finding envelopes (§ 65) by eliminating C from
F(x,y, C)=0 and ^F(x,y, C)=0. (7)
It should be borne in mind that in the eliminant of these two equations
there may occur some factors which do not represent envelopes and
which must be discarded from the singular solution. If only the singu-
lar solution is desired and the general solution is not known, this
method is inconvenient. In the case of Clairaut's equation, however,
where the solution is known, it gives the result immediately as that
obtained by eliminating C from the two equations
y=Cx-f/(0 and 0 = a- -f /'(O- W
It may be noted that as jo = C, the second of the equations is merely
the fivctor x -hf'(p) = 0 discarded from (6'). The singular solution may
therefore be found by eliminating p between the given Clairaut equa-
tion and the discarded factor x -^f\p)= 0.
A reexamination of the figure will suggest a means of finding the
singular solution without integrating the given equation. For it is seen
that when two neighboring curves of the family intersect in a point P
282 DIFFERENTIAL EQUATIONS
near the envelope, then through this point there are two lineal elements
which satisfy the differential equation. These two lineal elements have
nearly the same direction, and indeed the nearer the two neighboring
coryes are to each other the nearer will their intersection lie to the
envelope and the nearer will the two lineal elements approach coinci-
dence with each other and with the element upon the envelope at the
point of contact. Hence for all points (x, y) on the envelope the equa-
tion ♦ (ar, yiP)=0 of the lineal elements must have double roots for p.
Now if an equation has double roots, the derivative of the equation
must have a root. Hence the requirement that the two equations
^(«, y,i>)=0 and '^if;(x,y,p) = 0 (9)
have a common solution for p will insure that the first has a double
root for p ; and the points (a;, y) which satisfy these equations simul-
taneously must surely include all the points of the envelope. The rule
for finding the singular solution is therefore : Eliminate p from the
given differential equation and its derivative with respect to p, that is,
from (9). The result should be tested.
If the equation xp^ — 2 1^ + oa; = 0 treated above be tried for a singular solution,
the elimination of p is required between the two equations
acp* — 2yp-\-ax = 0 and xp — y = 0.
The result is y* = or*, which gives apair of lines through the origin. The substi-
tution of y = i Vox and p = ± Va in the given equation shows at once that
y* = ox* satisfies the equation. Thus y"^ = ox^ is a singular solution. The same
result is found by finding the envelope of the general solution given above. It is
clear that in this case the singular solution is not a particular solution, as the par-
ticular solutions are parabolas.
If the elimination had been carried on by Sylvester's method, then
0 X — y
X -2y a = -x(y2_ax2) = 0;
X — y 0
and the eliminant is the product of two factors x = 0 and y^ ^ ax^ = 0, of which
the second is that ju8t found and the first is the y-axis. As the slope of the y-axis
ti Infinite, the substitution in the equation is hardly legitimate, and the equation
ean hardly be said to be satisfied. The occurrence of these extraneous factors in
tba ellniinant is the real reason for the necessity of testing the result to see if it
tetoaUy repreaento a singular solution. These extraneous factors may represent
a graat variety of conditions. Thus in the case of the equation p^ + 2yp cot x = y*
praviottiiy treated, the elimination gives y^ csc^x = 0, and as esc x cannot vanish,
tba rMult reduces to y« = 0, or the x-axis. As the slope along the x-axis is 0 and y
la 0, the equation Is clearly satisfied. Yet the line y = 0 is not the envelope of the
general lolution ; for the curves of the family touch the line only at the points nir.
It It a particular solution and corresponds to C = 0. There is no singular solution.
ADDITIONAL ORDINABY TYPES 288
Many authors use a great deal of time and space dlacovliig Just what may and
what may not occur among the extraneous loci and bow many times it may ooour.
The result Ih a conwiderable number of statements which in their details are either
grossly incomplete or glaringly false or both (of. ff 66r-67). The rules here giveo
for finding singular solutions should not be regarded in any other light than as
leading to some expressions which are to be examined, the best way one can, to
find out whether or not they are singular solutions. One curve which may appear In
the elintiniitiun of p and which deserves a note is the tac-locus or locus of points of
tangency uf the particular solutions with each other. Thus in the system of circles
(z — Cy +y^ = r'^ there may be found two which are tangent to each other at any
assigned point of the x-axis. This tangency represents two coincident lineal
elements and hence may be expected to occur in the elimination of p between the
differential equation of the family and its derivative with respect to p ; but not in
the eliminant from (7).
EXERCISES
1. Integrate the following equations by solving for p = y':
(a) pa - 6p + 5 = 0, 03) p» - (2x +2/2)^2 + (j.^ - y2 + 2xy2)p- (x« -y*)y«=0,
(7) xi>2-2f^-x = 0, (3) pHx + 2y) -f SpHx + y) 4- P(y -i- 2x) = 0,
(.)j/a + p2 = l, (0p2-ax» = 0, (v) p = (a-x)Vl+p«.
2. Integrate the following equations by solving for y or x :
(a) ixp'^ + 2xp - y = 0, (fi) y = -xp + x*p\ (y) p + 2xy - x« - y« = 0,
(«) 2px-y + logp = 0, (€)x-yp = ap^, (f) y = x + a tan-»p,
(,) X = y + a logp, {0) X + py (2p2 + 3) = 0, (t) a^yp* - 2xp + y = 0,
(it) p» - 4xyp + 8 y« = 0, (X) x = p + logp, (m) p*(x* + 2ax) = a*.
3. Integrate these equations [substitutions suggested in (*) and («)] :
{a) xy2 (p2 + 2) =2py« + x», (/3) (nx + py)^ = (1 + P*) (y* + nx*),
(7) y* + xyp - xV = 0, («) v = yp^-^ 2px,
( «) V=px+ sin-ip, (f) y = p (X - 6) + a/p,
(17) y = px + P (1 - p2), (<9) y« - 2pxy - 1 = p« (1 -x%
(i) 4e2»'p2 + 2xp -1 = 0, 2 = e^y, {k) y = 2px + yV, y« = z,
(X) 4 €^vp^ + 2e2'p - e' = 0, (m) x* (y - px) = VP^-
4. Treat these equations by the p method (9) to find the singular solutions.
Also solve and treat by the C method (7). Sketch the family of solutions and
examine the significance of the extraneous factors as well as that of the factor
wlach gives the singular solution :
{a) p^-\-p{x-y)-x = 0, (fi) p^y* co8« a - 2pxy sin'a + y* - jt«sin«ar = 0,
(y) 4xp« = (3x-a)«, («) yp«x(x - a)(x - 6) = [8x«- 2x(a + 6) + a6]«,
(«) pa + xp- y = 0, (D 8a(l + p)« = 27 (x + y)(l -p)«,
(1,) x«p3 + x«yp + a» = 0, {$) y(3-4y)«p« = 4(l-y).
5. Examine sundry of the equations of Exs. 1, 2, 3, for singular solutions.
6. Show that the solution of y = X4p{p) +/(p) is given parametrically by the
given equation and the solution of the linear equation :
^ + x-?^ = ^!M_ Solve (a)y = mxp + n(l+M
dp ^{p)-p P-4^(P)
(/J) y = x(p-{- aVl+p«X (7) * = W> + ap\ (9) y = (1 +p)x + j»«
284 DIFFERENTIAL EQUATIONS
7. Ab any straight line is y = m« + 6, any family of lines may be represented as
p^wtx +/(m) or by the Clairaut equation y=px-{-f{p). Show that the orthog-
onal trajectorie* of any family of lines leads to an equation of the type of Ex. 6.
The Mine is true of the trajectories at any constant angle. Express the equations
of the following systems of lines in the Clairaut form, write the equations of the
oitbofonal trajectories, and integrate :
(a) tangents to a:« + y« = 1, (/S) tangents toy^ = 2 ax,
{y) tangenU to y« = x*, («) normals to y^ = 2 ax,
( « ) normals to y^ = x«, (f) normals to l^^ + aV = aVj^.
8. The evoluU of a given curve is the locus of the center of curvature of the
curve, or, what amounts to the same thing, it is the envelope of the normals of the
given curve. If the Clairaut equation of the normals is known, the evolute may be
obtained as its singular solution. Thus find the evolutes of
(a) y« = 4aa;, (p) 2xy = a^, (7) x^ -{- y^ = at,
(0^ + ^=1, (^)y' = ~ it)y = Ue^^e-^).
9. The involutes of a given curve are the curves which cut the tangents of the
given curve orthogonally, or, what amounts to the same thing, they are the curves
which have the given curve as the locus of their centers of curvature. Find the
involutes of
(a) z* + y* = a*, (fi) y^ = 2 mx, (7) y = a cosh (x/a).
10. As any curve is the envelope of its tangents, it follows that when the curve
is described by a property of its tangents the curve may be regarded as the singu-
lar solution of the Clairaut equation of its tangent lines. Determine thus what
conres have these properties :
(a) length of the tangent intercepted between the axes is i,
(p) sum of the intercepts of the tangent on the axes is c,
(7) area between the tangent and axes is the constant A;^,
(I) product of perpendiculars from two fixed points to tangent is A:^,
(«) product of ordinates from two points of x-axis to tangent is k^.
11. From the relation -3- = m VJtf 2 -|- N^ of Proposition 3, p. 212, show that as
an
the curve F = C is moving tangentially to itself along its envelope, the singular
solution of Mdx + Ndy = 0 may be expected to be found in the equation 1/^ = 0;
also the infinite solutions. Discuss the equation 1//* = 0 in the following cases :
(a) Vl - y«dx = Vl - x^dy, (/S) xdx + ydy - Vx^ -|- y2 _ a* dy.
102. Equations of higher order. In the treatment of special prob-
lems (1 82) it was seen that the substitutions
rendered the dififerential equations integrable by reducing them to in-
tegraUe equations of the first order. These substitutions or others like
them are useful in treating certain cases of the differential equation
ADDITIONAL ORDINARY TYPES 235
♦(-^j y» y\ y'\ •» t^*^)=0 of the nth order, namely, when one of the
variables and perhaps some of the derivatives of lowest order do not
occur in the equation.
Incase *(«, g, g^, ..., g) = 0, (11)
y and the first i — 1 derivatives being absent, substitute
g = , sothat ♦(.,,,|,...,^,) = 0. (ll-)
The original equation is therefore replaced by one of lower order. If
the integral of this be F(x, q) = 0, which will of course contain n — i
arbitrary constants, the solution for q gives
q=f{x) and y=j-Jf{x)(dxy. (12)
The solution has therefore been accomplished. If it were more con-
venient to solve F(x, y) = 0 for x = <^ (y), the integration would be
y =j-Ji^^y =f ■ •/?[*'(?)<='?]'; (IS-)
and this equation with x= if>(q) would give a parametric expression
for the integral of the differential equation.
X being absent, substitute p and regard /> as a function of y. Then
dy (Py dp d^y d / dp\
and ■ ♦,(3,,^,^,...,^j = 0.
In this way the order of the equation is lowered by unity. If this equa-
tion can be integrated as F(yy p) = 0, the last step in the solution may
be obtained either directly or parametrically as
J /(!/)
P =/(!/)> 7^=* (14)
or y
= «(P), .^fk^ji^. (U^
It is no particular simplification in this case to have some of the lower
derivatives of y absent from ♦ = 0, because in general the lower deriva-
tives of p will none the less be introduced by the substitution that
is made.
236 DIFFERENTIAL EQUATIONS
A.MexMnpleoonrider^x^-^j = \^j + 1»
TlMm g = «£±>J(|)%l and g = CiX±V5n^;
(or the equation is a Clairaut type. Hence, finally,
y =//[Ci* ± V^+T](dx)2 ^ I CiX» ± ix2 Vcf + 1 + C^x + C^.
Ai another example consider y" — y^ = y^ log y. This becomes
p^-p« = yMogy or ^^ _ 2i)2 = 2y2iogy.
dy ay
The equation is linear in jfl and has the integrating factor e-^v.
ip«e-««'= ry^e-avlogydy, —p = \e^»Jy^e-^vlogydy^ ,
and f- ^^^ , = V2x.
r ay
The integration is therefore reduced to quadratures and becomes a problem in
ordinary integration.
If an equation is homogeneous with respect to y and its derivatives^
that is, if the equation is multiplied, by a power of k when y is replaced
by kyy the order of the equation may be lowered by the substitution
y = «■ and by taking «' as the new variable. If the equation is homo-
geneous with respect to x and dx^ that is, if the equation is multiplied
by a power of k when x is replaced by kx, the order of the equation
may be reduced by the substitution x = e\ The work may be simplified
(Ex. 9, p. 162) by the use of
D:y = e-"'A(A - 1) . . . (A - ^ + 1) y. (15)
If the equation is homogeneous with respect to x and y and the dif~
ferentiaU <fo, rfy, cPy, • • -, the order may be lowered by the substitution
* " «*! y =» «*«» where it may be recalled that
I>:y = e-ACA -l)...{D,^n + l)y ....
= e-<-»>'(A + 1) A • • • (A - ^ + 2)«. ^ ^
Finally, if the equation is homogeneous with respect to x considered of
dimetuioru 1, and y considered of dimensions m, that is, if the equation
if multiplied by a power of k when kx replaces x and k'^y replaces y,
the robititution x = «*, y =s 6-«« will lower the degree of the equation.
It may be recalled that
/).-y - if -)'(A + m)(A 4- m - 1) . . . (A + m - 71 + 1)«. (15")
ADDITIONAL ORDINARY TYPES 287
Consider xyy" — xy^ = w' + tey^/ Va* — *■. If in this equation y be repUeed
by ky m that y' and y" are also replaced by *y' and Iry", it appears that the
equation in merely multiplied by Jc* and is therefore homogeneous of the lint
8ort mentioned. Substitute
V = e; / = eH\ y" = e«(«" + O-
Then t^* will cancel from the whole equation, leaving merely
xdxf 1 ^ todx
ar + toz^/Va* - x=» or -:;r---dx =
The equation in the first form is Bernoulli ; in the second form, exact. Then
*=6Vo«-«a+ C and d« = ****
The yariables are separated for the last integration which will determine f = logy
as a function of x.
Again consider x* — ^ = (x« + 2 xy) -^ — 4 y^. If x be replaced by kx and y by
dx* ox
l:*y so that y' is replaced by Jty' and y" remains unchanged, the equation is multi-
plied by Ar* and hence comes under the fourth type mentioned above. Substitute
z = e', y = e^'z, D^y = e'(A + 2)z, D^V = (A + 2) (A + \)z.
Then e*' will cancel and leave z" + 2 (1 — «)«' = 0, if accent* denote differentiation
with respect to i. This equation lacks the independent variable t and is reduced
by the substitution z" = zfdf/dz. Then
There remains only to perform the quadrature and replace z and thy x and y.
103. If the equation may be obtained by differentiation, as
/ dy d-y\ dn an , an , . , an , . ,._.
it is called an exact equation, and n (x, y,y\'--y y^" "*^) = C is an inte-
gral of 4^ = 0. Thus in ease the equation is exact, the order may be
lowered by unity. It may be noted that unless the degree of the nth
derivative is 1 the equation cannot be exact. Consider
where the coefficient of y^*^ is collected into <f>^. Now integrate ^j, par-
tially regarding only y^"-^) as variable so that
|^,rfy<--> = n, £ n. = g -f . . . + ^ y-- + ^y
*l
Then ♦__. = ^,|^__=./J+^,.
That is, the expression ^I' — n/ does not contain y<"> and may contain
no derivative of order higher than n — A*, and may be collected as
288 DIFFERENTIAL EQUATIONS
indicated. Now if ♦ was an exact derivative, so must * - n[ be. Hence
if m^l, the conclusion is that ^ was not exact. If m = 1, the process
of integration may be continued to obtain O^ by integrating partially
with respect to y*""*"*^- And so on until it is shown that ^ is not exact
or until ♦ is seen to be the derivative of an expression Q^ + n^-] = C.
Ab an example consider * = x V" + xy" + (2 xy - 1) y' + y^ = 0. Then
O, rzjxHy" = xV, * - ^{=-^r + (2xy - l)y' + y\
Q, = f- idTf^-xy', * - n; - Oa = 2«yy' + 2/2 = (xy2)'.
As the expresBion of the first order is an exact derivative, the result is
♦ -Oi'-Qa'-(xyV = 0; and *i = xV' - ajy' + xy2 _ Ci = 0
is the new equation. The method may be tried again.
Oj =fx^dy' = x2y', "ir^ - «i = -3xy' + xy^ - Ci-
This is not an exact derivative and the equation ^j = 0 is not exact. Moreover
the equation ^'j = 0 contains both x and y and is not homogeneous of any type
except when Cj = 0. It therefore appears as though the further integration of the
equation i^ = 0 were impossible.
The method is applied with especial ease to the case of
^0^ + ^,^^ + ■■■ + X.-i% + Xj/ - R(.x)=0, (17)
where the coeflScients are functions of x alone. This is known as the
linear equation^ the integration of which has been treated only when
the order is 1 or when the coefficients are constants. The application
of successive integration by parts gives
Oj = Jir^<-»>, n,= {x^ - J^o')y-% ^8 = (^. - x[ + x-)y^--^\ . . . ;
and aft«r n such integrations there is left merely
(^.-^.'-x-f---+(-l)"-^X,+(-l)"X,)2^-i2,
which is a derivative only when it is a function of x. Hence
X,- x:_, -f ... +(_i)"-ix^ +(_1)»X^ = 0 (18)
is the condition that the linear equation shall be exact, and
^y-'> + (J^» - -YOy<-«H(J^,- Jt; +j^-)y(-«) + ... = Cnd^ (19)
is tlie first solution in case it is exact.
As an ezanple Uke ir"' + iT cosx - 2y'8inx - y cosx = sin2x. The test
"'i "• -^i + ^l - ^i" =-CO8X+2cO8X-CO8X = 0
ADDITIONAL ORDINARY TYPES 239
Is satisfied. The integral is therefore y^'H- y'ooex* ysinx =— JootSz -f C,.
ThiH equation Htill satisfies the test for exactnees. Hence it may be iotegrmtcd
again witit tlie result / + yco8x = — ^8in3z-f C^x + C,. This belongi to Um
linear type. The final result is therefore
y = e-'
rJe'^'iC^x + C^^dx + C,e-*'«+ J(l - slnx).
BZERCISES
1. Integrate these equations or at least reduce them to quadratdtos :
(a) 2xy"V' = y"«- a*, (/S) (1 + a^*)/' + 1 + i^ = 0,
(7) 2/'^ + aV =0, (3) V^ - m«/" = e", (.) x*yi^ + flV = 0,
( n « VV = 2, (i») xy" + yl= 0, (0) v"'v" = 4,
( i ) (1 - x^O r - ^/ = 2, ( c) y»' = y/r\ (X) /' = /(y).
(m) 2(2a-y)y" = l + y^, (k) yy" _ y-^ - yV = 0,
(o) y/'+y^+ 1=0, (ir)2y" = 6i', (p) yV' = «•
2. Carry the integration as far as possible in these cases:
(a) a:V' = (w^'y'^ + ny^)^, 03) mxV = (y - xyO',
(7) x*y" = (y - «y')'» («) a^V - x»y' - xV* + 4 y« = 0,
(e) X- V + x-*y = iy'*, (r) ayy" + 6y^ = W{c* + a;*)"*.
3. Carry the integration as far as possible in these cases :
(a) (y« + X) y'" + 6 yy'y'' + y"^2y'^ = 0, (fi) y'y" - yx*y' = xy«,
(7) x^'" + Sx^y'y'' + Ox^yy" + 9xV^ + 18 xyy' + Sy* = 0,
(«) y + Sxy' + 2yy^ + (x« + ^y^')y" = 0,
(<) (2xV + ic*y)y" + ^xV'* + 2xyy' = 0.
4. Treat these linear equations:
(a) xy"+2y = 2x, (/S) (x« - l)y" + 4xy' + 2y = 2x,
(7) y" - y' cotx + y C8c«x = cos'x, («) (x* - x)y" + (3x - 2)y' + y = 0,
(e) (X - x»)y'" + (1 - 6x2) y" _ 22/ + 2y = 6x,
(f ) (x» + x2 - 3x + l)y'" + (9x2 + 6x - 9)y" + (18x + 6)y' +^6y = x»,
(77) (X + 2)2y"' + (X + 2)y" + y' = 1, {$) xV' + Sxy' + y = x,
(i) (x3-x)y'" + (8x2-3)y"+ 14xy'+4y = 0.
5. Note that Ex. 4 {$) comes under the third homogeneous tyx>ef and that Ex. 4
(17) may be brought under that type by multiplying by (x + 2). Test sundry of Exs.
1, 2, 3 for exactness. Show that any linear equation in which the coefficients are
polynomials of degree less than the order of the derivatives of which they are the
coefficient«, is surely exact.
6. Sometimes, when the condition that an equation be exact is not satisfied, it
is possible to find an integrating factor for the equation so that after multiplication
by the factor the equation becomes exact. For linear equations try x". Integrate
(a) xV + (2x* - x)y' - (2x» - l)y = 0, (/9) (x« - x*) y" - xV - « F = ©•
7. Show that the equation y" -f Py' + Qy'* = 0 may be reduced to quadratures
1° when /* and Q are both functions of y, or 2° when both are functions of x^ or 8*
when P is a function of x and Q is a function of y (integrating factor l/y^^ In
each ciuse find the general expression for y in terms of quadratures. Integrate
y" + 2y'cotx + 2y'atany = 0.
240 DIFFERENTIAL EQUATIONS
8. Find and diacun the curves for which the radius of curvature is proportional
to the nditui r of the curve.
9. If the nulius of curvature R is expressed as a function B = R{s) of the arc s
■Mtarnd from lome point, the equation /J = K («) or « = « {R) is called the intrinsic
eqwdUm of the curve. To find the relation between x and y the second equation
may be differentiated as ^ = «'(i?)dR, and this equation of the third order may be
aolvod. Show that if the origin be taken on the curve at the point 8 = 0 and if the
be tangent to the curve, the equations
the curve parametrically. Find the curves whose intrinsic equations are
(a) R-a, ifi) aR = 8'^ + a«, (7) R^ + s'^ = IQa^.
10. Given F = y («> + X^y(" -D + ^2^^" " 2) + • • • + X„ -ly' + X„y = 0. SI ow
that if M, a function of x alone, is an integrating factor of the equation, then
♦ = ^^ - (Xim)<-»> + (X2m)<«-2) + (- i)—HXn-i^y + (- 1)»X,^ = 0
la the equation satisfied by fi. Collect the coefficient of fj. to show that the condition
that the given equation be exact is the condition that this coefficient vanish. The
equation ♦ = 0 is called the adjoint of the given equation F = 0. Any integral n
of the adjoint equation is an integrating factor of the original equation. Moreover
Dotetbat
fuFdz = My<— 1) + (/iXj - ;i') y(»-2) 4. . . . + (_ l)nry^dx,
or d[MF - (- l)«y*] = d [m|/<« -D + (AtXj - fi')y(*'-^^ +-..]= da.
Hence if fiF is an exact differential, so is y*. In other words, any solution y of the
original equation is an integrating factor for the adjoint equation.
104. Linear differential equations. The equations
X^D'y + A,Z)"-V + . . . 4- X,_,Di/ + A.y = 0 ^""^^
are linear differential equations of the nth order ; the first is called the
eompUte equation and the second the reduced equation. If y^, y^, y^, • • •
are any solutions of the reduced equation, and C^, C^, C^, • • • are any
constants, then y = C^y^ + C^^ + Cg^g + . . . is also a solution of the
reduced equation. This follows at once from the linearity of the reduced
equation and is proved by direct substitution. Furthermore if / is any
solution of the complete equation, then y -f 7 is also a solution of the
complete equation (cf. § 96).
As the equations (20) are of the nth order, they will determine yf">
and, by differentiation, all higher derivatives in terms of the values of
*^P>y'»"'t y*'"*^ Hence if the values of the n quantities y^, y^, • •, yj""^^
which correspond to the value ar = x, be given, all the higher derivatives
arc determined (f f 87-88). Hence there are n and no more than n arbi-
trary conditions that may be imposed as initial conditions. A solution
ADDITIONAL ORDINARY TYPES 241
of the equations (20) which contains n distinct arbitrary constants if
called the general solution. By distinct is meant that the constantB
can actually be determined to suit the n initial conditions.
If l/^, i/^f • " } l/n ^® ^ solutions of the reduced equation, and
y'^c^i H-c^t 4---fcy„ (21)
then y is a solution and y', • • • , y<"-*> are its first n — 1 derivatives. If
Xq be substituted on the right and the assumed corresponding initial
values i/of l/oy '"i yo""^^ ^ substituted on the left, the above n equations
become linear equations in the n unknowns Cj, C,, • • •, C, ; and if they
are to be soluble for the C% the condition
^1 ^3 '•' 1/n
y\ y\ '•' y'n
^(jyvy%y"'^yn) =
y(n-l) y(-l) ... y(-l)
^0 (22)
must hold for every value of jr = x,,. Conversely if the condition does
hold, the equations will be soluble for the C^s.
The determinant W{y^^ y^, •••, //J is called the Wronskian of the n
functions y^, y^, •••, y^. The result may be stated as : If n functions
Vv yv ' " i y* w^ich are solutions of the reduced equation, and of which
the Wronskian does not vanish, can be found, the general solution of the
reduced equation can be written down. In general no solution of the
equation can be found, whether by a definite process or by inspection ;
but in the rare instances in which the n solutions can be seen by inspec-
tion the problem of the solution of the reduced equation is completed.
Frequently one solution may be found by inspection, and it is therefore
important to see how much this contributes toward effecting the solution.
If y^ is a solution of the reduced equation, make the substitution
y = y^z. The derivatives of y may be obtained by Leibniz's Theorem
(§ 8). As the formula is linear in the derivatives of «, it follows that
the result of the substitution will leave the equation linear in the new
variable z. Moreover, to collect the coefficient of z itself, it is necessary
to take only the first term y5*>« in the expansions for the derivative y<*\
"•^""^ (A><"> + A-^i"->) + . . . + A-.../, + .Yj,,)« = 0
is the coefficient of z and vanishes by the assumption that yj is a solu-
tion of the reduced equation. Then the equation for x is
P/") + V -^> -h • -f P. . a*" + P. .!«' = 0 ; (23)
i42 DIFFERENTIAL EQUATIONS
and if «' be taken as the variable, the equation is of the order n — 1.
It therefore appears that the knowledge of a solution y^ redtices the order
of the equation by one.
Now if y I y I • • • , y^ ^ere other solutions, the derived ratios
■>©■ ^-^s} ■■■■ -'-fey <-'
would be solutions of the equation in «' ; for by substitution,
are all solutions of the equation in y. Moreover, if there were a linear
relation C^z\ + C^z\ ^ h Cj,_^z^_^ — 0 connecting the solutions <,
an integration would give a linear relation
connecting the p solutions y,. Hence if there is no linear relation (of
which the coefficients are not all zero) connecting the p solutions y.- of
the original equation, there can be none connecting the p — \ solutions
z\ of the transformed equation. Hence a knowledge ofp solutions of
the original reduced equation gives a new reduced equation of which
p — 1 solutions are known. And the process of substitution may be
continued to reduce the order further until the order n — ^ is reached.
Ab an example consider the equation of the third order
(1 - X) y"' + (x2 - 1) y" - a; V+ xy = 0.
Here a simple trial shows that x and e' are two solutions. Substitute
Then (1 - x)zf" + (x* - 3x + 2)z" + (x2 _ 3aj + l)z' = 0
is of the second order in 7f. A known solution is the derived ratio (x/e*)'.
if = (xe-')' = c-'(l - x). Let z' = e-^(l - x)w.
From this, «" and r"' may be found and the equation takes the form
(l-x)u»" + (l + x)(x-2)w;' = 0 or *^' = xdx- -?-dx.
w' X — 1
This is a linear equation of the first order and may be solved.
loguT'zz Jx«-.21og(x-l) + C or «;' = Ciei*'(x - l)-a.
i0=C,Jci''(x-l)-2dx+C„
ADDITIONAL ORDINARY TYPES 243
The value for y is thus obtained in tenns of quadrature!!. It may be shown that In
ca^e the equation in of the nth degree with p known aolutiona, the final result will
call for p (n — j)) quatlratures.
105. If the general solution y = C^i/^ + C^^ H + C^, of the reduced
equation has been found (called the compUmentary function for the
complete equation), the general solution of the complete equation may
always be obtained in terms of quatlratures by the important and far-
reaching method of the variation of constants due to Lagrange. The
question is : Cannot functions of a; be found so that the expression
y = C,{x) y, -h Cjix) y, 4- • • • + C.(x) y. (24)
shall be the solution of the complete equation ? As there are n of these
functions to be determined, it should be possible to impose n — 1 condi-
tions uj)on them and still find the functions.
Differentiate y on the supposition that the C's are variable.
y' = c,y\ -h c^; -f • • + c^;+ yiC\ + y,c; + • • • + y.C
As one of the conditions on the Cs suppose that
yiC\ + y^c; + • • • -h y^c, = 0.
Differentiate again and impose the new condition
y\C\^y^C^ + '" + y:c',= ^,
so that y" = Ciy'( + C^'i + h C^tj;.
The differentiation may be continued to the (n — l)st condition
yS-^^c; + yf-^^c^ + • . . + yi— ^c; = 0,
and y<- "*> = C^y^;^ -^> + C^^^ -^ 4. . . . -h C^i» -«.
Then y^") = C^yS") -f C^?> + • • • + C^i"^
+ yi-^>c; + y^^-'K'^ + • • • -h yi-^^c;.
Now if the expressions thus found for y, y', y", •••, y^""*^ y^"^ be
substituted in the complete equation, and it be remembered that y^,
Uii • •• f I/m ^^6 solutions of the reduced equation and hence give 0 when
substituted in the left-hand side of the equation, the result is
y{''-''C[ + yi'^-'^C, -f . • • + yi^-'^c: = R-
Hence, in all, there are n linear equations
yiC{ +yac; 4....-hy.c; =o,
y'lC; +y,c, 4. .. + y;c; =o,
yf-^C\ + yi-'^^C, + . . + yi-«c: = 0,
(25)
244 DIFFERENTIAL EQUATIONS
connecting the derivatives of the C's ; and these may actually be solved
for those derivatives which will then be expressed iu terms of x. The
C*8 niay then be found by quadrature.
Ab an example consider the equation with constant coefficients
(D« + D)y = 8ecx with y = C^ + Cg cos x + CgSina;
a« the aoluiion of the reduced equation. Here the solutions y^ , 2/2 » Vz ™^y ^® taken
M 1 ootx, sin 2 respectively. The conditions on the derivatives of the C's become
l>7 diroct substitution in (25)
CTJ +cosxC'J +8inxCi =0, — sinxC^ + cosxC^ = 0, — cosxCg — sinxCj =8ecx.
Henoe Cj = sec x, C^ = — 1, Cg = — tan x
and Cj = logtan(ix+ Jir) + q, C^ = -x-\-c^, Cg = logcosx + Cj.
Henoe y^t^Jc logtan(Jx + Jir) + (Cg — x)cosx + (Cg + logcosx)sinx
Is the general solution of the complete equation. This result could not be obtained
by any of the real short methods of §§ 96-97. It could be obtained by the general
method of § 96, but with little if any advantage over the method of variation of
constants here given. The present method is equally available for equations with
variable coefficients.
106. Linear equations of the second order are especially frequent in
practical problems. In a number of cases the solution may be found.
Thus 1* when the coefficients are constant or may be made constant by
a change of variable as in Ex. 7, p. 222, the general solution of the
reduoed equation may be written down at once. The solution of the
complete equation may then be found by obtaining a particular integral
/ by the methods of §§ 95-97 or by the application of the method of
variation of constants. And 2° when the equation is exact, the solution
may be had by integrating the linear equation (19) of § 103 of the first
order by the ordinary methods. And 3° when one solution of the re-
duced equation is known (§ 104), the reduced equation may be com-
pletely solved and the complete equation may then be solved by the
method of variation of constants, or the complete equation may be
solved directly by Ex. 6 below.
Otherwise, write the differential equation in the form
The substitution y = w« gives the new equation
rf*« I2du \dz 1 R
^+i«S + ^js^+;i(''" + ^«' + ««)*=-• (26-)
If « be determined go that the coefficient of z' vanishes, then
ADDITIONAL ORDINARY TYPES 245
Now 4° if Q — i P' — J P' is constant, the new reduoed equation in
(27) may be integrated ; and 5' if it is A;/x*, the equation may also be
integrated by the method of Ex. 7, p. 222. The integral of the com-
plete equation may then be found. (In other cases this method may
be useful in that the equation is reduced to a simpler form where solu-
tions of the reduced equation are more evident.)
Again, 8upiK)se that the independent variable is changed to «. Then
Now &" ii z'^— ±Q will make «" + P«' = kz^, so that the coefficient
of dy/dz becomes a constant /:, the equation is integrable. (Trying if
«'* = ± Qz"^ will make z" 4- Pz* = kz'^/z is needless because nothing in
addition to G° is thereby obtained. It may happen that if z be deter-
mined so as to make «" -f- P«' = 0, the equation will be so far simpli-
fied that a solution of the reduced equation becomes evident.)
d^y 2 dy a*
Consider the example — - + --^+— y = 0. Here no solution is apparent,
djc* X dx X*
Hence compute Q— IP' — \P^. This is a^/x* and is neither constant nor propor-
tional to 1/x*. Hence the methods 4° and 5° will not work. From z^ = Q =z a*/3e*
or z' = a/i^y it appears that z" + Pz' = 0, and 6° works ; the new equation is
f^ + y = 0 with z = -?.
dz* X
The solution is therefore seen immediately to be
1/ = Cj cosz — Cj sin z or y = C^ cos(a/x) + C, sin (a/x).
If there had been a right-hand member in the original equation, the solution could
have been found by the method of variation of constants, or by some of the short
methods for finding a particular solution if R had been of the proper form.
EXERCISES
1. If a relation C^y^ + Cjj/, H + C^n — 0, with constant coeflBcients not all 0,
exists between n functions y^^y^, • • , y^oi x iov all values of z, the functions are
by definition said to be linearly dependent; if no such relation exists, they are said
to be linearly independent. Show that the nonvanishing of the Wronakian is a
criterion for linear independence.
2. If the general solution y = C^y^ + C^y^ + • • • + C^n is the same for
Xoi/<") + Xiy<— 1) + • • + X^ = 0 and P^y <»> + P,y(» -» + ••• + P«y = 0,
two linear equations of the nth order, show that y satisfies the equation
{X,P, - A',P,)yC-i) + . . . + {X\Po - A'oP.)y = 0
of the (n — l)st order ; and hence infer, from the fact that y contains n arbitrary
constants corresponding to n arbitrary initial conditions, the important theorem:
If two linear equations of the nth order have the same general solution, the oorrfr*
spouding coefficients are proportional.
246 DIFFERENTIAL EQUATIONS
3« M Fi I yt» • • • » y* *"* ** independent solutions of an equation of the nth ordelf,
•bow th»t the equation may be taken in the form Wiy^, Vz, " •, Vn, y) = 0, ;-
4. Show that if, in any reduced equation, Xn-\ + xXn = 0 identically, then x
Is a tolation. Find the condition that «■• be a solution ; also that e^ be a solution.
5. Find by inspection one or more independent solutions and integrate :
(a) (l + x*)y"-2x/ + 22/ = 0, (/9) xy'' -{■ {1 - x) y' - y = 0,
(7) (ox- 6ac«)i/"- 01^4-26^ = 0, («) iy" + xy'- (x + 2)y = 0,
(.)(iogx + i-i+i)r'+(iogx+i+i-i)r+(i-i)(.'-^)=o,
(f) y»'-xy'"+xy'-y = 0, (17) (4x2-x + l)y"'+8xV-4xy'-8y = 0.
6. If y| is a known solution of the equation y" + Py' + Qy = R ot the second
order, show that the general solution may be written as
y = C,y, + C,y,/e-/-g + Vri^f^'^fv.e^''^^
{dxf
7. Integrate:
(a) xy"-(2z+l)y'+(x + l)y = x2-a;-l,
{^) y" - xV + xy = X, (7) xy'' + (1 - x)y' - y = e^,
(') /' — «/ + (x — l)y = i2, (c) y^'sin^x + y'sinxcosx — y = x— ^inx.
8. After writing down the integral of the reduced equation by inspection, apply
the method of the variation of constants to these equations :
(a) (/>« + 1) y = tan x, (/S) (2)2 + 1) y = sec2 x, (7) (D - l)2y = e»(l - x)- 2,
(a) (l-x)y" + xy'-y = (l-x)2, (e) (l~2x + x2)(y-'-l)-x2y^'+2xy'-y = 1.
9. Integrate the following equations of the second order :
(a) 4xV + 4xV + (x2 + l)2y = 0, (/3) y'' - 2y'tanx - (a2 + 1)2/ = 0,
(7) xy" + 2y'-xy = 2e% (5) y"sinx + 2y'cosx + Sysinx = e%
(«) y" + y'tanx + y cos2x = 0, (f) (1 - x2)y" - xy' + 4y = 0,
(i|) y" + (2e'- l)y' + e^'y = e*^, (^) x«y" + 3xV + y = x-2.
10. Show that if X^" + ^^y' + X^y = 12 may be written in factors as
{X^ + XjD + X,)y = {p^B + g^) (p^D -{■ q^)y = R,
mhen the factors are not commutative inasmuch as the differentiation in one
factor la applied to the variable coefficients of the succeeding factor as well as
to D, then the solution is obtainable in terms of quadratures. Show that
9»P, + l)iPi+Pig2 = Xi and 91^2 +^1^2= ^2-
In this manner integrate the following equations, choosing p^ and p^ as factors of
X^ and determining q^ and g, by inspection or by assuming them in some form and
applying the method of undetermined coefficients :
(<») xy" + (1 - x)y' _ y = e^, 09) 8x2y" + (2 - 6x2)y' -4 = 0,
(»8x«y"+(2 + 6x-6x«)y'-4y = 0, (S) (x2- l)y"_ (3x + l)y'-x(x-l)y = 0,
(«) axy" + (8a + ««)y' + 86y = 0, (f) xy" - 2x(l + x)y' + 2(1 + x)y = x«.
11. Int^grato tbete equationa in any manner :
ADDITIONAL ORDINARY TYPES 247
(7) r + y'tanx + yco««x = 0, («) y"-2/n-?)y'+ /n«-2^\y=:^,
(e) (l-x«)r-a5y'-c«y = 0, (n (a«-«»)y"-8zy'-12y=b.
(i) /' + 2x-V-n*y = 0, (c) y"- 4x1^ + (4x« - 8)y = e^,
(X) y'' + 2 n/ cot nx + (m* - n*) y = 0, (m) /' + 2 (x-» + Bx-«) y' + Ax-^v = 0.
12. If i/i and y, are solutioua of /' + ly -^^ R = 0, show by ftHmlnatlng Q and
integrating that -
ViVi - ViV\ = Ce'J '*^.
What if (7 = 0? IfC^^O, note that y^ and y\ cannot vanish together ; uid if
l/i(a) = yi(^) = 0, use the rehitioii (yoy'i)a • (VtVW = *>0 to show that as yj. and
I/jj^ have opposite signs, y^a and 2^2 b have opposite signs and hence y^{^ = 0 where
a < ( < 6. Hence the theorem : Between any two roots of a solution of an equation
of the second order there is one root of every solution independent of the given
solution. What conditions of continuity for y and y' are tacitly assumed here t
107. The cylinder functions. Suppose that C^(x) is a function of x
whic'h is different lor different values of n and which satisfies the two
equations
C.-,(*)-C.„(x) = 2£c.(a:), C...(a;) + C.,.(x) = ^C.(x). (29)
Such a function is called a cylinder function and the index n is called
the order of the function and may have any real value. The two equa-
tions are supposed to hold for all values of n and for all values of x.
They do not completely determine the functions but from them follow
the chief rules of operation with the functions. For instance, by addi-
tion and subtraction,
(■:(^) = C,_,(x) - ^ C,(x) = ^ C.(x) - C. „(x). (30)
Other relations which are easily deduced are
i>,[.r"C\(a2-)] = ax-C,_,(ax), /),[.r-C,(ax)] = - «x-C,+i(ar), (31)
Z),[x^C,( V^)] = i V^^C,_,( V^), (32)
c; (x) = - C^(x), C_,(x) = (- iyc,(x), n integral, (33)
C,(x)K(x) - C:{x)K^(x) = C,^,(x)K^ix) - C.(a:)ir.^,(x) = ^, (34)
where C and K denote any two cylinder functions.
The proof of these relations is simple, but will be given to show the nee of (29).
In the first case differentiate directly and substitute from (29).
i>«[x"C«(nrx)]=x»
= x"
aD«C.(ax) + ^C.(ax)l
aCn-i{ax) - a— C^iax) + ^ C.(«)l.
ax * J
248 DIFFERENTIAL EQUATIONS
The ieoond of (81) is proved similarly. For (32), differentiate.
2 LVax vax J
Next (88) is obtained 1° by substituting 0 for n in both equations (29).
C.iix) - C^{x) = 2 Co' (X), C_i(x) + Cj(x) = 0, hence Cj (x) = - C^ix) ;
and 2** by substituting successive values for n in the second of (29) written in the
form xC,-i + «C,+i = 2 nC,. Then
«C-i + xCj = 0, xC-i + zCo = - 2 C_i, xCo + xCg = 2 Cj,
«C_8 + xC-i = - 4 C_2, xCi + xCg = 4 Cg,
xC_4 + xC_2 = - 6 Cj, xCg + xC^ = 6 Cg,
and so on. The first gives C_i = — C^. Subtract the next two and use C_i + C^ = 0.
Then C_s — C, = 0 or C_2 = (— 1)*C2. Add the next two and use the relations
already found. Then C_8 + Cg = 0 or C_8 = (— T^fC^- Subtract the next two,
and 80 on. For the last of the relations, a very important one, note first that the
two expressions become equivalent by virtue of (29) ; for
CnK - ^»^'» = - (^nKn - CnK^ +1 - - On^n + C„ +i^„ .
X X
Now £[X(C,+1£:. - CiKn+l)] = Cn+lKn - C„ir„+i + xKn(Cn - ^ C„+i)
+ xC^+i^^iTn - ^„+i^ - a^^«+i(^ C^« - Cn+l)
Hence x((!7a^iJra — C«Z'»+i) = const. = J., and the relation is proved.
The cylinder functions of a given order n satisfy a linear differential
equation of the second order. This may be obtained by differentiating
the first of (29) and combining with (30).
2 c; = c., - c^. = 2^ c... - 2 c. + ^ C.
Hotiw
Thb equation is known as BessePs equation; the functions C^(x)y which
have been called cylinder functions, are often called BesseVs functions.
Prom the equation it follows that any three functions of the same order
» are oonnected by a linear relation and there are only two independent
fnnotkmi of any given order.
ADDITIONAL ORDINARY TYPES 249
By a change of the independent variable, the Bessel equation may
take on several other forms. The easiest way to find them is to operate
directly with the relations (31), (32). Thus
= ~ a:— »C.^, + 2(n + l)a;— »C.^j - t/r-C.,
Hence ^^ + i_ll— i -^ + y = 0, y = x-C.(ar). (36)
Again g + (L:iMg4.y=0, y = x-C.(a;). (37)
Also xy" -I- (1 + n) y' -f y = 0, y = x" ^C,(2 V^. (38)
And xy" + (1 _ n) y' -f y = 0, y = x^ C.(2 Vi). (39)
In all these differential equations it is well to restrict x to positive values
■ ■
intosmuch as, if n is not specialized, the powers of x, as x", x' ", ar*, x~*, are
not always real.
108. The fact that n occurs only squared in (35) shows that both
C^{x) and C_^{x) are solutions, so that if these functions are inde-
pendent, the complete solution is y = aC^ -}- hC_^. In like manner the
equations (36), (37) form a pair which differ only in the sign of n.
Hence if H^ and H_^ denote particular integrals of the first and second
respectively, the complete integrals are respectively
y = aff^ + bH_^-^'' and y = aH_^ + bH^'"',
and similarly the respective integrals of (38), (39) are
y = a[^-\- hl_^x-'* and y = al_^ -f i/,x",
where /„ and I_^ denote particular integrals of these two equations. It
should be noted that these forms are the complete solutions only when
the two integrals are independent. Note that
/.(x) = x-i''C.(2 V^), C,(x) = (ix)-/,(J x^. (40)
As it has been seen that C^ = (— 1)*C_, when n is integral, it foUows
that in this case the above forms do not give the complete solution.
A particular solution of (38) may readily be obtained in series by the
method of undetermined coefficients (§ 88). It is
/.(.) = |;„^. «. = ,.,(„^,)(i-^>;...(„^.y (41)
as is derived below. It should be noted that /_, formed by changing
the sign of n is meaningless when n is an integer, for the reason that
250 DIFFERENTIAL EQUATIONS
from a certain point on, the coefficients a, have zeros in the denominator.
The determination of a series for the second independent solution when
n is integral will be omitted. The solutions of (35), (36) corresponding
to /,(x) are, by (40) and (41),
^-V.(a:) = 24,^(1 A (42')
where the factor n ! has been introduced in the denominator merely to
conform to usage.* The chief cylinder function C^(x) is ./»(a;) and it
always carries the name of Bessel.
To derive the series for /»(x) write
/, = ao + a^x + a^^ + • • • + a*_ix*-i + • • • ,
/; = a^ + 2 agX + 3 a^x^ + . . . + (fc - 1) a*_ix*-2 + . . . ,
r;= 2a^ +S-2a^ +"- + {k-l){k-2)ak-ixk-»-\--",
0 = K + a^in + 1)] + X [ai + a^2 (n + 2)] + x^ [a^ + a^S (n + 3)]
+ . . . + x*-i[aA_i + a*A:(n + A:)] + . • . .
Hence a^ + a^in + 1) = 0, a^ + 032 (n + 2) = 0, • • • , ak-i + a^A* (n + ^) = 0,
1
(1 + n)
X
"0
— a.
n + l' ^^ 2(n + 2) 2 ! (n + l)(n + 2) '
(-l)*«o
at =
A:!(n + l)...(n + A:)
If now the choice a^ = 1 is made, the series for 7„(x) is as given in (41).
The famous differential equation of the first order
xy' - ay + 6y2 _ c^n^ ^43^
known as RiccatVa equation^ may be integrated in terms of cylinder functions.
Note that If n = 0 or c = 0, the variables are separable ; and if 6 = 0, the equation
U linear. As these cases are immediately integrable, assume hen ?£ 0. By a suitable
change of variable, the equation takes the form
A compariaon of this with (89) shows that the solution is
n = AI_^{- 6cf) + BIa(- 6cf) . (- beer,
(430
which In terms of Bessel functions J becomes, by (40),
a
• If fi !• not Intoffral, both nl and (n + <) I must be replaced (§ 147) by Tin + 1) and
rCn + < + !).
ADDITIONAL ORDINARY TYPES 251
The value of y may be found by substitution and use of (29).
■xR-'
J.(2x«vClic/n) + il/ .(2x«vClc/ii)
m
where A denotes the one arbitrary constant of Integration.
It is noteworthy that the cylinder functions are sometimes expreHible in
of trigonometric f uiictionH. For when n = \ the equation (35) has the integrals
y = A sinx + ^cosx and y = xi[^Ci(x) + J5C_ i(x)].
Hence it is permissible to write the relations
xiCi(x) = 8inx, xic_ i(x) = coex, (45)
where C is a suitably chosen cylinder function of order \. From these equations
by application of (29) the cylinder functions of order p + i, where p is any integer,
may be found.
Now if Uiccati's ecjuation is such that h and c have opposite signs and a/n is
of the fonn p + i, the integral (44) can be expressed in terms of trigonometric
functions by using the values of the functions Cp + 1 just found in place of the /*s.
Moreover if b and c have the same sign, the trigonometric solution will still hold
formally and may be converted into exponential or hyperbolic form. Thus Riccati's
equation is integrable in terms of the elementary functions when a/n = p + | no
matter what the sign of be is.
EXERCISES
1. Prove the following relations:
{a) 4C;' = C,_2-2C«+ C« + 2, 03) xC, = 2(n + 1) C,+i - xC. + ,,
(7) 2'»C;" = C^_8-3C„_i + 3Cn+i-C« + 8, generalize,
(a) xC, = 2(n+l)C«+i-2(n + 3)C« + s + 2(n+6)C, + 6-xC. + «.
2. Study the functions defined by the pair of relations
F, _i (X) + F, +1 (X) = 2 A F,(x), Fn -1 (X) - F. +i (x) = ? F,(x)
ax X
especially to find results analogous to (30)-(36).
3. Use Ex. 12, p. 247, to obtain (34) and the corresponding relation in Ex. 8
4. Show that the solution of (38) is y = ^7, f + BJ,.
J x» + »/;
5. Write out five terms in the expansions of Iq, Jj, /_ i , J^, Ji.
/2 1
6. Show from the expansion (42) that \ I ^-Ji{x) = -sin x.
7. From (45), (29) obtoin the following :
xiC|(x) = '■ — cosx, xiC5(x) = /— — ijsinx — cosx,
xic_j(x) = -sinx-^^, xic_|(x) = ? sin x + (^- l) cosx.
252 DIFFERENTIAL EQUATIONS
8.ProTebylntegraUonbypart«:/^dx = ^ + 6^ + 6.8/!^.
•. Suppoee C,(x) and Kn{x) so choeen that ^ = 1 in (84). Show that
y = AC.{x) + BK.ix) + L[Kn{x)f^dx - C„(x) /^dx]
Is the Integral of the differential equation xV + xy' + {x^ - n^) y = Lx-^.
10. Note that the solution of Riccati's equation has the form
f{x)-\-Ag{x) andshowthat ^ + P{x)y + Q{z)y^ = B{x)
^ F(x) + ^0(x) dx ^' ' ^'
will be the form of the equation which has such an expression for its integral.
11. Integrate these equations in terms of cylinder functions and reduce the
results whenever possible by means of Ex. 7 :
(a) xy' - 6y + y2 + x2 = 0, (/3) xy' - 3y + y^ = x^,
\i) V" + ye2' = 0, (5) x^y" + rucy' + (6 + cx^w) y = 0.
18. Identify the functions of Ex. 2 with the cylindier functions of ix.
18. Let(x«-l)P: = (n + l)(P„+i-xP„), P;+i = xP; + (n + 1)P„ (46)
be taken as defining the Legendre functions Pn(x) of order n. Prove
(a) (x«-l)P: = n(xP»-P«_i), OS) (2n + l)xPn = (n + l)P„+i + nP„_i,
(7) (2n+l)P, = P;+i-P;_i, (5) (l-x2)P;'-2xP; + n(n + l)P„ = 0.
14. Show that P,q; - KQ,. = — ^ and P„Q„+i - P„+iQ„ = ^
x^ — 1 n + 1
where P and Q are any two Legendre functions. Express the general solution of
the differential equation of Ex. 13 (3) analogously to Ex. 4.
15. Let u = x^ — 1 and let D denote differentiation by x. Show
X>i+1m»+i = I>»+i(uu«) = uD^+iu" + 2(n + l)xi>»u« -\-n{n + l)l>»-iw»,
l>i-fiif»-i-i = D-Du^+i = 2 (n + 1) I>'(XM'') = 2 (n + l)xD«u« + 2 n (n + l)I>-^u\
Hence show that the derivative of the second equation and the eliminant of D^~^u^
between the two equations give two equations which reduce to (46) if
P (x) = ^ J*l /-a _ i\n fWhen n is integral these are
2"-n!dx* ' \Legendre'' 8 polynomials.
16. Determine the solutions of Ex. 13 (S) in series for the initial conditions
(a)P,(0) = l, P;(0) = 0, 03)P„(O) = O, P;(0) = 1.
17. Take P© = 1 and P^ = x. Show that these are solutions of (46) and compute
'*!♦'*»♦ '*4 '">n» Ex. 18 (^). If X = cos ^, show
P,s|oosS^ + |, P, = fco83^ + |cos^, P4 = ifcos4^ + |^cos2^ + A-
18. Wrtte Ex. 18 («) as £ [(1 - x") P^] + n(n + 1) P, = 0 and show
''-I ^/-i L dx dx J
ADDITIONAL ORDINARY TYPES 258
Integrate by part«, aaeume the f uncUona and their derivatlTet are finite, and ibow
J*^PnPJU = 0, if n:jtm.
19. By Bucceasive integration by part« and by reduction formula* ahow
^-1 •* 2^*{nl)*J-i dx« dx- 2».n!J-i ^ '
/»+i . 2
and / Ptdx = . n integral.
J -I 2n+l
20. Show
+L.„.,_ r+» *'(^«-i)"
r x"'P,dx= r a!«?-li_^ = 0, IffiKn.
Determine the value of the integral when m = n. Cannot the results of Exs. 18, 19
for m and n integral be obtained simply from these results ?
23 2* Z*
21. Consider (38) and its solution J^ = l-x + — ^- — + — j when
n = 0. Assume a solution of the form y = I^v + to ao that
cPw dvo , . „ dlf.dn . .. d*t) . d» ^
is the equation for lo if c satisfies the equation xo" + t' = 0. Show
r = ^ + 51ogx, x.,"+u,' + u, = 25-2^ + 2Bx^-^^ + ....
By assuming w = a^x + OjX* + • • • , determine the a's and hence obtain
and {A + TJlogx)/^ + tc is then the complete solution containing two constants.
As AIq is one solution, 2?logx • /(, + mj is another. From this second solution for
n = 0, the second solution for any integral value of n may be obtained by differ
entiation ; the work, however, is long and the result is somewhat complicated.
CHAPTER X
DIFFERENTIAL EQUATIONS IN MORE THAN TWO VARIABLES
109. Total differential equations. An equation of the form
P(x, y, z)dx -h Q{x, y, z)dy + R(x, y, z)dz = 0, (1)
involving the differentials of three variables is called a total differen-
tial equation. A similar equation in any number of variables would
also be called total; but the discussion here will be restricted to the
case of three. If definite values be assigned to x, y, «, say a, h, c, the
equation becomes
Adx + Bdy + Cdz = A(x - a) + B{y - h) + C (z - c) = 0, (2)
where a;, y, z are supposed to be restricted to values near a, b, c, and
represents a small portion of a plane passing through (a, b, c). From
the analogy to the lineal element (§ 85), such a portion of a plane may
be called 2^. planar elemerit. The differential equation therefore repre-
sents an infinite number of planar elements, one passing through each
point of space.
Now any family of surfaces F{xj y, z) = C also represents an infinity
of planar elements, namely, the portions of the tangent planes at every
point of all the surfaces in the neighborhood of their respective points
of tangency. In fact
dF = F'Jx + F'^dy + F'Jz = 0 (3)
is an equation similar to (1). If the planar elements represented by
(1) and (3) are to be the same, the equations cannot differ by more
than a factor /*(a;, y, «). Hence
f; = nP, f; = fiQ, f; = fiR.
If a function F{Xy y, z) = C can be found which satisfies these condi-
tions, it is said to be the integi-al of (1), and the factor fi (x, y, z) by
which the equations (1) and (3) differ is called an integrating factor
of (1). Compare § 91.
It may happen that /x = 1 and that (1) is thus an exact differential.
In this case the conditions
p; = q;, Q; = iz;, k = p:, (4)
254
MORE THAN TWO VAKIABLEB 856
which arise from F^; =. f;;, f;; - F^, /^ = F^, must be
Moreover if these conditions are satisfied, the equation (1) will be
an exact equation and the integral is given by
where x^^y^^x^ may be chosen so as to render the integration as simple
ius ])ossible. The proof of this is so similar to that given in the case of
two variables (§ 92) as to be omitted. In many cases which arise in
practice the equation, though not exact, may be made so by an obvious
integrating factor.
As ail example take zxdy — yzclx + x^dz = 0. Here the conditions (4) are not
fill tilled but the integrating factor l/x'^z is suggested. Then
xdy — ydx
t^ Z \x I
X*
is at once perceived to be an exact differential and the integral is y/x + log* = C.
It appears therefore that in this simple case neither the renewed application of the
conditions (4) nor the general formula for the integral was necessary. It often
happens that both the integrating factor and the integral can be recognized at once
as above.
If the equation does not suggest an integrating factor, the question
arises, Is there any integrating factor ? In the case of two variables
(§ 94) there always was an integrating factor. In the case of three
variables there may be none. For
da dp „ cti dQ
dn 8R .. da dp
If these equations be multiplied by R, P, Q and added and if the result
be simplified, the condition
KB-|)-(t-£)-K¥-2)-» «
is found to be imposed on P, Q, R if there is to be an integrating fac-
tor. This is called the condition of integrabUUy. For it may be shown
conversely that if the condition (6) is satisfied, the equation may be
integrated.
Suppose an attempt to integrate (1) be made as follows : First assume
that one of the variables is constant (naturally, that one which wilJ
256 DIFFERENTIAL EQUATIONS
make the resulting equation simplest to integrate), say z. Then
Pdx -I- Qiy = 0. Now integrate this simplified equation with an inte-
grating factor or otherwise, and let F(x, y, z) — <ji{z) be the integral,
where the constant C is taken as a function <^ of z. Next try to deter-
mine ^ so that the integral F{Xj y, z)= <(> (z) will satisfy (1). To do
this, differentiate ;
F^rfx -I- F^dy -h F^dz = d<l>.
Compare this equation with (1). Then the equations*
f; = xp, f; = \Qy (f; - xr) dz = d4»
must hold. The third equation (F^ — \R) dz = d<f> may be integrated
provided the coefficient S = F^ — XR of dz is a function of z and <^,
that is, of z and F alone. This is so in case the condition (5) holds. It
therefore appears that the integration of the equation (1) for which (5)
holds reduces to the succession of two integrations of the type discussed
in Chap. VIII.
Ab an example take {2x^ + 2xy ■\' 2xz^ -{- l)dx + dy + 2 zdz = 0. The condition
(2z« + 2xy + 2xz^ + 1)0 + 1 (- 4:xz) + 2z{2x) = 0
of integrability is satisfied. The greatest simplification will be had by making x
constant. Then dy + 2 zdz = 0 and y -\- z^ = <p (x) . Compare
dy + 22dz = d0 and (2x^ ■}■ 2xy + 2xz^ -{■l)dx -\- dy + 2zdz = 0.
Then X = 1, - {2 x^ -\- 2 xy + 2 xz^ ■\- 1) dx = dip ;
or -(2a;« + l + 2x^)da; = d0 or dip + 2x<pdx =- {2x^ + l)dx.
This is the linear type with the integrating factor e^. Then
c**(d^ + 2x^dx)=-e'*(2x2 + l)da; or e^^p =- C e=^{2x^ + l)dx + C.
Hence y + «* + «-«• Je«'(2x«+l)dx=Ce-^ or e^{y -{■ z^) + fe'^{2x^ + l)dx= C
is the soluUon. It may be noted that e*" is the integrating factor for the original
equation :
«^[(2a!« + 2xy + 2xz* + l)dx + dy + 2zdz] = dle^{y + z^) + fe^{2x^ + l)dxl.
Tooomplete the proof that the equation (1) is integrable if (5) is satisfied, it is
aaoMsary to show that when the condition is satisfied the coefficient S = F^- \R
U A function of t and F alone. Let it be regarded as a function of x, F, z instead
of X, y, t. It is necessary to prove that the derivative of 5 by x when F and z are
oooMant Is tero. By the formulas for change of variable
/5S\ ^/asx /dS\dF /es\ _/ss\ dF
Vte/,.. K^Jr.u \dFj ex' W,,."VaFJ^.ei;"*
^A +*S^^' X is not an integrating factor of (1), but only of the reduced equation
MORE THAN TWO VARIABLES 257
But f; = XP and f; = XQ, and hence q(^ -^(~) =Q(^ •
WV, dx\dz / dzdx dx H
fds\ - /?^_?^\ . T>«x „ax
\5z/y,a \^ 5X/ d« to'
Then Q>(^
W/x.. Vto ay/ ^to dy
«(si.r'['e-s)-(S-s)-(s-g)]
L to ay J
where a term has been added in the first bracket and subtracted in the
Now a« X is an integrating factor for Pdx + Qdy, it follows that (XQ)^ = (XP)J ;
only the first bracket remains. By the condition of integrability this, too, vi
and hence iS as a function of 2, F, z does not contain x but is a function of F and
z alone, as was to be proved.
110. It has been seen that if the equation (1) is integrable, there is
an integrating factor and the condition (5) is satisfied ; also that con-
versely if the condition is satisfied the equation may be integrated.
Geometrically this means that the infinity of planar elements defined
by the equation can be grouped upon a family of surfaces F(a;, yyX)=iC
to which they are tangent. If the condition of integrability is not satis-
fied, the planar elements cannot be thus grouped into surfaces. Never-
theless if a surface G (x, y, «) = 0 be given, the planar element of (1)
which passes through any point (x^, y^^ z^ of the surface will cut the
surface G = 0 in a certain lineal element of the surface. Thus upon the
surface G (a;, y^ z)=0 there will be an infinity of lineal elements, one
through each point, which satisfy the given equation (1). And these
elements may be grouped into curves lying upon the surface. If the
equation (1) is integrable, these curves will of course be the intersections
of the given surface 6? = 0 with the surfaces F= C defined by the
integi-al of (1).
The method of obtaining the curves upon G (x, y, «) = 0 which are
the integrals of (1), in case (5) does not possess an integral of the form
-^(^j //) -) = C, is as follows. Consider the two equations
Pdx 4- Qdy -f Rdz = 0, G'^dx + G'^dy -J- C^dz = 0,
of which the first is the given differential equation and the second is
the differential equation of the given surface. From these equations
258 DIFFERENTIAL EQUATIONS
one of the differentials, say rf«, may be eliminated, and the correspond-
ing variable x may also be eliminated by substituting its value obtained
by solving G {x, y, z) = 0. Thus there is obtained a differential equa-
tion Mdx + Ndy = 0 connecting the other two variables x and y. The
integral of this, F(x, y) = C, consists of a family of cylinders which cut
the given surface G = 0 in the curves which satisfy (1).
Consider the equation ydx + xdy — (x + y + «) dz = 0. This does not satisfy the
condition (5) and hence is not completely integrable ; but a set of integral curves
may be found on any assigned surface. If the surface be the plane 2; = x + 2/, then
ydx + xdy — (x + y + z) dz = 0 and dz = dx + dy
give (« + z)dx + (y + z)dy = 0 or (2x + y)dx + (2y + x)dy = 0
by eliminating dz and z. The resulting equation is exact. Hence
x' + xy + y2 = c and z = x + y
give the curves which satisfy the equation and lie in the plane.
If the equation (1) were integrable, the integral curves may be used to obtain
tlie integral surfaces and thus to accomplish the complete integration of the equa-
tion by Mayer's method. For suppose that F(x, y,z) = C were the integral surfaces
and that F(x, y, z) = F(0, 0, Zq) were that particular surface cutting the z-axis at Zq.
The family of planes y = Xx through the z-axis would cut the surface in a series
of curves which would be integral curves, and the surface could be regarded as
generated by these curves as the plane turned about the axis. To reverse these
considerations let y = Xx and dy = \dx ; by these relations eliminate dy and y from
(1) and thus obtain the differential equation Mdx + Ndz = 0 of the intersections
of the planes with the solutions of (1). Integrate the equation as/(x, z,\) = C and
determine the constant so that/(x, z, X) =/(0, Zq, X). For any value of X this gives
the intersection of F{x, y, z) = F(0, 0, Zq) with y = Xx. Now if X be eliminated by
the relation X = y/x, the result will be the surface
f(x, 2, 1) =/(o, Zo, 1^, equivalent to ^(x, y, z) = F(0, 0, Zo),
which is the integral of (1) and passes through (0, 0, Zq). As Zq is arbitrary, the
solution contains an arbitrary constant and is the general solution.
It is clear that instead of using planes through the z-axis, planes through either
of the other axes might have been used, or indeed planes or cylinders through any
line parallel to any of the axes. Such modifications are frequently necessary owing
to the fact that the substitution /(O, Zq, X) introduces a division by 0 or a log 0 or
some other impossibility. For instance consider
lf^ + «fy-ydz = 0, y = Xx, dy = \dx, X^x^dx + Xzdx - Xxdz = 0.
Then Xdx-t-'^"'^^ = 0, and Xx - - =/(x, z, X).
* X
But here /(O, Zq, X) is impossible and the solution is illusory. If the planes (y - 1) = Xx
pfturing through a line parallel to the z-axis and containing the point (0, 1, 0) had
been used, the result would be
dv=i\dx, (1 + Xx)adx + Xzdx-(1 + Xx)dz = 0,
MORE THAN TWO VARIABLES 269
Hence z = — «o or x =— «a = C,
1 + Xz y
is the tM)lution. The same rcKult could have been obtained with z = Xr or y = X (z — a)
In the latter case, however, care should be taken to U8e/(z, «, X) =/(a, t^ X).
EXERCISES
1. Test these equations for exactness ; if exact, integrate ; if not exact, find an
integrating factor by inspection and integrate :
(a) (y + z)dz + (z -f x) dy -f- (z + y)dz = 0, (/3) y^^ + zdy-ydz = 0,
{y) xdx + ydy - Va« - z* - y^dz = 0, (8) 2z{dx - dy) ■{■ (x ^ y) dz = 0»
(e) (2x + y'^ + 2xz)dx + 2xydy + x^dz = 0, (r) zydx = zxdy + yHz,
(i,) x(y - 1) (z - l)dx + y (z - 1) (z - l)dy + 2(x - 1) (y - l)dz = 0.
2. Apply the test of integrability and integrate these:
(a) (x2 - y« - z^)dx + 2zydy + 2zzdz = 0,
(/3) (X + y2 + z'-» + l)dx + 2ydy + 2zdz = 0,
(7) (y + «)*dz + zdy = (y + a)dz,
(a) (1 _ x2 _ 2 y«2) dz = 2 xzdz + 2 yz^dy,
(e) z^dx^ + yHy^ - z^dz* + 2 xydxdy = 0,
(f) z(xdz + ydy + zdz)^ = (z^ - z* - y*) (zdz + ydy + zdz)dz.
3. If the equation is homogeneous, the substitution x = uz^ y = vz^ frequently
shortens the work. Show that if the given equation satisfies the condition of inte-
grability, the new equation will satisfy the corresponding condition in the new
variables and may be rendered exact by an obvious integrating factor. Int^^rate :
{a) (y* •k-yz)dx + (xz + z^)dy + (y^ -xy)dz = 0,
(/9) (x2y - y8 - y^z) dx + (xy^ - xH - x») dy + {xy^ + x^y) dz = 0,
(7) {y^ + yz-\-z^)dx + (x« + XZ + z^)dy + {x^ -^^ xy + y*) dz = 0.
4. Show that (5) does not hold ; integrate subject to the relation imposed :
(a) ydx + xdy — (z -f y + z)dz = 0, z + y + z = fc or y = kx,
(/5) c (xdy + ydy) + Vl - a^x* - t^y^dz = 0, a^x^ + 6*y« + c*z* = 1,
(7) dz = aydx + 6dy, y = kx or z'* + y* + 2* = 1 or y =/(z).
5. Show that if an equation is integrable, it remains integrable after any change
of variables from z, y, z to it, c, 10.
6. Apply Mayer's method to sundry of Exs. 2 and 8.
7. Find the conditions of exactness for an equation in four yariables and write
the formula for the integration. Integrate with or without a factor :
(a) (2 z + y2 + 2 xz) dx + 2 xydy + x^dz + du = 0,
{p) yzudx + xzudy + xyudz + xyzdu =0,
(7) {y-\-z-\- u)dx i.{x-\-z-\- u)dy -^ {x + y •\- u) dz -{■ {x -^ y -i- z)du = 0,
(a) u{y + z)dx -i-uiy -^z-^- l)dy -^ udz - (y -^^ z)du = 0.
8. If an equation in four variables is integrable, it must be so when any one of
the variables is held constant. Hence the four conditions of int^rability obtained
by writing (5) for each set of three coefficients must hold. Show that the conditions
860 DIFFERENTIAL EQUATIONS
ta9 Mtlffled in the following cases. Find the integrals by a generalization of the
Bietbod In the text by letting one variable be constant and integrating the three
nnaining terms and determining the constant of integration as a function of the
foaith in such a way as to satisfy the equations.
(a) z{y ■{■ z)dx -{- z (u - x)dy -h y {x - u)dz •{■ y {y •{■ z)du = 0,
(P) uyzdx -^uzxlogxdy -{-uxy log xdz — xdu = 0.
9. Try to extend the method of Mayer to such as the above in Ex. 8.
10. If 0{x, y^z) = a and H{x, y, z) = b are two families of surfaces defining a
family of curves as their intersections, show that the equation
(o;f; - g:h;) dx + (g:k - ^X) dy + (g;^; - g-^K) dz = o
is the equation of the planar elements perpendicular to the curves at every point
of the curves. Find the conditions on G and H that there shall be a family of sur-
faces which cut all these curves orthogonally. Determine whether the curves below
have orthogonal trajectories (surfaces) ; and if they have, find the surfaces :
(a) y = X + a, z = z + 6, (fi) y = ax + 1, z = bx,
{y) X* + y' = a*, « = 6, (5) xy = a, xz = 6,
(«) X* + y« + 2* = a2, xy = 6, (r) x^ + 2y^ + Sz^ = a, xy-\-z = h,
(ti) \ogxy = az, X + y -\-z = b, (0) y = 2 ax -{■ a^, z = 2bx + U^.
11. Extend the work of proposition 3, § 94, and Ex. 11, p. 234, to find the normal
derivative of the solution of equation (1) and to show that the singular solution may
be looked for among the factors of pr^ = 0.
12. If F = Pi + Qj + Kk be formed, show that (1) becomes F.dr = 0. Show
that the condition of exactness is VxF = 0 by expanding VxF as the formal vector
product of the operator V and the vector F (see § 78). Show further that the condi-
tion of integrability is F.(VxF) = 0 by similar formal expansion.
13. In Ex. 10 consider VG and V-ff. Show these vectors are normal to the sur-
faces G = a, H = b, and hence infer that (VG')x(V-H') is the direction of the inter-
section. Finally explain why dr.(VGxVH) = 0 is the differential equation of the
orthogonal family if there be such a family. Show that this vector form of the family
reduces to the form above given.
111. Systems of simultaneous equations. The two equations
S=/(a^,y,«), ^ = g(x,y,z) (6)
In the two dependent variables y and z and the independent variable x
oonstitute a set of simultaneous equations of the first order. It is more
onstomary to write these equations in the form
dx dy dz
X{x,y,z)- Y(x,y,z)^ Z(x,y,zy ^^^
which is symmetric in the differentials and where X:Y: Z = l:f: g.
At any assigned point a?^, y^, z^ of space the ratios dxidyidz of the
difbrentiaU ate determined by substitution in (7). Hence the equations
MORE THAN TWO VARIABLES 261
fix a definite direction at each point of space, that is, they determine a
lineal element through each point. The problem of integration is to
combine these lineal elements into a family of conres P(x, y, «)» C,,
G(xy y, z) = Cj, dejjending on two parameters C, and C,, one curve pass-
ing through each point of space and having at that point the direction
determined by the equations.
For the formal integration there are several allied methods of pro.
cedure. In the first place it may happen that two of
dx _dy dy _dz dx dx
are of such a form as to contain only the variables whose dififerentiaU
enter. In this case these two may be integrated and the two solutions
taken together give the family of curves. Or it may happen that one
and only one of these equations can be integrated. Let it be the first
and suppose that F(x, y) = Cj is the integral. By means of this inte-
gral the variable x may be eliminated from the second of the equations
or the variable y from the third. In the respective cases there arises
an equation which may be integrated in the form G (y, «, C^) = C, or
G(x, Zy F) = C^y and this result taken with F(Xj y) = C^ will determine
the family of curves.
Consider the example — = ^-^ = — Here the two equations
yz xz y
xdx ydy . xdx ,
— = ^-^ and — = dz
y X z
are integrable with the results x* — y« = C^, x'* — z* = C,, and these two integrals
constitute the solution. The solution might, of course, appear in very different
form ; for there are an indefinite number of pairs of equations F(x, y, «, C^) = 0,
G (x, y, 2, Cj) = 0 which will intersect in the curves of intersection of x* — y* = C, ,
and x« - z« = C2 . Iti fact (y« + C,)* = (2* + Cj)« is clearly a solution and could
replace either of those found above.
Consider the example — = — ^ = Here
xa-y«-z« 2xy 2xz
— = — , with the integral y = C,«,
y z
is the only equation the integral of which can be obtained directly. If y be elimi-
nated by means of this first integral, there results the equation
. = — or 2x«ix + r(C«+l)««-z«]d« = 0.
This is homogeiieous and may be integrated with a factor to give
x« + (Cf + l)z* = C^ or x« + y* + «* = C^.
Hence y = C^z, x« + y« + z« = C^
is the solution, and represents a certain family of circles.
262 DIFFERENTIAL EQUATIONS
Another method of attack is to use composition and division.
dx_dy_dz__ Xdx + yidy + vdz
X" Y" Z" \X^fxY-^vZ ' ^ ^
Her© X, /I, V may be chosen as any functions of (x, y, z). It may be
possible 80 to choose them that the last expression, taken with one of
the first three, gives an equation which may be integrated. With this
first integral a second may be obtained as before. Or it may be that
two different choices of X, /a, v can be made so as to give the two desired
integrals. Or it may be possible so to select two sets of multipliers that
the equation obtained by setting the two expressions equal may be
solved for a first integral. Or it may be possible to choose A., /a, v so
that the denominator XX -}- ftF-f- v-^ = 0, and so that the numerator
(which must vanish if the denominator does) shall give an equation
\dx -f i»dy -\-vdz = ^ (9)
which satisfies the condition (5) of integrability and may be integrated
by the methods of § 109.
Consider the equations — = — ^ = Here take X, u, v
x^^y^^yz x^ + y^-xz (x + y)z
a« 1, — 1, — 1 ; tlien \X + /lY + vZ = 0 and dx — dy — dz = 0 is integrable as
X — y — « = Cj . This may be used to obtain another integral. But another choice
of X, /i, v as z, y, 0, combined with the last expression, gives
xdz + vdx dz
Hence x^y — z = C^ and x^ + y2 _ q^^^
will serve as solutions. This is shorter than the method of elimination.
It will be noted that these equations just solved are homogeneous. The substi-
tution z = uz, 1/ = w might be tried. Then
udz -^ zdu _ vdz ■{■ zdv _ dz _ zdu _ zdv
u* + o* + r~u* + t>« — w~ u + 0 ~ »a — uw + » ~ 'a;i^uv-'U*
or <^u _ dv _ dz
»* — UD + 0 ~ u* — M» — u "" T*
Now the first equations do not contain z and may be solved. This always happens
in the homogeneous case and may be employed if no shorter method suggests itself.
It need hardly be mentioned that all these methods apply equally to
the case where there are more than three equations. The geometric
picture, however, fails, although the geometric language may be contin-
ued if one wishes to deal with higher dimensions than three. In some
the introduction of a fourth variable, as
r
MORE THAN TWO VABIABLES 268
is useful in solving a set of equations which originally contained only
three variables. This is particularly true when X, Y, Z are linear with
constant ccH'fficients, in which case the methods of § 98 may be applied
witli / as inch'ixindent variable.
112. Simultaneous differential equations of higher order, as
S-'(S;-«("*'s;'f> ;5(-f)-(-*f'f>
especiiilly those of the second order like these, are of constant occur-
rence in mechanics; for the acceleration requires second derivatives
with respect to the time for its expression, and the forces are expressed
in terms of the coordinates and velocities. The complete integration of
such equations requires the expression of the dependent variables as
functions of the independent variable, generally the time, with a num-
ber of constants of integration equal to the sum of the orders of the
equations. Frequently even when the complete integrals cannot be
found, it is possible to carry out some integrations and replace the
given system of equations by fewer equations or equations of lower
order containing some constants of integration.
No special or general rules will be laid down for the integration of
systems of higher order. In each case some particular combinations of
the equations may suggest themselves which will enable an integration
to l)e performed.* In problems in mechanics the principles of energy,
momentum, and moment of momentum frequently suggest combinations
leading to integrations. Thus if
x" = X, y" = r, «" = Z,
where accents denote differentiation with respect to the time, be multi-
plied by dx^ dyj dz and added, the result
x"dx -t- y''dy + z"dz = Xdx -f Ydy + Zdz (11)
contains an exact differential on the left ; then if the expression on the
right is an exact differential, the integration
i {x"' + y"' -h z"') =fxdx + Ydy + Zdz-^C (ll*)
* It is possible to differentiate the griven equations repeatedly and eliminate all Um
depentlent variables except one. The resulting; differential equation, say in Zand (. may
then be treated by the methods of previous chapters ; but this is rarely i
when the equation is linear.
yx
^64 DIFFERENTIAL EQUATIONS
(mn be performed. This is the principle of energy in its simplest form.
If two of the equations are multiplied by the chief variable of the other
and subtracted, the result is
yx^^-xy^'=yX-xY (12)
and the expression on the left is again an exact differential; if the
right>hand side reduces to a constant or a function of tj then
•-xy<=Jf{t) + C (12')
is an integral of the equations. This is the principle of moment of
momentum. If the equations can be multiplied by constants as
Ix" 4- my" + nz" = IX -\- mY -\- nZ , (13)
so that the expression on the right reduces to a function of t, an inte-
gration may be performed. This is the principle of momentum. These
three are the most commonly usable devices. .
As an example : Let a particle move in a plane subject to forces attracting it
toward the axes by an amount proportional to the mass and to the distance from
the axes ; discuss the motion. Here the equations of motion are merely
Then dx^-¥dy^=-k{xdx + ydy) and (^^\ (^^''=^k{x^ + y^) + C.
In thia case the two principles of energy and moment of momentum give two
integrals and the equations are reduced to two of the first order. But as it happens,
the original equations could be integrated directly as
dt*
(|)' = -...C«,
VC2 - kx^
^..=-^. gy=-^»..« ^=|L==..
The coneUinte C* and K^ of integration have been written as squares because they
are neoeenrily positive. The complete integration gives
Vkx = Csin (Vkt + Cj), V^ = ^sin (Vkt + K^).
Am another example : A particle, attracted toward a point by a force equal to
r/m* + AV* per unit mass, where m is the mass and h is the double areal velocity
and r is the distance from the point, is projected perpendicularly to the radius vec-
tor at the distance Vmh ; discuss the motion. In polar coSrdinates the equations
of notion are
MORE THAN TWO VAKIABLE8 266
The second Integrates directly a« r*d^/dt = A where the concUmt of Inte^nMion k
is twice the areal velocity. Now subetitute in the flxat to eliminate 4.
(fir A«
Now as the particle is projected perpendicularly to the radius, dr/di = 0 at the
start when r = VmA. Hence the constant C is h/m. Then
dr ,. , rM^ .. . VriOydr
= (tt and — ^ = dt give == = df.
\in m« \ Am
Hence ^K^\^^^C or i^ _ -L = l^L±i2! .
\r* A r" Am mA
Nuw if it be assumed that 0 = 0 at the start when r = VmA, we find C = Q
mA
Hence r* = is the orbit
1 + 0*
To find the relation between 0 and the time,
ii^iiip - hdt or ^ ;= di or ( = mtan-U,
if the time be taken sm t = 0 when 0 = 0. Thus the orbit is found, the expraakm
of <p ius a function of the time is found, and the expression of r as a function of the
time in obtainable. The problem is completely solved. It will be noted that the
conKUintB of integration have been determined after each integration by the initial
cumlitions. This 8implities the subsequent integrations which might In fact be
Impossible in terms of elementary functions without this simplification.
EXERCISES
1. Integrate these equations :
dx dy dz dx dy dz
yz xz , xy y^ x^ x*y*z*
xz yz xy yz xz x -^ y
dx dy dz ,>.. ^ ^V
2. Integrate the equations: (a)
y X 1 + 2* ' -1 Sy + iz 2y + 6»
dx dy dz
bz — cy ex — az 'ay-^bx
dx dy dz dx dy dz
y-\-z x + « « + y
x^-^y^ 2xy xz -^^ yz* '
. dx dy dz dx dy dz
^*^y«x-2x*^2i/*-x«y^«(x«-y»)' ^*^ x(y- «) "^ y(«-x) "^ «(x-y)
/>\ ^ _ ^y _ ^ ^ _ "" ^ _ <fe
x(y« - 2«) y (z« - x«) 2(x« - y«) ' x(y« - z«) y (r« + x*) «(x« + y«)
_d^^_dy_^_dz_^ _dj^^ dy ^_j£_^^
y-« x+y x+« y-» 2+y+t «+f+<
266 DIFFERENTIAL EQUATIONS
S Show that the differential equations of the orthogonal trajectories (curves
of the* family of surfaces F(x, v.z) = C are dx:dy:dz = F;,:F;: F; . Find the curves
which cut Uie following families of surfaces orthogonally :
(a) a»x« + b^y^ + cH* = C, (/3) xyz = C, (7) y^ = Cxz,
(8) y = xtan(z + C), (e) y = xtanCz, (f) z = Cxy,
4. Show that the solution oi dx.dy :dz = X : Y : Z, where X, F, Z are linear
expressions in x, y, «, can always be found provided a certain cubic equation can
be solved.
5. Show that the solutions of the two equations
^+r(ax + 6y) = ri, §+T{a^x + ¥y)=T,,
dt oi
where T, Tj, T, are functions of t, may be obtained by adding the equation as
^ix^ly)-^\T{x-\-ly)=T^^-lT^
dt
after multiplying one by i, and by determining \ as a root of
X2 - (a + 6')^ + «^' - a'd = 0.
6. Solve: (a) t^ + 2{x-y) = t, t^ + x + 5y = t\
at dt
{P) tdx = {t- 2x)dt, tdy = {1x + ty + 2x- t)dt,
Idx _ mdy _ ndz _dt
mn{y — z)~nl{z — x) Im {x — y) t
7. A particle moves in vacuo in a vertical plane under the force of gravity alone.
Integrate. Determine the constants if the particle starts from the origin with a
velocity V and at an angle of a degrees with the horizontal and at the time t = 0.
8. Same problem as in Ex. 7 except that the particle moves in a mediimi which
retlsts proportionately to the velocity of the particle.
9. A particle moves in a plane about a center of force which attracts proportion-
ally to the distance from the center and to the mass of the particle.
10. Same as Ex. 9 but with a repulsive force instead of an attracting force.
11. A particle is projected parallel to a line toward which it is attracted with
a force proportional to the distance from the line.
12. Same as Ex. 11 except that the force is inversely proportional to the square
of the distance and only the path of the particle is wanted.
13. A particle is attracted toward a center by a force proportional to the square
of the distance. Find the orbit.
14. A particle is placed at a point which repels with a constant force under
which the particle moves away to a distance a where it strikes a peg and is
ddlected off at a right angle with undiminished velocity. Find the orbit of the |
■obwquent motion.
15. Show that equatioiui (7) may be written in the form drxF = 0. Find the
ooodiUuu on F or on JT, F, Z that the integral curves have orthogonal surfaces.
MORE THAN TWO VARIABLES 267
113. Introduction to partial differential equations. An equation
which contains :i dcjHmdent variable, two or more independent varia'
bles, and one or more partial derivatives of the dependent variable
with res{)ect to the inde})endent vai'iables is CBkiled A partial differential
equation. The equation
is clearly a linear partial differential equation of the first order in one
dependent and two independent variables. The discussion of this equi^
tion preliminary to its integration may be carried on by means of the
concept of j)lanar elements^ and the discussion will immediately suggest
the method of integration.
When any point {x^^ y^, z^ of space is given, the co('t!i<irnts /'. Q, R
in the equation take on definite values and the derivativrs y; ami y
are connected by a linear relation. Now any planar element through
(*o> ^o ^o) ^^y ^ considered as specified by the two slopes p and q ; for
it is an infinitesimal portion of the plane » — ^q = p{x — x^-{- q{y — y^
in the neighborhood of the point. This plane contains the line or lineal
element whose direction is
dx:dy:dz=P:Q:R, (15)
because the substitution of P, Q, R for dx = x — x^j dy = y — y^,
dz = z — z^ in the plane gives the original equation Pp -\- Qq = R,
Hence it appears that the planar elements defined by (14), of which
there are an infinity through each point of space, are so related that all
which pass through a given point of space pass through a certain line
through that point, namely the line (15).
Now the problem of integrating the equation (14) is that of grouping
the i)lanar elements which satisfy it into surfaces. As at each point
they are already grouj^d in a certain way by the lineal elements through
which they pass, it is first advisable to group these lineal elements into
curves by integrating the simultaneous equations (15). The integrals
of these equations are the curves defined by two families of surfaces
F(xy y, z) = Cj and G (ar, y, z) = C^. These curves are called the charac-
teristic curves or merely the characteristics of the equation (14). Through
each lineal element of these curves there pass an infinity of the planar ele-
ments which satisfy (14), It is therefore clear that if these curves be in
any wise grouped into surfaces, the planar elements of the surfaces must
satisfy (14) ; for through each point of the surfaces will |)ass one of the
curves, and the planar element of the surface at that point must there-
fore pass through the lineal element of the curve and hence satisfy (14).
968 DIFFERENTIAL EQUATIONS
To group the curves F(x, y, «) = C'j, G(x, y, z) = C^ which depend
on two parameters Cj, C^ into a surface, it is merely necessary to intro-
duce some functional relation C^=f(C^ between the parameters so
that when one of them, as Cj, is given, the other is determined, and
thus a particular curve of the family is fixed by one parameter alone
and will sweep out a surface as the parameter varies. Hence to integrate
{lA)f first integrate (15) and then write
G(x,y,z) = ^lF(x,y,z)-] or $(F, 6^) = 0, (16)
where ♦ denotes any arbitrary function. This will be the integral of
(14) and will contain an arbitrary function ^.
As an example, integrate {y—z)p + {z — x)q = x — y. Here the equations
J^ = j!L- = -^ give a;2 + y2 + z2 = Ci, x+y-{-z = C^
y^t «— X x—y
M the two integrals. Hence the solution of the given equation is
x + y + 2 = *(x^ + y^ + 2;^) or *(x2 4■y2 + 2^x + y + 2;) = 0,
where ♦ denotes an arbitrary function. The arbitrary function allows a solution
to be determined which shall pass through any desired curve ; for if the curve be
/(x, y, z) = 0, (7 (x, y, z) = 0, the elimination of x, y, z from the four simultaneous
equations
F(x, y, z) = Ci, G (X, y, z) = Cj, /(x, y, z) = 0, g (x, y,z) = 0
will express the condition that the four surfaces meet in a point, that is, that the
curve given by the first two will cut that given by the second two ; and this elimi-
nation will determine a relation between the two parameters Cj and C^ which will
be precisely the relation to express the fact that the integral curves cut the given
curve and that consequently the surface of integral curves passes through the given
curve. Thus in the particular case here considered, suppose the solution were to
through the curve y = x*, z = x ; then
x« + y« + 2« = Ci, x-|-y + z = C2, y = x2, z = x
2x«-l-x* = Ci, x2 + 2x = C2,
whence (C,« + 2 C, - C^)* -^-BC}- 24 C^ - 16 C^C^ = 0.
The ■abedtution of Cj = x« + y" + z^ and C^ = x-\-y -^ z in this equation will
give the •olution of (y — z)p + (« — x) g = x — y which passes through the parabola
y = «•»« = «.
114. It will be recalled that the integral of an ordinary differ-
ential equation /(x, y, y', • • •, y^"^) = 0 of the nth order contains n con-
•lantSy and that conversely if a system of curves in the plane, say
P(Xf y, Cj, . . . , C,) = 0, contains n constants, the constants may be
eliminated from the equation and its first n derivatives with respect
to & It has now been seen that the integral of a certain partial
differential equation contains an arbitrary function, and it might be
MORE THAN TWO VARIABLES 269
inferred that the elimination of an arbitrary function would give
rise to a partial differential equation of the first order. To show
this, suppose F(ar, y, «) = *[6'(ar, y, «)]. Then
follow from partial differentiation with respect to x and y ; and
(f:g; - f;g:;)p + (f^g: - F;c?;)y = f;^; - f^g;
is a partial differential equation arising from the elimination of ♦'.
More generally, the elimination of n arbitrary functions will give rise
to an equation of the nth order; conversely it may be believed that
the integration of such an equation would introduce n arbitrary func-
tions in the general solution.
Ab an example, eliminate from z = ♦ (xy) + * (x + y) the two arbitrary func-
tions ♦ and '^. The first differentiation gives
p = *'.y + >!'', 5 = *'.x + *', p — 9 = (y — «)♦'.
Now differentiate again and let r = — z i « = 1 1 = — - . Then
dx^ dxdy dy^
r -«=-*' + (y - x)*" . y, 8 - t = *' + (y - x)*" • X.
These two equations with p — g = (y — x) *' make three from which
, . , . , x + y . , S^z ^ , ^ d^z , S'z x-\-y/dz dz\
xr-(x + y)8 + ye= -(p-q) or x— -- (x + y) — — + y-- = r--ir)
X — y ax* dxdy dy^ x — y\dz dy/
may be obtained as a partial differential equation of the second order free from
* and 4^. The general integral of this equation would be z = * (xy) + ♦ (x + y).
A partial differential equation may represent a certain definite type
of surface. For instance by definition a conoidal surface is a surface
generated by a line which moves parallel to a given plane, the director
plane, and cuts a given line, the directrix. If the director plane be taken
as « = 0 and the directrix be the 2;-axis, the equations of any line of
the surface are
z = C^y y = C^, with C^ = ^(c;)
as the relation which picks out a definite family of the lines to form a
particular conoidal surface. Hence z = *(y/a*) may be regarded as the
general equation of a conoidal surface of which « = 0 is the director
plane and the «-axis the directrix. The elimination of * gives px-\-qy = 0
as the differential equation of any such conoidal surface.
Partial differentiation may be used not only to eliminate arbitrary func-
tions, but to eliminate constants. For if an equation /(or, y, «, Cj, C^ = u
contained two constant's, the equation and its first derivatives with respect
to X and y would yield three equations from which the constants could
270 DIFFERENTIAL EQUATIONS
be eliminated, leaving a pai-tial differential equation F(x, y, z, p,q) = 0
of the first order. If there had been five constants, the equation with
its two first derivatives and its three second derivatives with respect
to X and y would give a set of six equations from which the constants
could be eliminated, leaving a differential equation of the second order.
And 80 on. As the differential equation is obtained by eliminating the
constants, the original equation will be a solution of the resulting dif-
ferential equation.
For example, eliminate from z = Ax^ + 2 Bxy + Cy^ + Dx + Ey the five con-
sUnts. The two first and three second derivatives are
p = 2i4x + 25y + A q = 2Bx + 2Cy-\-E, r = 2A, s = 2 B, t = 2C.
Hence z =~ ^rx^ - ^ty^ - sxy -^ px + qy
is the differential equation of the family of surfaces. The family of surfaces do
not constitute the general solution of the equation, for that would contain two
arbitrary functions, but they give what is called a complete solution. If there had
been only three or four constants, the elimination would have led to a differential
equation of the second order which need have contained only one or two of the
second derivatives instead of all three ; it would also have been possible to find three
or two simultaneous partial differential equations by differentiating in different ways.
115. If /(a:,y,«, C„C^=0 and F(x, y, z, p, q) = 0 (17)
are two equations of which the second is obtained by the elimination of
the two constants from the first, the first is said to be the complete solu-
tion of the second. That is, any equation which contains two distinct
arbitrary constants and which satisfies a partial differential equation of
the first order is said to be a complete solution of the differential equa-
tion. A complete solution has an interesting geometric interpretation.
The differential equation F = 0 defines a series of planar elements
through each point of space. So does f(x, y, «, C^, C^ = 0. For the
tangent plane is given by
^1
dx
(^-a^o) +
(.-.o)4-|
(«-^o) = 0
^ith /(x„ 2/^, z^, Cj, C^ = 0
M the condition that C^ and C^ shall be so related that the surface
puses through (x^, y^^ z^. As there is only this one relation between
the two arbitrary constants, there is a whole series of planar elements
through the point As /(«, y, «, C,, Q = 0 satisfies the differential equa-
tion, the planar elements defined by it are those defined by the differen-
tial equation. Thus a complete solution establishes an arrangement of
the planar elements defined by the differential equation upon a family
of sarfaoes dependent upon two arbitrary constants of integration.
MORE THAN TWO VARIABLES 271
From the idea of a solution of a partial differential equation of the
first order as a surface pieced together from planar elements which
satisfy the e(iuation, it appears that the envelope (p. 140) of any family
of solutions will itself U; a solution ; for each point of the envelope is
a point of tangency with some one of the solutions of the family, and
the planar element of the envelope at that point is identical with the
planar element of the solution and hence satisfies the differential equa-
tion. This observation allows the general solution to be determined from
any comj)lete solution. For if in /(x, y, «, Cj, C^) = 0 any relation
C\ = ^((\) is introduced between the two arbitrary constants, there
arises a family depending on one parameter, and the envelope of the
family is found by eliminating C^ from the three equations
C. = *(C.), ^ + ^^^ = 0, /=0. (18)
As the relation C^ = *(Cj) contains an arbitrary function ♦, the result
of the elimination may be considered as containing an arbitrary func-
tion even though it is genei*ally impossible to carry out the elimination
except in the case where * has been assigned and is therefore no longer
arbitrary.
A family of surfaces /(a;, y, «, Cj, C^) = 0 depending on two param-
eters may also have an envelope (p. 139). This is found by eliminat-
ing Cj and Cj from the three equations
f(x,y,z,C„C.;) = 0, ^ = 0, ^ = 0.
This surface is tangent to all the surfaces in the complete solution.
This envelope is called the singtilar solution of the partial differential
equation. As in the case of ordinary differential equations (§ 101), the
singular solution may be obtained directly from the equation ; • it is
merely necessary to eliminate p and q from the three equations
dF dF ^
F(x,i/,z,p,q) = 0, ^ = 0, ^ = 0.
The last two equations express the fact that F(/>, q) = 0 regarded as
a function of p and q should have a double point (§ 67). A reference
to § 67 will bring out another point, namely, that not only are all the
surfaces represented by the complete solution tangent to the singular
solution, but so is any surface which is represented by the general
solution.
• It is hardly necessary to point out the fact that, as in the caM of ordinary equationa,
extraneous factors may arise in the elimination, whether of Cf, C^ or of p, q.
272 DIFFERENTIAL EQUATIONS
EXERCISES
1. Integrate these linear equations :
(tf) aap + yrg = ly, (/3) a (p + g) = «, {y) x^p + y^q = z^,
(8) -VP-\-xq + l-¥z* = 0, {€) yp-xq = x^- y^, (f) (x + z)p = y,
<f) ««p-ajyg + 1/2 = 0, {$) {a-x)p + {b-y)q = c-z,
(i) ptanx + jtany = tanz, {k) {y^ -\- z^ - x^)p - 2xyq + 2xz = 0.
2. Determine the integrals of the preceding equations to pass through the curves :
for (a) xa + y« = 1, z = 0, for (/S) y = 0,x = z,
for (7)y = 2x, z = l, for (e) x = z, y = z.
3. Show analytically that if F(x, y, z) = C^i8 & solution of (16), it is a solution
of (14). State precisely what is meant by a solution of a partial differential equa-
tion, that is, by the statement that ^(x, y, z) = C^ satisfies the equation. Show that
the equations
Pg+«| = B and p|?+q|? + B^ = o
dx &y dx dy dz
are equivalent and state what this means. Show that if F=C^ and G = C^ are
two solutions, then F = *(6?) is a solution, and show conversely that a functional
relation must exist between any two solutions (see § 62).
4. Generalize the work in the text along the analytic lines of Ex. 3 to estab-
lish the rules for integrating a linear equation in one dependent and four or n
independent variables. In particular show that the integral of
+ ... + P, =P„^i dependson _1 = . . . = -^ = ^ — ,
^l ^ Pi Pn Pn+l
and that if F^ = Cj, • • • , F„ = C„ are n integrals of the simultaneous system, the
integral of the partial differential equation is ^{F^, • • •, F„) = 0.
.Integrate: (a) x— + y \-z—z=xyz,
dx dy dz
OS) (y + z + M) ^ + (z + u + X) ?!^ + (u + X -I- y) - = X + y + z.
ox dy dz
6. Interpret the general equation of the first order ^(x, y, «, p, g) = 0 as deter-
mining at each point (Xq, y,,, Zq) of space a series of planar elements tangent to a
certain cone, namely, the cone found by eliminating p and q from the three simul-
taneous equations
^(«o» Voi «o» P. 9) = 0, (X - Xq)p + (y - yo) g = z - Zo»
7. Eliminate the arbitrary functions :
(«) « + y + t = ♦(x« + ya + z% (^) *(x2 + y2, « _ xy) = 0,
(7) X = ♦(« + y) + ♦ (X - y), («) z = e«»* (x - y),
(f)*=:y« + 2*(x-i + Iogy), (f) */?, V, ^) = o.
\y « x/
MORE THAN TWO VARIABLES 278
8. Find the difFerential equatioiui of theae types of surf Met:
(a) cylindetH with generators parallel to the line z = ax^y = ht^
(/3) conical surfaces with vertex at (a, 6, c),
(7 ) surfaces of revolution about the line z : y : x s a : 6 : e.
9. Eliminate the constants from these equations :
(a) z = (a; + a)(y + 6), 0^) a (x« + y«) + &e« = 1,
(7) (x-a)'» + (y-6)« + (z-.c)« = l, (a) (x-a)« + (y-6)«+(z-c)«=<l«.
(« ) Ax^-\- Bxy + Cj/« + J>J2 + Eyz = t«.
10. Show geometrically and analytically M>:if Fu-. y, «) + aG(x, y, x) = 6 Is a
complete solution of the linear equation.
11. Plow many constants occur in the complete solution of the equation of the
third, f(Jiirth, or nth onler ?
12. Discuss the complete, general, and singular solutions of an equation of the
first order F(x, y, «, u, u^, u^, u^) = 0 with three independent variables.
13. Show that the planes « = ax + &y + C, where a and b are connected by the
relation F(a, b) = 0, are complete solutions of the equation F(p, q) = 0. Integrate:
{a)pq = l, 09)9=P^ + 1, {y)p^ + q^ = m^,
(«) Pq = k, (e) A:logg + P = 0, (f) 3p2 - 2ry2 = 4p7,
and determine also the singular solutions.
14. Note that a simple change of variable will often reduce an equation to tb»
type of Ex. 13. Thus the equations
f(?.?) = 0, f(xp,,) = 0, i^(f.f) = 0,
with z = e'\ X = e^, « = c»', x = C, y = ci^,
take a simpler form. Integrate and determine the singular solutions :
{a) q = z-\-px, 09) x^p^ + y^q^ = z^ (7) z = pq,
(«) 9 = 2yp2, (e) (p_y)2 + (y_x)2 = l, (f) z=p-V-.
15. What is the obvious complete solution of the extended Clairaut equation
z = xp + yg +/(p, 7) ? Discuss the singular solution. Integrate the equations :
(a) z = xp-\-yq-{- Vp^ + ^'^ + 1, (/3) z = xp + yg + (p + g)»,
(7) z = xp + y« + pg, («) z = xp + yg - 2 VS.
116. Types of partial differential equations. In addition to the
linear eciuation and the tyj)es of Exs. 13-15 above, there are several
ty{)es which should be mentioned. Of these the first is the general
equation of the first order. If F(Xj y, «, ^, y) = 0 is the given equation
and if a second equation <I>(j*, y, «, p^ y, a) = 0, which holds simultane-
ously with the first and contains an arbitrary constant can be found,
the two equations may be solved together for the values of p and g, and
the results may l^e substituted in the relation dz = pdx -f- qdy to give a
total differential equation of which the integral will contain the con-
stant a and a second constant of integration b. This integral will then
274
DIFFERENTIAL EQUATIONS
be a complete integral of the given equation ; the general integral may
then be obtained by (18) of § 116. This is known as CharpWs method.
To find a relation * = 0 differentiate the two equations
F{xy y, «, py q) = 0, *(x, y, «, p, q,a) = 0
with reepeot to x and y and use the relation that dz be exact.
(19)
r + rv + F-'^ + F'^
0,
/ f^P
dx
*; + *;i? + *;:t^ + *;^ = o,
dq
dx
*;.
-K,
f:^:
^'F\
K^i-(pK+^K)t = 0'm
(21)
dy dx '
Multiply by the quantities on the right and add. Then
(F-.+pK^^nF'.+.F^^-K-^ -ay
Now this is a linear equation for * and is equivalent to
dp dq dx dy dz d^
f: +pF: = /•; + yir; = IT^ = 37^ = _(^i.; + ^p^ - T
Any integral of this system containing p ov q and a will do for ^, and
the simplest integral will naturally be chosen.
As an example take «p(» + y) + j)(g — p) — 2^ = 0. Then Charpit's equa-
tions are
dp dq dx
- «p +p«(x + y) ~ zp - 2zg + i>5(x + y) ~ 2p-q-z{x-\-v)
_ dy _ dz
~ - p ~ 2p2 _ 2pq — pz{x + y)'
How to combine these so as to get a solution is not very clear. Suppose the sub
stitution « = e*', p = e«'p', q = e«Y be made In the equation. Then
P'(« + V)+P'(g'-P')-1 = 0
is the new equation. For this Charpit's simultaneous system is
<^_<V_ dx ^ dy dz
jK ]/ 2 p' - g' - (X + y) ~ - p' " 2 p'a - 2p9 - p' (X + y) *
The flnt two equations give at once the solution dp' = dg' or g' = p' + a. Solving
P'(« + y) + J)'(«^-pO-l = 0 and q' = p' + a,
1 1
l^ =
a + « + y
7' =
+ a, dz^ = J^±^ + adv,
a + x + y a-\-x-\-y
MORE THAN TWO VARIABLES 275
Then z = log (a + x + y) + ay + 6 or logc = log (a 4- « •(- y) •!• ay -f 6
Is a complete solution of the given equation. ThiB will determine ths gOMiil
integral by eliminating a between the three equations
« = c»i' + *(a + X + y), b =f(a), 0 = (y +/'(a))(a + * + y) + 1,
where /(a) denotes an arbitrary function. The rules for determining the sIngaUr
Holution give 2 = 0; but it is clear that the surfaces in the comi^etfe aolation can-
nut be tangent to the plane z = 0 and hence the result z = 0 must be not ft dngulAr
Holution but an extraneous factor. There is no singular solution.
The method of solving a partial differential equation of higher order
than the first is to reduce it first to an equation of the first order and
then to complete the integi*ation. Frequently the form of the equation
will suggest some method eiusily applied. For instance, if the deriva-
tives of lower order corresponding to one of the independent variables
are absent, an integration may be performed as if the equation were
an ordinary equation with that variable constant, and the constant of
integration may be taken as a function of that variable. Sometimes a
change of variable or an interchange of one of the independent variables
with the dependent variable will simplify the equation. In general the
solver is left mainly to his own devices. Two special methods will Ije
mentioned below.
117. If the equation is linear with constant coefficients and all the
derivatives are of the same order, the equation is
{a^D^ + a^Dr'D^ + • + a^-^DJ);-' + a,D;)z = R{x, y). (22)
Methods like those of § 95 may be applied. Factor the equation.
%{!>, - «A) {D, - «A) ■■■{!>,- a.D,) z = R(x, y). (22')
Then the equation is reduced to a succession of equations
each of which is linear of the first order (and with constant coefficients).
Short cuts analogous to those previously given may be developed, but
will not be given. If the derivatives are not all of the same order but
the polynomial can be factored into linear factors, the same method will
apply. For those interested, the several exercises given below will serve
as a synopsis for dealing with these types of equation.
There is one equation of the second order,* namely
• This is one of the important diffprential equations of physics; other Important equip
tions and methods of treating them are discussed in Chap. XX.
276 DIFFERENTIAL EQUATIONS
which occurs constantly in the discussion of waves and which has there-
fore the name of the wave equation. The solution may be written down
by inspection. For try the form
uix, y, «, t) = F{ax •\-by-^cz- Vt) + G{ax ^-hy + cz + Vt). (24)
Substitution in the equation shows that this is a solution if the relation
fl« ^ ft« 4- c* = 1 holds, no matter what functions F and G may be. Note
that the equation
ax-\-by-\-cz-Vt^O, a^ + b^ -{- c^ = 1,
is the equation of a plane at a perpendicular distance Vt from the origin
along the direction whose cosines are a, b, c. If t denotes the time and
if the plane moves away from the origin with a velocity F, the function
F(ax -\-by-\-cz— Vt) = F{0) remains constant ; and if G = 0, the value
of u will remain constant. Thus uz=zF represents a phenomenon which
is constant over a plane and retreats with a velocity F, that is, a plane
wave. In a similar manner u = G represents a plane wave approaching
the origin. The general solution of (23) therefore represents the super-
position of an advancing and a retreating plane wave.
To Monge is due a method sometimes useful in treating differential equations
of the second order linear in the derivatives r^ s^t-, it is known as Mange's method.
Let Br + -Ss + 7Y = F (25)
be the equation, where B, 5, T, F are functions of the variables and the derivatives
p and q. From the given equation and
dp = rdx + sdy, dq = sdx + idy,
the elimination of r and t gives the equation
B {Rdy* - Sdxdy + Tdx^) - {Rdydp + Tdxdq - Vdxdy) = 0,
and this will surely be satisfied if the two equations
Rdy^ - Sdxdy + Tdx^ = 0, Rdydp + Tdxdq - Vdxdy = 0 (250
oan be satisfied simultaneously. The first may be factored as
^V -/i (35, y, z, p,q)dx = 0, dy - f^ («, y, z, p,q)dx = 0. (26)
The problem then is reduced to integrating the system consisting of one of these fac-
tor* with (25') and dz =pdx + qdy, that is, a system of three total differential equations.
If two independent solutions of this system can be found, as
«i («, y, z, p, q) = Ci, u^ (X, y, z, p, q) = C^,
then u, = ♦<!<,) la a first or intermediary integral of the given equation, the general
integral of which may be found by integrating this equation of the first order. If
the two factors are distinct, it may happen that the two systems which arise may
both be Integrated. Then two first integrals u, = * (u^) and v^ = <if {v^) will be found,
and instead of Integrating one of these equations it may be better to solve both for
p and 9 and to 8ub«Utute In the expression dz=pdx-{-qdy and integrate. When,
howe? er, it U not poeslble to find even one first integral, Monge's method fails.
MORE THAN TWO VARIABLES 277
As an example Uke (z + y) (r — t) = — ip. The equ&UoiM are
(x + y)dy*-(x + y)dx« = 0 or dy — dz = 0, dy-f<t7 = 0
and (z + y) dydp — (z + y) dxdq + 4pdzdy = 0. (A)
Now the equation dy — dx = 0 may be integrated at once to give y = x + C\. The
second equation (A) then takes the form
2zdp + 4pdz-2zd9+ C^{dp ^ dq) = 0;
but as dz = pdx + qdy = (p + 9)dz in tliia case, we have by combination
2 {xdp + pdx) -2(xdq + qdx) + Cj (dp - dy) + 2d« = 0
or (2z+Ci)(p-9) + 2« = C, or (z + y)(p - «) + 2« = C,.
Hence (x + y) (p - g) + 2« = ♦(!/- z) (27)
is a first integral. This is linear and may be integrated by
dx dy dz ,^ dx dg
- - or x + y = K^, -
z + y z + y ♦(y-z)-2« " K^ ♦(ir,-2z)-2»
This equation is an ordinary linear equation in z and x. The Integration gives
f
e^^*(iri-2z)dz + ir,.
ax /^ ix
Hence (z + y)2^ +i'- I e^»*(Ji - 2z)dz = K^ = ^(K{) = ♦(z + V)
is the general integral of the given equation when K^ has been replaced by z •(- y
after integration, — an integration which cannot be performed until ♦ is given.
The other method of solution would be to use also the second system containing
dy + dz = 0 instead of dy — dx = 0. Thus in addition to the first integral (27) a
second intermediary integral might be s<3Ught. The substitution of dy •\- dx = 0^
y + z = Cj in (A) gives C^ (dp + dq) + 4 pdx = 0. This equation is not integrable,
because dp + d^ is a perfect differential and pdx is not. The combination with
dx = pdx + qdy = {p — q)dx does not improve matters. Hence it is impossible to
determine a second intermediary integral, and the method of completing the
solution by integrating (27) is the only available method.
Take the equation pa — qr = 0. Here S=p, R=—q, r=F = 0. Then
— qdy^ — pdxdy = 0 or dy = 0, pdx + qdy = 0 and — qdydp = 0
are the equations to work with. The system dy = 0, qdydp = 0, dt =pdX'^ qdy^
and the system pdx + qdy = 0, qdydp = Oy dz = pdx + qdy are not very satisfactory
for obtaining an intermediary integral u^ = 4>(Uj), although p = ♦(«) Is anobvioua
solution of the first set. It is better to use a method adapted to Uila q>eeial
<luation. Note that
By (11), p. 124, -=-(^); then ^=-/(y)
P \cy/u cy
Md x=-ff{y)dy + ♦(x) = ♦(y) + *(«).
278 DIFFERENTIAL EQUATIONS
EXERCISES
1. Integrate these equations and discuss the singular solution:
(a) p* + 9* = 2x, 03) (p2 + 9^) X = pz, (y) {p + q){px + qy) = l,
(«) W = J» + «/y, (e) p2 + ^'^ = X + y, (n xp2 _ 202) + xy = 0,
(,) ^ = z*(p-q), {B) q{pH + q'^) = \, (t) pU + 5^) = g(z - c),
(«) Xp(l + 9) = g«, (X) V^ iP^ - 1) = X2i)2, (;,) 22 (p2 + ^2 ^ 1) 3, ^2,
(r)p = (« + yg)«, (o)l)z = l + 9^ (T)z-pg = 0, (/>) 9 = xp + ^2.
2. Show that the rule for the type of Ex. 13, p. 273, can be deduced by Charpit's
method. How about the generalized Clairaut form of Ex. 15 ?
S. (a) For the solution of the type /j (a;, p) ^f^iy^ q)i the rule is : Set
Mx,p)=f2{y,q) = a,
and aolve for p and g as p = g^{x^ a), q = g^iVj a) ; the complete solution is
<= /fl'i(^> a)dx + Joziy, a)dy + h.
(fi) For the type F(z, p, 9) = 0 the rule is : Set X = x ■\- ay, solve
the complete solution is a; + ay + 6 =/(z, a). Discuss these rules in the light of
Charpit's method. Establish a rule for the type F{x + y, p, q) = 0. Is there any
advantage in using the rules over the use of the general method ? Assort the exam-
ples of Ex. 1 according to these rules as far as possible.
4. What is obtainable for partial differential equations out of any characteristics
of homogeneity that may be present ?
5. By differentiating p =/(x, y, z, q) successively with respect to x and y show
that the expansion of the solution by Taylor's Formula about the point {Xq, y^, Zq)
may be found if the successive derivatives with respect to y alone,
^ Sh. &^ £»z
dy' ay2* Qyi* ' ^'
are assigned arbitrary values at that'point. Note that this arbitrariness allows the
solution to be passed through any curve through {Xq, y^, Zq) in the plane x = Xq.
6. Show that F{x, y, z, p, g) = 0 satisfies Charpit's equations
*, = ^ = J«L = ^ = "P = ^ , (28)
where u is an auxiliary variable introduced for symmetry. Show that the first
three equations are the differential equations of the lineal elements of the cones of
Ex. 6, p. 272. The integrals of (28) therefore define a system of curves which have
a planar element of the equation F = 0 passing through each of their lineal tan-
gentUl elemenU. If the equations be integrated and the results be solved for the
variables, and If the constants be so determined as to specify one particular curve
with the Initial conditions Xq. Vot «o» Po» </o» then
« = «(M»Xo, y^,«o.Po»Vo). y = y(- ••),« = «(••), p=p(---)» q = q{:")'
MORE THAN TWO VARIABLES 279
Note that, along the curve, q=/{p) and that ooiiMqiiently the pUnar
just mentioned must lie upon a developable surface oontaining the curve (| 67). The
curve and the planar elemenUs along it are called a characterUtic and a charaeUritUe
strip of the given differential equation. In the caae of the linear equation the
characteristic curves afforded the integration and any planar element through
their lineal tangential elements satisfied the eijuation ; but here it la onlj thoee
planar elements which constitute the characteristic strip that satisfy the equation.
What the complete integral does is to piece the cliaracterisUc strips into a family
of surfaces dependent on two parameters.
7. By simple devices integrate the equations. Check the answers :
(«) a + pf{x) = g{y), («) ar = xy, (f) xr = (n^ l)p.
8. Integrate these equations by the method of factoring:
(a) (Dl - a^Dl) 2 = 0, (p) {D, - D,)* z = 0, (7) (D^ - D^ « = 0,
(«) (Dj + 3Dx^^ + 2Z^)z = x + y, (e) (Z^ - D,I>^ - 6 1^) z = xy,
(f) (Dj-2);-8D, + 82),)2 = 0, in) (Dj-l^ + 2i).+ l)« = e— .
9. Prove the operational equations :
(a) e«^y0(y) = (1 + axD„ + \ a^x^Dj + • • .)0(y) = 0(y + arx),
(/5) 7^— ^— ?r^ = «"^»^^ = ^'"^'^<y) = ^(y + '")'
Vx — aJJp Ux
(7) ^— ^^^K(x, V) = e''^t,f'e-'iOpR{i, y)d^ = f'R{(,v^ax-ai)d(.
10. Prove that if [(Dx - a^Dy)'»i • • • (Dx - atDy)'**] 2 = 0, then
z = *ii(y + a^x) + ar*i2(y + «!«) + • • • + «"^-**imi(y + a^x) + • • •
+ *ti(y + atx) + x*w(y + crtx) + • • • + x'^k-^*hm^{y + apr),
where the <I>'8 are all arbitrary functions. This gives the solution of the reduced equa-
tion in the simplest case. What terms would correspond to (Dx — aD, — /S)"z = 0 f
11. Write the solutions of the equations (or equations reduced) of Ex.8.
12. State the rule of Ex. 9 (7) as : Integrate R{x,y — ax) with respect to x and
in the result change y to y + ax. Apply this to obtaining particular solutions of
Ex. 8 (5), (e), (ri) with the aid of any short cuts that are analogous to thoee of
Chap. VIII.
13. Integrate the following equations :
(a) (7^-7>J^ + Dy-l)2=cos(x + 2y) + cr, (^) x*r« + 2 xy« + y V = x« + y«,
(7) (7;i+/)^+l>v-l)z = 8in(x + 22/), («) r-«-8p + 8« = e' + «»,
(e) (Dj-2DxDj + D;)z = x-a, (f) r-< + p + 3g-2« = e«-i'-xV
(,) (DJ- DxDy- 27Jj + 2Dx+ 2Dy)z = e«' + »r+ 8in(2x + y) + xy.
14. Try Mongers method on these equations of the second order :
(a) q^r - 2p9S + P»« = 0, (/S) r - a't = 0, (7) r + » = - p,
(S) g(l + (7)r-(p + g + 2p<7)a + p(l + p)« = 0, (f) xV + 2 «y« + y«t = 0,
(f) (6 + C(?)V-2(6 + C9)(a + cp)s + (a + cp)«t = 0, (1,) r-^kaH = %a$,
li any simpler method is available, state what it is and apply it also.
280 DIFFERENTIAL EQUATIONS
15. Show that an equation of the form Br + /Sa + Tf + U{rt ^ s^) = V neces-
■arily ari«efl from the elimination of the arbitrary function from
«^(«, y, 2, P, Q) =f[u2{x, y, z, p, g)].
Note that only such an equation can have an Intermediary integral.
16. Treat the more general equation of Ex. 15 by the methods of the text and
thui abow that an intermediary integral may be sought by solving one of the systems
Udv + Xj Tdx + \ Udp = 0, Udx + \Rdy -\- \Udq = 0,
Udx + \Rdy ■\- \Udq = 0, Udy ■}■ \Tdx + \Udp = 0,
dz = pdx + qdy^ dz = pdx + qdy^
where \ and X, are roots of the equation \^{RT + UV) + \US + U^ = 0.
17. Solve the equations : (a) 8^ — rt = 0, (/3) s^ — rt = a*,
(7) ar + ba + ct-i- e{H-8^) = h, (5) xqr+ypt + xy{s^ - H)-pq.
PART III. INTEGRAL CALCULUS
CHAPTER XI
ON SIMPLE INTEGRALS
118. Integrals containing a parameter. Consider
^(a)=P/(x,a)cte,
(1)
a definite integral which contains in the integrand a parameter a. If
the indefinite integral is known, as in the case
/
cos axdx = - sin ax.
a
cos axdx = - sm aac
a
i 1
it is seen that the indefinite integral is a function of x and or, and that
the definite integral is a function of a alone because the variable x
disappears on the substitution of the limits. If the limits themselves
depend on a, as in the case
s:
cos aixdx = - sin ax
a
= - (sin a* — sin 1),
a^ ^
the integral is stilj a function of a.
In many instances the indefinite integral
in (1) cannot be found explicitly and it then
becomes necessary to discuss the conti-
nuity, differentiation, and integration of the
function <f>(a) defined by the integral with-
out having recourse to the actual evaluation
of the integral; in fact these discussions
may Im? required in order to effect that
evaluation. Let the limits x^ and x^ be taken
as constants indei)endent of a. Consider the range of values x^^x^x^
for X, and let a^^a^a^ be the range of values over which the func-
tion <^ (a) is to be discussed. The function /(a?, a) may be plotted as
the surface z =/(ar, a) over the rectangle of values for («, a). The
281
282 INTEGRAL CALCULUS
value ^(<r<) of the function when a = a,, is then the area of the section
of this surface made by the plane a = a^. If the surface /(x, a) is con-
tinuous, it is tolerably clear that the area <^ (a) will be continuous in a.
The function ^ (a) is continuotis iff(x, a) is continuous in the two varior
bUs (x, a)
To discuss the continuity of </> (a) form the difEerence
0(a + Aa) -4»{a)= f\f^^^ ^ + ^^) -•^(^' ^)1^- (2)
Now ^{a) will be continuous if the difference 0 (a + Aar) — 0(a) can be made as
small as desired by taking Aa sufficiently small. If /(a;, y) is a continuous func-
tion of (x, y), it is possible to take Ax and Ay so small that the difference
|/(x + Ax, y + Ay) -/(x, y)| < e, |Ax| < 5, |Ay| < 8
for all points (x, y) of the region over which /(x, y) is continuous (Ex. 3, p. 92).
Hence in particular if /(x, a) be continuous in (x, a) over the rectangle, it is pos-
sible to take Aa so small that
|/(x, a + Aa)-/(x, a)I<e, IAa|<a
for all values of x and a. Hence, by (65), p. 25,
|0(a + Aa) - 0(a)| = I P[/(x, a + Aa) -/(x, a)] dx|< f'^'edx = €(Xi - Xo).
It i« therefore proved that the function <t>{a) is continuous provided /(x, a) is con-
tinuous in the two variables (x, a) ; for e (Xj — x^) may be made as small as desired
if € may be made as small as desired.
As an illustration of a case where the condition for continuity is violated, take
1
^ . . /•! adx ^ , X
♦ («) = I -;; :: = tan-i —
^^ ' Joa3 + x2 a
= cot-la if a^% and 0 (0) = 0.
0
0(a) =/'-
•/o 1
Here the integrand fails to be continuous for (0, 0); it becomes infinite when
(x, a) = (0, 0) along any curve that is not tangent to a = 0. The function 0 (a) is
defined for all values of a ^ 0, is equal to cot-^a when a ?£ 0, and should there-
fore be equal to | ir when a = 0 if it is to be continuous, whereas it is equal to 0.
The importance of the imposition of the condition that /(x, a) be continuous is
clear. It should not be inferred, however, that the function 0(a) will necessarily
be discontinuous when/(x, a) fails of continuity. For instance
dx 1 / / /— ^ 1
This function is continuous in a for all values a ^ 0 ; yet the integrand is dis-
oontinuous and indeed becomes infinite at (0, 0). The condition of continuity
Imposed on /(x, a) in the theorem is si^fficient to insure the continuity of 0 (a)
but 6y no means necessary ; when the condition is not satisfied some closer exami-
nation of the problem will sometimes disclose the fact that <f> (a) is still continuous.
In case the limits of the integral are functions of a, as
f(x,a)dx, a^^ama, (3)
ON SIMPLE INTEGRALS
283
the function ^(or) will surely be continuous if /(«, a) is oontiniioiii
over the region bounded by the lines a = or^, a^a^ and the corres
^0 = ^o(*)» ^i = ^iW> ^"^ ^^ *^^® functions ^/a) and ^/a) are continnoos.
For In this case
fix, ar + Aa)dx
tr,(« + ^«)
/(x, a + Aa)dx
[/(2, a + Aa)-/(x, ar)]dz.
yo(«)
The absolute vahies may be taken and the inte-
grals reduced by (06), (65'), p. 25.
|0(a + Aa)-^(a)|<e|firj(a)-sro(«)I + l/«ii« + Aa)||AtyJ+|/(fo»a + Aa)||Aa,|,
where (^ and (, are values of x between g^ and g^ + Aflr^, and ^, and (7, + A^i* By
taking Anr small enough, p,(a + Aa) - gr,(a) and i7o(a: + Aa) - ^^(tr) may be nude
as small as desired, and hence A^ may be made as small as desired.
119. To find the derivative of a function ^(a) defined by an integral
containing a parameter y form the quotient
Aa Aa
= 1^ / /(:r,a + Aa)rfaj- / /(x,a:)^,
A^^ p.'"'/(x,ar + A«)-/fe«)^.^ p /(x,<t + A«)^
+ p-^Vfr.'T + Aa)^
/ Aa
The transformation is made by (63), p. 25. A further rcdiiftion may
be made in the last two integrals by (foo'), p. 25, which is the Theorem
of the Mean for integrals, and the integrand of the first integral may be
modified by the Theorem of the Mean for d ives (p. 7, and Ex. 14,
p. 10). Then
A^
Aa
and
^ffa
^(fi
f;(:r,a + eAa)dx-f{$^,a-\-Aa)^-hf($,,a + ^a)'^
da
A critical examination of this work shows th^t the derivative 4' (a)
exists and may be obtained by (4) in case ^ts and is continuous
284 INTEGRAL CALCULUS
in (x, a) and gJa), ffi(«) are differentiable. In the particular case that
the limits g^ and g^ are constants, (4) reduces to Leibniz^s Rule
which states that the derivative of a function defined hy an integral
with fixed limits may he obtained by differentiating under the sign of
integration. The additional two terms in (4), when the limits are varia-
ble, may be considered as arising from (66), p. 27, and Ex. 11, p. 30.
This process of differentiating under the sign of integration is of
frequent use in evaluating the function <t>(a) in cases where the indefi-
nite integral of f(xj a) cannot be found, but the indefinite integral of
/; can be found. For if
*(«) = r /(«» «)^, then ^ = r y^dx = ^(a:).
Now an integration with respect to a will give <^ as a function of a
with a constant of integration which may be determined by the usual
method of giving a some special value. Thus
^ ^ Jo logo; da J^ logx X
^-j-^, <^(a:)=log(a;-f-l)-|.C.
But <t>(0)= j Odx = 0 and <^ (0) = log 1 + C.
X^ of — 1
-y— 6?a; = log(a-f 1).
In the way of comment upon this evaluation it may be remarked that the func-
tlona (x« — l)/loga; and x' are continuous functions of (x, a) for all values of x in
the interval O^x^l of integration and all positive values of a less than any
MBlgned value, that is, O^a^K. The conditions which permit the differen-
tUtlon under the sign of integration are therefore satisfied. This is not true for
negative values of a. When a <0 the derivative a;« becomes infinite at (0, 0). The
method of evaluation cannot therefore be applied without further examination.
Ai A matter of fact <p{a) = log (a + 1) is defined for a:>- 1, and it would be
natural to think that some method could be found to justify the above formal
evalumUon of the integral when -l<a^K (see Chap. XIII).
To niu«trat« the application of the rule for differentiation when the limits are
function* of a, let It be required to differentiate
Hence ^ = -^af+^
da a + 1
*fm logx da Ja
ar«« — 1 __ oc« — \
log a log a
ON SIMPLE INTEGEAIiS 286
This formal reHiiIt \h only good Hubject to the conditionii of continuity. Cleariy a
muHt be greater than zero. This, however, is the only restriction. It might ■Mm at
firKt AH though the value z = 1 with logz = 0 in the denominator of (r*— l)/logs
would cause difficulty ; but when z = 0, this fraction is of the form 0/0 and has a
finite value which pieces on continuously with the neighboring values.
120. The next problem would be to find the integral of a function
defined by an integral containing a jyarameter. The attention will be
restricted to the case where the limits x^ and x^ are constants. Consider
the integrals ^a ^t
f ^(a)da=C • rf(x,a)d.r../u,
where a may be any point of the interval oTq ^ er ^ or^ of valaes ovei
which ^(a) is treated. Let
*(a)= I •/ f(xja)da'dx.
Then *' (a)= j-^ j f{x, a)da'dx^ / /(«, d)dx=^^ (a)
by (4'), and by (66), p. 27; and the differentiation is legitimate if /(x, or)
be assumed continuous in (a;, a). Now integrate with respect to a. Then
But *(ar^)= 0. Henoe, on substitution,
\ . I f{x,a)da'dx= I <^(a:)</a:= / • / f{x,a)dX'd€U (6)
X, Ja^ Ja^ Ja^ J x^
Hence appears the rule for integration, namely, integrate under the
sign of integration. The rule has here been obtained by a trick from
the previous rule of differentiation; it could be proved directly by
considering the integral as the limit of a sum.
It is interesting to note the interpretation of this integration on the
figure, p. 281. As 4»{<i) is the area of a section of the surface, the
product <^(a)da is the infinitesimal volume under the surfaoe and
included between two neighboring j)lanos. The integral of ^(a) is
therefore the volume * under the surface and boxed in by the four
• For the " volume of a solid with parallel bases and variable croM Mctkm " tea
Ex. 10, p. 10, and $ 35 with Exs. 20, 23 thereunder.
286 INTEGRAL CALCULUS
planes « = tf^, a =: a, x = x^, x — x^. The geometric significance of
the reversal of the order of integrations, as
I * / /(^> a)da'dx= j -I f(x, a) dx • da,
is in this case merely that the volume may be regarded as generated
by a cross section moving parallel to the ;s;a:-plane, or by one moving
parallel to the »c-plane, and that the evaluation of the volume may
be made by either method. If the limits x^ and x^ depend on a, the
integral of ^(a) cannot be found by the simple rule of integration
under the sign of integration. It should be remarked that integration
under the sign may serve to evaluate functions defined by integrals.
As an illustration of integration under the sign in a case where the method leads
to a function which may be considered as evaluated by the method, consider
^(a)= rVdiC = -^, f%{a)da= f ^-^ = log -^-±1 .
^' Jo a + 1 Ja^ ' Jaa + 1 ^a + 1
<f>{a)da= f • f x<'da.dx= f -^- dx = f -^ -dx.
a Jo Ja Jo lOgX\a=a Jo log X
J'^x* — z<* 6 + 1
dx = log — !— - = ^ (a, 6), a ^ 0, 6 ^ 0.
0 logx a + 1 ^ ^ ' "
In this case the integrand contains two parameters a, 6, and the function defined
is a function of the two. If a = 0, the function reduces to one previously fdund.
It would be possible to repeat the integration. Thus
X 1^^ "^ ^^^^"^ "^ ^^' lo^^^^^ "^ ^^^^ = (« + 1) log (or + 1) - a.
Thi« is a new form. If here a be set equal to any number, say 1, then
— -^dx = 21og2-l.
0 (log x)2 ^
In this way there has been evaluated a definite integral which depends on no
parameter and which might have been difficult to evaluate directly. The introduc-
tion qf a parameter and Us subsequent equation to a particular value is of frequent use
in evaluating d^nite integrals.
EXERCISES
1. Evaluate directly and discuss for continuity, 0 ^ a ^ 1:
» o^dx /•! dx , ^ r'^ xdx
i^\ C ^^ /«v r^ dx /»i
Va2 + x2 ^'o Va2 + x^
u. If /(x, a, p) is a function containing two parameters and is continuous in
the three variables (x, or, /S) when x^^x^x^, a^^ a ^ a^, p^^ p^ p^, show
J fix, a, p)dx = ^(a, p) is continuous in (a, p).
ON SIMPLE INTEGRALS 287
3. Differentiate and hence evaluate and state the valid range for a t
(a) r'log(l + aco8z)da: = irlogli-^!l=-^,
«/o 2
4. Find the derivatives without previously Integrating :
-tano^dx, (/3) / tan-J — dx, (7) f e «• d».
5. Extend the assumptions and the work of Ex. 2 to find the partial deriva*
lives 0^ and 01 and tlie total differential d0 if x^ and X| are constants.
6. Prove the rule for integrating under the sign of integration by the direct
method of treating the integral as tlie limit of a sum.
7. From Ex. 0 derive the rule for differentiating under the sign. Can the com-
plete rule including the case of variable limits be obtained this way ?
pg{x, <r)
8. Note that the integral 1 /(x, a) dx will be a function of (x, a). Derive
formulas for the partial derivatives with respect to x and a.
fj pax d /• \/Z
9. Differentiate : (a) — / sin (x + a)dx, (/3) — / x«dx.
da t/o dx Jo
10. Integrate under the sign and hence evaluate by subsequent differentiation:
ir
{a) j x«logxdx, (/3) j '^x sin axdx, (7) T z sec* oxdx.
11. Integrate or differentiate both sides of these equations :
Jrtl 1 /» 1 u J
x^dx = to show / X* (log x)»dx = (— 1)" ,
0 a + 1 Jo ^ "^ ' ^ ' (rt+l)»+i'
„, /"» dx IT , , /»• dx ir 1.8.6. ..(2n— 1)
fl) / — = — — to show / — = ^ '- ,
"^^ Jo x^ + a 2V^ •'0 (a;2 + a)"-»-i 2 2- 4- 6...2 n- o-^*
7) I c-**co8mxdx = — to show I dx = -log(^-- -1.
'Jo a^-\-m^ Jo xsecmx 2 \a«+mV
/»» ffi pto g—ax — g— S* /S -CC
3) / e-«=^sinmxdx = — to show / dx = tan-» — — tan-« - ,
Jo a^A-m^ Jo xcacmx m m
b— cosx
cosz
^0 a: — cosx Va* — 1 •'<> (<x— cosx)* •/o a —
''o 1 + x sinira Jo 1 + x Jo (l + z)logx
Note that in (^)-(8) the integrals extend to infinity and that, as the rules of
he text have been proved on the hypothesis that the interval of integration is
finite, a further justitication for applying the rules is necessary ; this will be
reated in Chap. XIII, but at this \Mnut. tlu> rules may be applied formally
without justification.
2gg INTEGRAL CALCULUS
12. ETaluate by any means these integrals :
(a) //V^m5cos-i^cto=a«g + i).
/»;iog(l-t- cosacosx) _ 1 /^^ _ ^aV
<^) Jo ^^ . 2U /
^ ' Jo a — ftsinxsinx a
'0 C0S2
121. Curvilinear or line integrals. It is familiar that
ydx= j f(x)dx
is the area between the curve y =f(x)y the a;-axis, and the ordinates
x=: aj X = b. The formula may be used to evaluate more complicated
areas. For instance, the area between the parabola y^=x and the semi-
cubical parabola 1/^ = x^ is
A= f x^dx — I x^dx = I ydx — I ydx^
where in the second expression the subscripts P and S denote that the
integrals are evaluated for the parabola and semicubical parabola. As
a change in the order of the limits changes the sign of
the integral, the area may be written
f ydx + I ydx = — j ydx — j ydx,
0 sJi pJ\ sJo
Ri
^S
and is the area bounded by the closed curve formed
of the portions of the pai-abola and semicubical parabola from 0 to 1.
In considering the area l)Ounded by a closed curve it is convenient to
arrange the limits of the different integrals so that they follow the curve
in a definite order. Thus if one advances along P from 0 to 1 and re-
tarns along S from 1 to 0, the entire closed curve has been described
in a uniform direction and the inclosed area has been constantly on the
righUhand side; whereas if one advanced along ^' from 0 to 1 and
ON SIMPLE INTEGRALS
returned from 1 to 0 along P, the curve would have been described
in the opposite direction and the area would have been constantlj
on the left-hand side. Similar considerations apply to more general
dosed curves and lead to the definition : If a closed curve which
nowliere crosses itself is described in such a direction as to keep the
inclosed area always upon the left, the area is considered as positive ;
whereas if the description were such as to leave the area on the right,
it would be taken as negative. It is clear that to a person standing in the
inclosure and watching the description of the boundary, the desorip*
tion would appear counterclockwise or positive in the first case (| 76).
In the case above, the area wl^en positive is
^ = - / ydx-\- I ydx\==- I ydx,
LsJa pJ\ J Jo
(«)
where in the last integral the symbol O denotes that the integral is to
be evaluated around the closed curve by describing the
curve in the positive direction. That the formula holds
for the ordinary case of area under a curve may be
verified at once. Here the circuit consists of the con-
tour ABB' A' A. Then
J/» pB r»B* pA' /%A
ydx = j ydx + | ydx -f I ydx + / ydx,
O J A Jb J B' J A'
The first integral vanishes because y = 0, the second and fourth vanish
because x is constant and dx = 0. Hence
Jr% pA' pB'
I ydjx, = — I ydx — I ydx,
O Jb' J A'
It is readily seen that the two new formulas
A = f xdy and A = \ I {xdy — ydx) (7)
Jo Jo
also give the area of the closed curve. The first is proved as (6) was
proved and the second arises from the addition of the two. Any one
of the three may 1x3 used to compute the area of the closed curve ; the
last lias the advantage of symmetry and is i)articularly useful in finding
the area of a sector, because along the lines issuing from the origin
y:x = dy: dx and xdy — ydx = 0 ; the previous form with the integrand
xdy is advantageous when part of the contour consists of lines parallel
to the avaxis so that rfy = 0 ; the first form has similar advantages
when parts of the contour are parallel to the y-axis.
290 INTEGKAL CALCULUS
The connection of the third formula with the vector expression for
the area is noteworthy. For (p. 175)
dA^^Txdt, A = i fixdr,
Jo
and if r = ari + yj, dr = idx + jdy,
then ^ = / ^""^^ = h^ f {^^V "" V^^)-
Jo Jo
The unit vector k merely calls attention to the fact that the area lies
in the xy-plane perpendicular to the «-axis and is described so as to
appear positive.
These formulas for the area as a curvilinear integral taken around
the boundary have been derived from a simple figure whose contour
was cut in only two points by a line parallel to the axes. The exten-
sion to more complicated contours is easy. In the first place note that
if two closed areas are contiguous over a part of their contours, the inte-
gral around the total area following both contours, but omitting the part
in common, is equal to the sum of the integrals. For
/ ^/ =/+/^/^/=/ '
JPRSP JPQRP J PR JrSP JpQR J RP J QRSP
since the first and last integrals of the four are in oppo-
site directions along the same line and must cancel. But
the total area is also the sum of the individual areas and hence the
integral around the contour PQRSP must be the total area. The for-
mulas for determining the area of a closed curve are therefore applicable
to such areas as may be composed of a finite number of areas each
bounded by an oval curve.
If the contour bounding an area be expressed in parametric form as x =/(0>
y = ^ (t), the area may be evaluated as
fm'P'{t)dt=-f4>{t)r{t)dt = if[f{t)<f>'{t) -<f>{t)r{t)]dt, (7o
whtre the limite for t are the value of t corresponding to any point of the contour
Aod the Talue of t corresponding to the same point after the curve has been
dMcribed once in the positive direction. Thus in the case of the strophoid
y* = «"-^^, the line y = tx
a + x
cttU the curve In the double point at the origin and in only one other point ; the
ooOrdlnatee of a point on the curve may be expressed as rational functions
X = a (1 - <«)/(! + t\ y = at(l^ t^)/{l + fi)
of I byaolving the itrophold with the line ; and when t varies from - 1 to + 1 the
(«, y) deecribes the loop of the strophoid and the limits for t are - 1 and + 1.
ON SIMPLE INTEGRALS 291
122. Consider next the meaning and the evaluation of
/ [^(a^,y)^^ + Q(x,y)rfy], where y =/(x). (8)
Ct/u.fc
This is called a curuUinear or line Intefjral along the curve (J or y 9tf(x)
from the point (a, b) to (ar, y). It is jx^ssible to eliminate y by the r«*la.
tion y =f(x) and write
f
[^(^,/W) + Q(^,/(^))/'(a^)]^. (9)
d
The integral then becomes an ordinary integral in x alone. If the curve
had been given in the form x —f(i/), it would have been better to con-
vert the line integral into an integral in y alone. The method of evaluat-
ing the integral is therefore defined. The differential of the integral
may be written as
r ' \pdx + Qdy) = Pdx-ir Qdy, (10)
Ja,h
where either x and dx ov y and dy may be eliminated by means of the
equation of the curve C For further particulars see § 123.
To get at the meaning of the line integral^ it is necessary to con-
sider it as the limit of a sum (compare § 16). Suppose that the curve
C between (a, h) and (a;, y) be divided into n parts, that A-r, and Ay,
are the increments corresponding to the ith part, and that {f^^ ly^) is
any point in that part. Form the sum
If, when n becomes infinite so that Aa: and Ay each
approaches 0 as a limit, the sum o- approaches a
definite limit independent of how the individual
increments Ao*. and Ay,- approach 0, and of how the
point (^,, »;,) is chosen m its segment of the curve,
then this limit is defined as the line integral
lim <r = P V (x, y)dx-\-Q («, y) rfy]. (12)
Cja,h
It should be noted that, as in the case of the line integral which giyes
the area, any line integral which is to be evaluated along two curves
which have in common a portion described in opposite directions may
be replaced by the integral along so much of the curves as not repeated ;
for the elements of o- corresponding to the common portion are equal
and opposite.
292 INTEGRAL CALCULUS
Thai 9 doee approach a limit provided P and Q are continuous functions of (x, y)
and proTided the curve C is monotonic, that is, that neither Ax nor Ly changes its
is easy to prove. For the expression for <r may be written
bj Qilng the equation y =/(x) or x =/-i (y) of C. Now as
J^'p(x,/(x))dx and fjQ{f-Hy),y)dy
are both existent ordinary definite integrals in view of the assumptions as to con-
tinuity, the sum v must approach their sum as a limit. It may be noted that this
proof does not require the continuity or existence oif\x) as does the formula (9).
In practice the added generality is of little use. The restriction to a monotonic
curve may be replaced by the assumption of a curve C which can be regarded as
made up of a finite number of monotonic parts including perhaps some portions of
lines parallel to the axes. More general varieties of C are admissible, but are not
very useful in practice (§ 127).
Further to examine the line integral and appreciate its utility for
mathematics and physics consider some examples. Let
F(x,y) = X{x,y)+iY(x,y)
be a complex function (§ 73). Then
/ F(x,y)dz= f \x(x,y)-{-iY(x,y)2ldx + idy2
7«/c-e cJa,b
(Xdx--Ydy)-\-i / (Ydx-hXdy)
b cJa, h
(13)
It is apparent that the integral of the complex function is the sum of two
line integrals in the complex plane. The value of the integral can be
computed only by the assumption of some definite path C of integra-
tion and will differ for different paths (but see § 124).
By definition the work done hy a constant force F acting on a particle,
which moves a distance s along a straight line inclined at an angle B to
the force, is W = Fs cos B. If the path were curvilinear and the force
were variable, the differential of work would be taken
aa rfir = Fcos BtU, where ds is the infinitesimal arc
and B is the angle between the arc and the force.
Henoe
^ =/''^*' = r*" Vcos Bih = r F.rfr,
where the path must be known to evaluate the integral and where
the but expression is merely the equivalent of the others when the
ON SIMPLE INTEGRALS 298
notations of vectors axe used (p. 164). These expressions may be con-
verted into the ordinary form of the line integral. For
F = A'i 4- KJ, (lT = idx-\' yiij, F*dT = Xdx^ Ydy,
Fcos$€U= I (Xdx + Ydt/),
b %/a,b
where X and V are the components of the force along the axes. It is
readily seen that any line integral may be given this same inter-
pretation. If
f Pdx-\-Qdy, form F = Pi + QJ.
a,b
f Pdx + Qdy= I FconOds.
0,6 *Ja,b
To the principles of momentum and moment of momentum ($ 80) may imow be
added the principle of work and energy for mechanics. Colder
m— - = F and m -— . dr = F»(ir = d fT.
Then
dt \2 dt'dt) ~ 2 dt'i'dt 2 dt' dl^ ~ dt^' dt*
or d(-v^)z=—.dx and d(-mA = dW,
\2 / dt^ \2 J
Hence 1 mc^ - i m»« = f V.dr = W,
2 2 ' Jr„
In words : Tfie change of the kinetic energy \ mv^ of a particle moving under the
action of the resultant force F is equal to the work done by the force, that is, to the line
integral of the force along the path. If there were several mutually interacting
particles in motion, the results for the energy and work would merely be added as
S \ mv^ — 2 J mrj = S >r, and the total change in kinetic energy is the total work
done by all the forces. .The result gains its significance chiefly by the consideration
of what forces may be disregarded in evaluating the work. As d>r= F»dr, the
work done will be zero if dr is zero or if F and dr are perpendicular. Hence in
evaluating IT, forces whose point of application does not move may be omitted
(for example, forces of support at pivots), and so may forces whose point of appli-
cation moves normal to the force (for example, the normal reactions of smooth curves
or surfaces). When more than one particle is concerned, the work done by the
mutual actions and reactions may be evaluated as follows. Let ij , r, be the vectors
to the particles and r^ — r, the vector joining them. The forces of action and re-
action may be written as i c (r, — r^), as they are equal and opposite and in the line
joining the particles. Hence
dW=d}\\-\-dn\ = c (rj - ro).dri - c (tj - r2)^r,
= c (rj - r,).d (r^ - r,) = J cd [(r^ - r,).(ri - r^] = J cdrj,,
where r,, is the distance between the particles. Now dlTyaaishes when and only
when dr^^ vanishes, that is, when and only when the distance between the particles
294 INTEGRAL CALCULUS
remains conrtant. Hence when a system of particles is in motion the change in the
total kinetic energy in passing from one position to another is equal to the work done by
tKe forces, where, in eveUuating the work, forces acting at fixed points or normal to the
line qf motion of their points of application, and forces due to actions and reactions of
paHielea rigidly connected, may be disregarded.
Another important application is in the theory of thermodynamics. If U, p, v
are the energy, pressure, volume of a gas inclosed in any receptacle, and if dtT" and
dv are the increments of energy and volume when the amount dH of heat is added
to the gas, then „ /» „
dH = dU-\-pdv, and hence S= j dU + pdv
\a the total amount of heat added. By taking p and v as the independent variables,
H = f^^dp + (^ +p)doj =f[fip, v)dp + g{p, v)dv].
The amount of heat absorbed by the system will therefore not depend merely or
the initial and final values of {p, v) but on the sequence of these values between
thoee two points, that is, upon the path of integration in the pw-plane.
123. Let there be given a simply connected region (p. 89) bounded by
a closed curve of the type allowed for line integrals, and let P (x, y) and
Q{x, y) be continuous functions of (x, y) over this region. Then if the
line integrals from (a, h) to (ic, y) along two paths
I Pdx + Qdy = / Pdx + Qdy
a,b TJa,b
are equal, the line integral taken around the combined path
' + / = I Pdx+Qdy = 0
a, ft tJx,v Jo
vanishes. This is a corollary of the fact that if the order of description
of a curve is reversed, the signs of Aaj^ and Ay,- and hence of the line
integral are also reversed. Also, conversely, if the in-
tegral around the closed circuit is zero, the integrals f p J5^
from any point (a, h) of the circuit to any other point / (^ )
(jt, y) are equal when evaluated along the two different \}(^[^^
parts of the circuit leading from (a, h) to (ic, y).
The chief value of these observations arises in their application to
the case where P and Q happen to be such functions that the line inte-
gral around any and every closed path lying in the region is zero. In
this case if (a, ^») be a fixed point and (x, y) be any point of the region,
the line integral from (a, b) to (x, y) along any two paths lying within
the region will be the same; for the two paths may be considered as
forming one closed path, and the integral around that is zero by hy-
pothesis. The value of the integral will therefore not depend at all on
ON SIMPLE INTEGRALS 296
the patli of integration but only on the final point (x, y) to which the
integration is extended. Hence the integral
/■■
«/a,6
[P(x, y)rfaj + Q(x, y)dy']^ F(x, y), (14)
extended from a fixed lower limit (</, /;) to a variable upper limit (x, y),
must Ije a function of (x, y).
Tliis result may be stilted as the theorem : The naooftory and •uffi-
cient condition that the line inttfjral
f
J a, I
[P(x,y)^-hQ(aJ,y)rfy]
define a single valued function of (x, y) over a simply etmneeted rtgiim
U that the circuit integral taken around any and every closed curve in
the region shall be zero. This theorem, and in fact all the theoreniB on
line integi-als, may be immediately extended to the case of line integrals
in space,
...
«/a, 6, c
[P (x, y,z)dx + Q (x, y, z)dy -\- R (a-, y, z) dz]. (15)
If the integral about every closed path is zero so that the inteyralfnm
a fixed lower limit to a variable upper limit
' P{x,y)dx^Q(x,y)dy
a,b
defines a function F(Xj y), that function has continuous first partial
derivatives and hence a total differential^ namely,
dF dp
a^"=^' a^=^' ^F^Pdx + Qdy. (16)
To prove this statement apply the definition of a derivative.
Pdx -f Qdy - I Pdx + Qfiy
h Ja.h
dp ,. AF ,.
•5- = lim -7— = lim
Ox AxAoAX Ax^o ^^
Now as the integi-al is independent of the path, the integral to
(x + Ax, ?/) may follow the same i)ath as that to (ar, y), except for
the passage from (x, y) to (x + Ar, y) which may be taken along the
straight line joining them. Then Ay = 0 and
c/x, y ^^
Ax
296 INTEGRAL CALCULUS
by the Theorem of the Mean of (66'), p. 25. Now when Aa :£= 0, the
value ( intermediate between x and x + ^x will approach x and P (^, y)
will approach the limit P(xy y) by virtue of its continuity. Hence
^F/^ approaches a limit and that limit is P(Xf y) = dF/dx. The other
derivative is treated in the same way.
If the integrand Pdx -f Qdy of a line integral is the total differential
dF of a single valued function F(x, y), then the integral about any closed
eireuit is zero and
r 'pdx + Q,dy = r 'dF = F(x, y) - F(a, h). (17)
%Ja,h *Ja,b
If equation (17) holds, it is clear that the integral around a closed path
will be zero provided F(x, y) is single valued; for F(xy y) must come
bock to the value F(a, b) when (x, y) returns to (a, b). If the function
were not single valued, the conclusion might not hold.
To prove the relation (17), note that by definition .
fdF=fpdx ■¥Q^y = lim^^ [P(f,-, vi)^i + Q(fi, %)Ay/I
and AF.- = P (f.-, ,;.•) Axi + Q (f,-, vi) ^Vi + ^i^i + egAi/,-,
where «j and e, are quantities which by the assumptions of continuity for P and Q
niay be made uniformly (§ 25) less than c for all points of the curve provided Ax<
and Lvi are taken small enough. Then
|2j(P.AXi + Q,-Ay.-)-2j AF.-|< e5j(|AxfI + lAy.l);
and since 2AF,- = P(x, y) - F{a, 6), the sum SP.Aa;, + Q.Ay,- approaches a limit,
and that limit is
UmV [P.Ax» + Q,Ay,] = r^^'Pdx ■{■ Qdy = F{x, y) - P(a, 6).
^^ *f a,b
EXERCISES
1. Find the area of the loop of the strophoid as indicated above.
2. Find, from (6), (7), the three expressions for the integrand of the line inte-
gr&Ii which give the area of a closed curve in polar coordinates.
3. Given the equation of the ellipse a; = a cos «, y = 6 sin t. Find the total area,
the area of a segment from the end of the major axis to a line parallel to the minor
axis and cutting the ellipse at a point whose parameter is t, also the area of a sector.
4. Find the area of a segment and of a sector for the hyperbola in its parametric
form x=ia coiib t, y = 6 sinh t.
». ExpreM the folium «• + y» = 8 oa^ in parametric form and find the area of
thaloopc
6. What area Is given by the curvilinear integral around the perimeter of the
dOMd ourre r = a«ln«4^? What in the case of the lemniscate ra = a» cos 2 0
diioHbed as In making the figure 8 or the sign «?
ON SIMPLE INTEGRAX8 297
7. Write f(jr y the analogous form to (0) for z. Show that In oanriUaMr
coordinates x = 0 (u, o), y = ^ (u, o) the area is
8. Compute these line integrals along the paths awfgned :
x'^ydx + y^dy, y* = x or y = x or y* = x*,
0.0
(/3) r ' (x« + y)dx + (x + y«)dy, y« = x or y = x or y« = ««,
-dx + dy, y = logx or y = 0 and x = e,
1. 0 3f
X 8in ydx + y cos xdy, y = mx or x = 0 and y = y,
0,0
Jr^l + i
(X — ty) dz, y = X or x = 0 and y = 1 or y = 0 and « = 1,
(x* — (1 + t)xy + y^)d2, quadrant or straijfht line.
9. Show that fPdx + Qdy = fVP^ + Q^ cos^da by working directly with the
figure and without the use of vectors.
10. Show that if any circuit is divided into a number of circuits by drawing
lines within it, as in a figure on p. 91, the line integral around the original circuit is
equal to the sum of the integrals around the subcircuits taken in the proper order.
11. Explain the method of evaluating a line integral in space and evaluate :
Jr* 1.1,1
xdx + 2 ydy + zdz, y* = x, z' = x or y = « = x,
0.0.0
(/3) I y logxdx + yHy + -dz^ y = x — 1, z = x* or y = iQgx, « = «.
»^i.o. 1 2;
12. Show that fPdx + Qdy + Rdz = f VP" + <? + B* cos Ms.
13. A bead of mass m strung on a frictionless wire of any shape falls from one
point (Xj,, yy, Zq) to the point (Xj, y^, Zj) on the wire under the influence of grarity.
Show that ing{zQ — z^) is the work done by all the forces, namely, gravity and
the nonnal reaction of the wire.
14. If X =/(0» y = 9{t), and /'(«), g'{t) be assumed continuous, show
£'P(Z, y)a. + Q(x, y)^v=£^{p^ + «f )d<.
where /{Iq) = a and g (t^) = 6. Note that this proves the statement made on pi^ MO
in regard to the possibility of substituting in a line integral. The theorem is also
needed for Exs. 1-8.
15. Extend to line integrals (15) in space the results of ( 188.
16. Angle as a line integral. Show geometrically for a plane enrre that
d0 = cos(r, n)ds/r^ where r is the radius vector of a curve and dM th* el— wnt of
298 INTEGRAL CALCULUS
arc Mid (r, n) the angle between the radius produced and the normal to the curve,
Is the angle subtended at r = 0 by the element ds. Hence show that
J r J rdn J dn
where the integrals are line integrals along the curve and dr/dn is the normal
derivative of r, is the angle 0 subtended by the curve at r = 0. Hence infer that
ri]2£rd. = 2x or rli^<b = 0 or C^-^d. = 0
Jo dn «'o dn •/o dn
according as the point r = 0 is within the curve or outside the curve or upon
the curve at a point where the tangents in the two directions are inclined at the
angle $ (usually ir). Note that the formula may be applied at any point (f, 17) if
r* = ({ _ x)* + (17 — vY where (x, y) is a point of the curve. What would the inte-
gral give If applied to a space curve ?
17. Are the line integrals of Ex. 16 of the same type j P{x,y)dx+ Q(x, y)dy
as those in the text, or are they more intimately associated with the curve ? Cf . § 165.
J^ 0, 1 /» 0, 1
(x — y) ds, (fi) I xyds along a right line, along a quad-
1,0 •/-i,o
rant, along the axes.
124. Independency of the path. It has been seen that in case the
integral around every closed path is zero or in case the integrand
Pdx H- Qdy is a total differential, the integral is independent of the
path, and conversely. Hence if
' Pdx + Qdy, then ^ = P, ^ = Q,
a, 6
^x~ ' dy
and ^L.^^A, i!£. = ?^ £f-^
dxdy dx dydx dy dy ex
provided the partial derivatives PJ and Q^ are continuous functions.*
It remains to prove the converse, namely, that: If the two partial
derivatives P^ and Q^J are continuous and equal, the integral
J
J a, I
Pdx + Qdy with P; = q; (18)
u independent of the path, is zero around a closed path, and the quantity
Pdx + Qdy is a total differential.
To show that the integral of Pdx + Qdy around a closed path is zero
if P^ «a (t, consider first a region R such that any point {x, y) of it may
* See 1 02. In particular observe the comments there made relative to differentials
which are or which are not exact. This difference corresponds to integrals which are
and which are not Independent of the path.
ON SIMPLK INTEGRALS 299
F(x, y) = rV(x, h)dx+ ['qCx, y)dy (19)
Ja Jb
be reached from (a, b) by following the lines y = 6 and ar = ar. Then
define the function Fix^ y) as
for all ]>oints of that region R. Now "3
dF dF d r'
But ^jf «(x,y)rfy=jf ^rfy=jf g^rfy = /'(.,y)
This results from Leibniz's rule (4') of § 119, which may be applied
since Q^ is by hypothesis continuous, and from the assumption Q^ — P'g.
Then ^p
-^ = P(x, h) + />(x, y) - P(ar, b) = P(ar, y).
Hence it follows that, within the region specified, Pdx 4- Qdy is the
total differential of the function F(a*, y) defined by (19). Hence along
any closed circuit witliin that region R the integral of Pdx + Q4y is
the integral of dF and vanishes.
It remains to remove the restriction on the type of region within which the
integral around a closed path vanishes. Consider any closed path C which lies
within the region over which P^ and Q'^ are equal continuous functions of (x, y).
As the path lies wholly within li it is possible to nile It so finely that any little
rectangle which contains a portion of the path shall lie wholly within R. The
reader may construct his own figure, possibly w ith reference to that of § 128, where
a finer ruling would be needed. The path C may thus be surrounded by a signg
line which lies within R. Each of the small rectangles within the zigzag line It a
region of the type above considered and, by the proof above given, the integral
around any closed curve within the small rectangle must be zero. Now the circuit
C may be replaced by the totality of small circuits consisting either of the perim-
eters of small rectangles lying wholly within C or of portions of the curve C and
portions of the perimeters of such rectangles as contain parts of C. And if C be so
replaced, the integral around C is resolved into the sum of a large number of inte-
grals about these small circuits ; for the integrals along such parts of the small
circuits us are portions of the perimeters of the rectangles occur in pairs with oppo-
site signs.* Hence the integral around C is zero, where C is any circuit within R,
Hence the integral of Pdx + Qdy from (a, 6) to (x, y) is indei)endent of the path
and defines a function F(x, y) of which Pdx + Qdy is the total differential. As
this f tmction is continuous, its value for points on the boundary of R may be defined
as the limit of F(x, y) as (x, y) approaches a ]>oint of the Iwundary, and it may thereby
be seen that the line integral of (18) around the boundary is also 0 without any fur-
ther restriction than that P^ and Q'^ bo e(iual and continuous within the boondarj.
• See Ex. 10 above. It is well, In connection with §§ Y2:\-V2&, to read earefnUy the
work of §§ 44--15 dealing with varieties of regions, reducibllity of circoits, etc
800 INTEGRAL CALCULUS
It should be noticed that the line integral
r 'pdx + Qdy= C P {x, h)dx-{- Tq (x, y) dy, (19)
when Pdx 4- Qdy is an exact differentialj that is, when P^ = Q^, may be
evaluated by the rule given for integrating an exact differential (p. 209),
provided the path along y = ^ and x=:x does not go outside the region.
If that path should cut out of iJ, some other method of evaluation would
be required. It should, however, be borne in mind that Pdx + Qd-
is best integrated by inspection whenever the function F, of which
Pdx 4- Qdy is the differential, can be recognized ; if F is multiple valued,
the consideration of the path may be required to pick out the par-
ticular value which is needed. It may be added that the work may be
extended to line integrals in space without any material modifications.
It was seen (§ 73) that the conditions that the complex function
F(x, y) = X (x, y) + iY{x, y), z = x-{- iy,
be a function of the complex variable z are
j?; = -y; and x^=r;. (20)
If these conditions be applied to the expression (13),
F(Xyy)= j Xdx-Ydy + ij Ydx + Xdy,
ioT the line integral of such a function, it is seen that they are pre-
cisely the conditions (18) that each of the line integrals entering into
the complex line integral shall be independent of the path. Hence
the integral of a function of a complex variable is independent of the
path of integration in the complex plane* and the integral around a
closed path vanishes. This applies of course only to simply connected
regions of the plane throughout which the derivatives in (20) are equal
and continuous.
If the notations of vectors in three dimensions be adopted,
jXdx -f Ydy + Zdz = C'F.dx,
where F = J:i + Kj + Zk, dx = idx + ^dy + k<f«.
In the particular case where the integrand is an exact differential and
the integral around a closed path is zero,
Xdx 4- Ydy + Zdx = F.rfr = dU= dr.VU,
ON SIMPLE INTEGRALS 801
where U is the function defined by the integral (for VU see p. 172).
When F is interpreted as a force, the function K = — IT such that
F=-VK or X = -'-f, r = -^. z^.^y
cx By a«
is called the potential function of the force F. Tike nsf^ive of the
slope of the potential function is the force F and the ne^aiwee of the
partial derivatives are the component forces along the axee*
If the forces are such that they are thiut derivable from a potential fondloii,
they are said to be conaeroative. In fact if
'"0 = '=-'^' '»^'-<"=-*-vr=-dr.
r^\ d^T , mdi di
1 m — •ar = • —
Jt, dt^ 2dt dt
"=-F \
fW-foO = ro-Fi or ^
^f + n = |r« + F,,
and
or
Thus the sum of the kinetic energy | mr* and the potential energy F is the
at all times or |x)8itionR. This is the principle of the conaeroatUm q^ et^erg^ for the
simple case of the motion of a particle when the force is oonaervatlTe. In cam the
force is not conservative the integration may still be performed ws
^W-»o')=/"F.<& = Tr.
where W stands for the work done by the force F during the motion. The result is
that the change in kinetic energy is equal to the work done by the force ; but d W
is then not an exact differential and the work must not be r^arded as a function
of (x, y, z), — it depends on the path. The generalization to any number of particles
as in § 123 is immediate.
125. The conditions that P'^ and Qj be continuous and equal, which
insures independence of the path for the line integral of Pdx -f Qrfy,
need to be examined more closely. Consider two examples :
where
It appears formally that P^ = Q^. If the integral be calculated around a sqiian of
side 2 a surrounding the origin, the result is
/» + « + ndj , r + " ody r-^-^adx r~* — ady _^ /*•*•• aeti
Pdx.
-\-Qdy
-h
. , ^
'x^ + l^
dP
_ V'-
x«
dQ_
_ y«-x«
^
(x« + i^)«'
az
"(x« + yV
802 INTEGRAL CALCULUS
The integral fails to vanish around the closed path. The reason is not far to seek,
the derivatives P' and Q^ are not defined for (0, 0), and cannot be so defined as
to be continuous functions of (x, y) near the origin. As a matter of fact
',p^ydx ^ xdy__ ^^•»'^,^„_ly_,,„_lyh'^
Jr. X, If _ ydx, xdy __ r^^v
dtan-i^ = tan-i^
X X\a,b
and tan -* (y/x) is not a single valued function ; it takes on the increment 2 v when
one traces a path surrounding the origin (§ 45).
Another illustration may be found in the integral
/d£_ rdx-\-idy _ r xdx + ydy . r
z ~ J x + iy ~ J «2 ^ y2 J
— ydx + xdy
X2 + 2/2
taken along a path in the complex plane. At the origin z = 0 the integrand l/z
becomes infinite and so do the partial derivatives of its real and imaginary parts.
If the integral be evaluated around a path passing once about the origin, the
result is
r ^= rilog(x2 + y2) + £tan-i?^]'''*' = 2iri. (21)
Jo Z \_2 XAa,b ^ '
In this case, as in the previous, the integral would necessarily be zero about any
closed path which did not include the origin ; for then the con-
ditions for absolute independence of the path would be satisfied.
Moreover the integrals around two different paths each encircling
the origin once would be equal ; for the paths may be considered
as one single closed circuit by joining them with a line as in the
device (§ 44) for making a multiply connected region simply con-
nected, the integral around the complete circuit is zero, the parts
due to the description of the line in the two directions cancel,
and the integrals around the two given circuits taken in opposite directions are
therefore equal and opposite. (Compare this work with the multiple valued nature
of log z, p. 161.)
Suppose in general that P{xj y) and Q(x, y) are single valued func-
tions which have the first partial derivatives Py and Q^ continuous
and equal over a region R except at certain points A, B, -". Surround
these points with small circuits. The remaining portion of 72 is such
that P'^ and Q*^ are everywhere equal and continuous ; but the region
is not simply connected, that is, it is possible to draw in the region
circuits which cannot be shrunk down to a point, owing to the fact
that the circuit may surround one or more of the regions which have
been cut out. If a circuit can be shrunk down to a point, that is, if it
is not inextricably wound about one or more of the deleted portions,
the integral around the circuit will vanish ; for the previous reasoning
will apply. But if the circuit coils about one or more of the deleted
regions so tliat the attempt to shrink it down leads to a circuit which
consists of the contours of these regions and of lines joining them, the
integral need not vanish ; it reduces to the sum of a number of integrals
ON SIMPLE INTEGRALS 808
taken around the contours of the deleted portions. If one ffirenit
can be slirunk into another, the integrals around the two cironits §n
equal if the direction of description is the same ; for a line oonneoting
the two circuits will give a combined circuit which can be shrunk down
to a point.
The inference from these various observations is that in a moltiply
connected region the integral around a circuit need not be zero and
the integi-al from a fixed lower limit («, i) to a variable npper limit
(a-, y) "lay not be absolutely indej)endent of the path, but may be dif-
ferent along two paths which are so situated relatively to the excluded
regions that tlie circuit formed of the two paths from (a, b) to (aj, y)
cannot be shrunk down to a point. Hence
^(^>y)= r^P^l-^-^-Qdy, p; = q; (generally),
/'■■
the function defined by the integral, is not necessarily single valued.
Nevertheless, any two values of F(x, y) for the same end point will
differ only by a sum of the form
Fi{^, y) — F^(x, y) = mi/j -f rriih + •
where /j, /j, . . . are the values of the integral taken around the con-
tours of the excluded regions and where mj, m^ , , . are positive or
negative integers which represent the number of times the combined
circuit formed from the two paths will coil around the deleted regions
in one direction or the other,
126. Suppose that f{z) = X{x, y) -f iY(x, y) is a single valued funo-
tion of z over a region R surrounding the origin (see figure above), and
that over this region the derivative /'(«) is continuous, that is, the
relations A'J" = — F^ and A'_; = l',; are fulfilled at every jx)iut so that
no points of R need be cut out. Consider the integral
CfJ^dz^ C^^^^idx^-idy) (22)
Jo * Jo^+^^
over paths lying within R, The function f(z)/x will have a contin-
uous derivative at all points of R except at the origin « = 0, where the
denominator vanislies. If then a small circuit, say a circle, be drawn
al)out the origin, the function /(«)/« will satisfy the requisite condi-
tions over the region wliich remains, and the integral (22) taken around
a circuit which does not contain the origin will vanish.
The integral (22) taken around a circuit which coils once and only
once about the origin will bo ec^ual to the integral taken around the
JJ04 INTEGRAL CALCULUS
small circle about the origin. Now for the circle,
where the assumed continuity of f(z) makes \rj(z)\ < c provided the
circle about the origin is taken sufficiently small. Hence by (21)
- rm,.=
Ci^ dz = C'-^ dz = 2 7ri/(0) -f ^
Jo Jo
with 1^1 = 1 r^c^«]^ Cl'llldzl^c C 'd$ = 2 7r€.
\Jq 1 Jo I I Jo
Hence the difference between (22) and 2 7rif(0) can be made as small
as desired, and as (22) is a certain constant, the result is
Jo
'•M.,=
dz = 27rif(0). (23)
A function f(z) which has a continuous derivative f'(z) at every
point of a region is said to be analytic over that region. Hence if the
region includes the origin, the value of the analytic function at the
origin is given by the formula
/(O)
Jo
where the integral is extended over any circuit lying in the region and
passing just once about the origin. It follows likewise that it z = a is
any point within the region, then
/(«)
j_ r/M
27ri
Jo
where the circuit extends once around the point a and lies wholly within
the region. This important result is due to Cauchy.
A more convenient form of (24) is obtained by letting t = z repre-
sent the value of z along the circuit of integration and then writing
ass z and regarding z as variable. Hence Cauchy's Integral :
Jo
This stales that if any circuit be drawn in the region over which f{z)
if analytic^ the value of f(z) at all points within that circuit may be oh-
by evaluating Cauchi/s Integral (26). Thus f(z) may be regarded
ON SIMPLE INTEGRALS
806
as defined by an integral containing a parameter «; for many pur-
poses this is convenient. It may be remarked that when the yalnet of
f{z) are given along any circuit, the integral
may l)e regarded as defining /(«) for all points
within tliat circuit.
To find the successive derivatives of /(z)f it
is merely necessary to differentiate with respect
to z under the sign of integration. The condi-
tions of continuity which are required to justify
the differentiation are satisfied for all points z
actually within the circuit and not upon it. Then
As the differentiations may be performed, these formulas show that an
analytic function has continuous derivatives of all orders. The definition
of the function only required a continuous first derivative.
Let a be any j)articular value of z (see figure). Then
1 ^ 1 ^1 1
t — z {t ^ a) ^ {z — a) t — a z — a
t — a
t — a (t — ay (t — a)" * __ z — a
t — a_
with R = - — : I ^ { f-^^^ dt.
" 27riJ^(t-ay ^_z-at-a
t — a
Now t is the variable of integration and « — a is a constant with respect
to the integration. Hence
This is Taylor^s Formula for a function of a complex variable.
306 INTEGRAL CALCULUS
EXERCISES
1. If P' — <3i, Qj = ^» ^ = ^« 3,nd if these derivatives are continuous, show
that Pdx + Qdy + Rdz is a total differential.
2. Show that r'* P(x, y, a)dx + Q{x, y, a)dy, where (7 is a given curve,
CJa,b
defines a continuous function of or, the derivative of which may be found by differ-
entiating under the sign. What assumptions as to the continuity of P, Q, P^ , Qa
do you make ?
• ,^ 1 r'dz r''Vxdx-{-ydy , . r^^v — ydx + xdy , ^ , ^,
S. If loff 2 = I — = I — :; — ^ + * I — ^-^^ :r^ "^ taken as the
definition of log «, draw paths which make log (J + I V— 3) = jfri, 2| tti, — If irt.
4. Study r ~ with especial reference to closed paths which surround + 1,
«/o z^ — 1
— 1, or both. Draw a closed path surrounding both and making the integral vanish.
5. If /(z) is analytic for all values of z and if \f{z) \ < K, show that
taken over a circle of large radius, can be made as small as desired. Hence infer
that/(«) must be the constant /(z) =/(0).
6. If G (z) = Oq + ajZ + . . . + anZ^ is a polynomial, show that/(z) = 1/6? (z) must
be analytic over any region which does not include a root of G{z) = 0 either within
or on its boundary. Show that the assumption that G{z) = 0 has no roots at all
leads to the conclusion that /(z) is constant and equal to zero. Hence infer that
an algebraic equation has a root.
7. Show that the absolute value of the remainder in Taylor's Formula is
1 7^ ML
Al = '^-"'
«| r f{t)dt
\Jo{t-a)^{t-
2ir fi^ p — r
2ir \Jo{t-aY{t-z)
for all points z within a circle of radius r about a as center, when p is the radius
of the largest circle concentric with a which can be drawn within the circuit about
which the integral is taken, M is the maximum value of f{f) upon the circuit, and
L is the length of the circuit (figure above).
8. Examine for independence of path and in case of independence integrate :
(a) jx'^ydx + xyHy, (/3) j xy^dx + x^ydy, (7) C xdy + ydx,
(«) J (x« + a;y) dx + {y^ + xy) dy, («) / 2/ cos xdy + i 2/^ gin xdx.
9. Find the conservative forces and the potential :
(a) X = —±— Y = —l—, Z = _l_,
(x« + !/«)» (x« + y^)\ (xa + y^)h
ifi) X=^nx, Y=^ny, (7) X = yx, Y=y/xi
ON SIMPLE INTEGRALS 807
10. If /2(r, 0) and ♦(r, 0) are the component force* reeoWed along the nuUuc
vector and perpendicular U) the radiuH, sliow that dW = Hdr -f r^di^ to the dUbN
ential of work, and express the condition that the forces /f, ^ be eoneenraUre.
11. Show that if a particle is acted on by a force £ = ~/(r) directed towaid
the origin and a function of the distance from the origin, the force ia coneerratlve.
12. If a force follows the Law of Nature, that is, acts toward a point and Tarica
inverriely U8 the square r' uf the distance from the point, ahow that the
is — k/r.
13. From the results F = -VForF=- C ¥*dr = T-Ydx + Ydv + Zig
that if Fi is the potential of Fj and Fj of F.^ then V=l\'^ F, will be the
potential of F = F^ + F2, that is, show that for conservative forces the addition of
potentials is equivalent to the parallelogram law for adding forces.
14. If a particle is acted on by a retarding force — kw proportional to the
velocity, show that R = \ Icv^ is a function such that
Ovx CVp CVg
dW=- ky*dx = - A: {v^ + vyiy + tj^).
Here R is called the dissipative function ; show the force is not conservative.
15. Pick out the integrals independent of the path and integrate :
(a) J yzdx + xzdy + xydz^ (Ji) j ydx/z + xdy/z — xydz/ifl^
(7) J xyz {dx + dy ■{- dz), (3) f log {xy) dx + xdy + ydz.
16. Obtain logarithmic forms for the inverse trigonometric functions, analopous
to those for the inverse hyperbolic functions, either algebraically or by considering
the inverse trigonometric functions as defined by integrals as
1 C dz . . r* dz
tan-»z=i I -, 8in-»z = / _——__»....
•/o 1+z^ Jq Vl — z*
17. Integrate these functions of the complex variable directly according to the
niles of integration for reals and determine the values of the integrala by
substitution :
ze^'^dz, 03) J cosSzdz, {y)J (1 + ««)-*<^
In the case of multiple valued functions mark two different paths and give two values.
18. Can the algorism of integration by parts be applied to the definite (or indefi-
nite) integral of a function of a complex variable, it being underBtood that the
integral must be a line intej;ral in the complex plane ? Consider the proof of
Taylor's Formula by integnition by parts, p. 67, to ascertain whether the proof is
valid for the complex plane and what the remainder means.
808 INTEGRAL CALCULUS
19. Suppose that in a plane at r = 0 there is a particle of mass m which attracts
according to the law F = m/r. Show that the potential is F = m log r, so that
p = _VV. The induction or flux of the force F outward across the element ds of
a curve in the plane is by definition — Fcos(F, n)ds. By reference to Ex. 16,
p. 297, show that the total induction or flux of F across the curve is the line integral
(along the curve)
-/Fcos(F,„)d. = m/^'da = /^<fc;
— Ir 1 r dV ,
•nd wi = — - / F cos (F,n)ds = -— j -—ds,
2 IT Jo 2'irJodn
where the circuit extends around the point r = 0, is a formula for obtaining the
mass m within the circuit from the field of force F which is set up by the mass.
20. Suppose a number of masses 7Wj , wig, • • • , attracting as in Ex. 19, are situated
at pointe ({j, Vih (^2* ^i)^ * ' ' i" *^® P^^"^- ^®*
F = Fj + F, + . . ., F= Fi + Fg + • • ., Vi = m.log[(^i- x)2 + {vi-y)^]i
be the force and potential at (x, y) due to the masses. Show that
=i/^Fcos(F,„)da = l-2;//^<t^=X'»' = ^.
where 2 extends over all the masses and S' over all the masses within the circuit
(none being on the circuit), gives the total mass M within the circuit.
127. Some critical comments. In the discussion of line integrals
and in the future discussion of double integrals it is necessary to speak
frequently of curves. For the usual problem the intuitive conception
of a curve suffices. A curve as ordinarily conceived is continuous, has
a continuously turning tangent line except perhaps at a finite number
of angular points, and is cut by a line parallel to any given direction in
only a finite number of points, except as a portion of the curve may
coincide with such a line. The ideas of length and area are also appli-
cable. For those, however, who are interested in more than the intuitive
presentation of the idea of a curve and some of the matters therewith
connected, the following sections are offered.
If 0 (t) and f (t) are two single valued real functions of the real variable t defined
for all values in the interval t^ ^ t ^ tj, the pair of equations
x = ^(0, y = ^(<), t^^t^t,, (27)
will be said to define a curve. If 0 and rj/ are continuous functions of t, the curve
will be called continuous. If <f>{t{) = 0(y and ^(<^) = \('(<o), so that the initial and
end points of the curve coincide, the curve will be called a closed curve provided
It is continuous. If there is no other pair of values t and t' which make both
0(t) = 0(r) and }ff(t) = ^(r), the curve will be called simple; in ordinary language,
the curve does not cut itself. If t describes the interval from t^ to «, contiiniously
and ooncUntly in the same sense, the point (x, y) will be said to describe the curve
In agifen lenae ; the opposite sense can be had by allowing t to describe the interval
In the oppodte direction.
ON SIMPLE INTEGRALS 809
Let the interval t^^t^t^ be divided into anj number n of »«hfntiinb
At^ A^^ • • • , ^mt. There will be n corresponding incremenu for x tad y,
A|X, V, . . ., A^, and A,y, A,y, . . ., A«|f.
Then A.-c = V(A.x)a + (A<i/)« ^|A<x| + |A<y|, |A<jc|SA4C, |A^|SA^
are obvious inequalities. It will be necessary to consider the three hum
(r,=2^1A,xI, (r,=]g|A,yI, ^, = ^ A,c = JJ ^^^(WToSJp.
1 1 11
For any diviHion of the interval from t^ to t^ each of these sums has a definite
positive value. When all posHible modes of division are cunsidered for any and
every vahie of n, the Kuniri 0-^ will form an infinite set of numbers which may be
either limited or unlimited above (§22). In case the set is limited, tha upper
frontier of the set is called the variation of x over the curve and the curve is said
to be of limited variation in x ; in case the set is unlimited, the curve is of nnHfqtf4»t|
variation in x. Similar observations for the sums 0-,. It may be remarked that the
geometric conception corresponding to the variation in x is the sum of the projee-
tions of the curve on the x-axis when the sum is evaluated arithmetically and not
algebraically. Thus the variation in y for the curve y = sin 2 from 0 to %w \» 4.
The curve y = sin(l/x) between these same limits is of unlimited Tarlation in y.
In both cases the variation in x is 2ir.
If both the sums 0-^ and <r^ have upper frontiers L^ and L,, the sum r, will hava
an upper frontier X3 ^ L^ + L„ ; and conversely if a^ has an upper frontier, both
0, and ff.^ will have upper frontiers. If a new point of division is intercalated in A/,
the sum a-, cannot decrease and, moreover, it cannot increase by more than twice
the oscillation of x in the interval A,<. For if A^x + Aa,x = A«x,
|Ai,-x| + |A2.-x|^|A.x|, |Ai,-x| + |A2,-x|^2(3r,-m,).
Here Aut and A-^tt are the two intervals into which A,< is divided, and Iff — m< is the
oscillation in the interval A,t. A similar theorem is tnie for v^. It now remains to
show that if the interval from t^ to t^ is divided sufficiently fine, the sums r, and #,
will differ by as little as desired from their frontiers Lj and L,. The proof b Ilka
that of the similar problem of § 28. First, the fact that L^ is the frontier of r,
that some method of division can be found so that L^ — r^ <\t. Suppose the
ber of |x)ints of division is n. Let it next be assumed that ^{t) is continooos; it
must then be uniformly continuous (§ 25), and hence it is possible to find a I so
small that when A,« < 8 the oscillation of x is 3f, — »j,- < t/An. Consider then any
method of division for which A,i < 3, and its sum o-J. The superposition of the former
division with n points upon this gives a sum ff\' ^ ff[. But ^i — #1 < 2iu/4m = J*,
and a[' ^ <r^. Hence L^ — o'{ <\t and i^ — «rj < «. A similar demonstration may
be given for 0-5 and L^.
To treat the sum 0-, and its upper frontier X, note that here, too. the intercalation
ef an additional point of division cannot decrease «>, and, as
V(Ax)« + (Ai/)«^|Ax| + |Ay|,
It cannot increase 0-3 by more than twice the sum of the oscillations of z and y in
the interval M. Hence if the curve is continuous, that is, If both x and y are con-
tinuous, the division of the interval from t^ to i, can he tiken ao fine that #, shall
810 INTEGRAL CALCULUS
differ from its upper frontier X, by less than any assigned quantity, no matter how
small. In this case X, = « is called the length of the curve. It is therefore seen that
the necesaar]/ and 8uffi.cient condition that any continuous curve shall have a length is
that its Cartesian coordinates x and y shall both he of limited variation. It is clear that
if the frontiers L^{t), L^{t), L^{t) from t^, to any value of t be regarded as functions
of f, they are continuous and nondecreasing functions of t, and that L^{t) is an
increasing function of t; it would therefore be possible to take s in place of < as
the parameter for any continuous curve having a length. Moreover if the deriva-
tives x' and y* oix and y with respect to t exist and are continuous, the derivative s'
exists is continuous, and is given by the usual formula s' — Vx'^ + y"^. This will
be left as an exercise; so will the extension of these considerations to three
dimensions or more.
In the sum Xj — x^ = 2A,-x of the actual, not absolute, values of AjX there may
be both positive and negative terms. Let rr be the sum of the positive terms and
9 be the sum of the negative terms. Then
Xj - Xo = ir -K, (Tj = TT + »', 2 IT = Xi - Xq + o-j, 2 V = Xq - Xi + (rj.
As o^ has an upper frontier Xj when x is of limited variation, and as x^ and Xj are con-
stants, the sums rr and v have upper frontiers. Let these be IT and N. Considered
as functions of t, neither 11 (f) nor N(t) can decrease. Write xif) = x^ -f- 11 (i) — N(i).
Then the function x (<) of limited variation has been resolved into the difference of
two functions each of limited variation and nondecreasing. As a limited non-
decreasing function is integrable (Ex. 7, p. 54), this shows that a function is integrable
over any interval over which His of limited variation. That the difference x = x" — x"
of two limited and nondecreasing functions must be a function of limited variation
follows from the fact that [Ax] ^ | Ax''| + | Ax'|. Furthermore if
x = Xo+n-N be written « = [Xq + n + [xj + i - y -[N + |XoI+ «- g,
it is seen that a function of limited variation can be regarded as the difference of two
positive functions which are constantly increasing, and that these functions are con-
tinuous if the given function x (t) is continuous.
Let the curve C defined by the equations x = 4>{t), y = \f/{t), t^^t^t^, be
continuous. Let P(x, y) be a continuous function of (x, y). Form the sum
^ P (f .-, vi) A.-X = 2^ P (f .-, ,;,•) A,-x- - ^ P (f .-, vi) AiX', (28)
where AjX, A^x, ... are the increments corresponding to A^^, A^t, • • • , where (f,-, rn)
is the point on the curve which corresponds to some value of t in A,i, where x is
assumed to be of limited variation, and where x" and x' are two continuous increas-
ing functions whose difference is x. As x" (or x') is a continuous and constantly
increasing function of (, it is true inversely (Ex. 10, p. 45) that t is a continuous and
constantly increasing function of x" (or x'). As P(x, y) is continuous in (x, y), it
is continuous in t and also in x" and x\ Now let Ait = 0 ; then A,x" = 0 and
A^^O. Also
lim^P,AfX''=J'*Pdx" and lim^^P.A.x' = J'^^Pdx'.
Tlie llmiu exist and are integrals simply because P is continuous in x" or in x'.
liencu the sum on the U^ft of (28) has a limit and
lim
SPAiX = f'Pdi = f'Pdx'^ - r^Pdxf
ON SIMPLE INTEGRALS
811
may be d^ned as the line integral qf P along the curse C t^ UmMtd waH^ttm im c
The aM8uiuptiun that y in of limited variation and that Q(x, y) Ic eoDlinnoni would
lead to a correKiM>ii<iiiig line integral. The OMiimpMoii thai both x at^ yv^ tfcnifrf
variation, that is, that the curve ia reetifitMe^ and that P and Q art eimtimmomt MNMld
lead to the existence of the litie integrcU
r'''*''P(x, y)dx + P^J- "\dy.
C^x,.„,
Me curvM may be oon-
A conHiderable tlu'ory of line integrals over giMui
structtHl. The subject will not be carried further .. nf.
128. The <|ueHtion of the area of a curve requirei* cuit- (iil '.a
tirst pliice note that the intuitive closed plane curve which u u.
tively believed to divide the plane into two regiona, one interior, one exterior to tbt
curve ; and thene regions have the property that any two points of thei
may bo connected by a continuous curve which does not cut the
whereas any continuous curve which connects any point of one region to a point
of the other must cut the given curve. The first question which arinee with regard
to the general closed simple curve of page 308 is : Does such a curve diride the plane
into juKt two regions with the properties indicated, that is, is there an interior and
exterior to the curve ? The answer is affirmative, but the proof is somewhat dilBcult —
not because the statement of the problem is involved or the proof repleCe with
advanced mathematics, but rather because the statement is so simple and
tary that there is little to work with and the proof therefore requires the
and most tedious logical analysis. The theorem that a closed simple plane curre
has an interior and an exterior will therefore be assumed.
As the functions x{t), y{t) which define the curve are continuous, they are lim-
ited, and it is possible to draw a rectangle with sides x = a, x = b, y = r, y = dso
as entirely to surround the curve. This rectangle may next be nde«l wltJi a
ber of lines parallel to its sides, and thus be
divided into smaller rectangles. These little rec-
tangles may be divided into three categories, those
outside the curve, those inside the curve, and
those upon the curve. By one upon the curve is
meant one which has so much as a single point
of its perimeter or interior upon the curve. Let
A, Ai, Au, Ae denote the area of the large rec-
tangle, the sum of the areas of the small rectan-
gles, which are interior to the curve, the sum of
the areius of those upon the curve, and the sum of
those exterior to it. Of coui-se A=Ai -{■ A^-^-A^.
Now if all methods of niling be considered, the
quantities Ai will have an upper frontier L,-, the quan
frontier !,«, and the quantities Au will have a lower f i
of ruling new rulings be added, the quantities At and
ihe conditions .4;. ^ At, A^ ^ A^, and hence A'^ ^ A^. i- >'.... ihis It follows that
^ = L, -h ^ + Le. For let there be three moties of nding which for the reipecdva
cases Ai, A^, Au make these three t|uantities «liffer inm\ their frontiers Lt^ I^lm
by less than J e. Then the superposition of the three systems of rulings gives rise
to a ruling for winch -4 J, A',, A'^ must differ from the frontier Talues bj le«
ve aa
any
a: and a: wilb
812 INTEGRAL CALCULUS
1 *, and hence the sum !.< + f« + i«, which is constant, differs from the constant A
by less than e, and must therefore be equal to it.
It is now possible to d^ne as the (qualified) areas of the curve
Li = inner area, l^ = area on the curve, if + ^ = total area.
In the case of curves of the sort intuitively familiar, the limit Ui is zero and
Li = A — Lt becomes merely the (unqualified) area bounded by the curve. The
question arises : Does the same hold for the general curve here under discussion ?
This time the answer is negative; for there are curves which, though closed and
simple, are still so sinuous and meandering that a finite area l^ lies upon the curve,
that is, there is a finite area so bestudded with points of the curve that no part of
it is free from points of the curve. This fact again will be left as a statement with-
out proof. Two further facts may be mentioned.
In the first place there is applicable a theorem like Theorem 21, p. 61, namely :
It is possible to find a number S so small that, when the intervals between the
rulings (both sets) are less than 5, the sums ^„, At, Ae differ from their frontiers
by less than 2 e. For there is, as seen above, some method of ruling such that these
sums differ from their frontiers by less than €. Moreover, the adding of a single
new ruling cannot change the sums by more than AD, where A is the largest inter-
val and D the largest dimension of the rectangle. Hence if the total number of
intervals (both sets) for the given method is N and if 5 be taken less than c/NAD,
the ruling obtained by superposing the given ruling upon a ruling where the inter-
vals are less than S will be such that the sums differ from the given ones by less
than e, and hence the ruling with intervals less than S can only give rise to sums
which differ from their frontiers by less than 2 c.
In the second place it should be observed that the limits Li, lu have been obtained
by means of all possible modes of ruling where the rules were parallel to the x- and
y-axes, and that there is no a priori assurance that these same limits would have
been obtained by rulings parallel to two other lines of the plane or by covering the
plane with a network of triangles or hexagons or other figures. In any thorough
treatment of the subject of area such matters would have to be discussed. That
the discussion is not given here is due entirely to the fact that these critical com-
ments are given not so much with the desire to establish certain theorems as with
the aim of showing the reader the sort of questions which come up for considera-
tion in the rigorous treatment of such elementary matters as " the area of a plane
curve," which he may have thought he " knew all about."
It is a common intuitive conviction that if a region like that formed by a square
be divided into two regions by a continuous curve which runs across the square
from one point of the boundary to another, the area of the square and the sum of
the areas of the two parts into which it is divided are equal, that is, the curve
(counted twice) and the two portions of the perimeter of the square form two
iilinple cloned curves, and it is expected that the sum of the areas of the curves is
the area of the 8<juare. Now in case the curve is such that the frontiers f„ and t^
formed for the two curves are not zero, it is clear that the sum i,- 4- L^ for the
two curves will not give the area of the square but a smaller area, whereas the
mm (L< + /«) -♦- (Lf + tj will give a greater area. Moreover in this case, it is not
et«7 to formulate a general definition of area applicable to each of the regions and
•uch that the sum of the areas shall be equal to the area of the combined region.
Uut if l^ and t^ both vanish, then the sum Li + X,' does give the combined area.
ON SIMPLE INTEGRALS 818
It iH therefore cuBtomary to ruMet the appUeaUom i^tkt term **«f«a** to mt
cloned curves aa have /m = 0, ami to my thai the quadraturt <^ iMeA CMiiWi {m
but that tlie quadrature of curveH for which (« ;e 0 b impoMlbto.
It may be proved that : If a curve is rectifiable or even \f one qf the /kmttkmM g{(^
or y {t) vi of limited variation, the limit l^ (a tero and the quadraimre qf the ame <•
poMsUUe. Fur let the interval t^^^t^t^ be divided into lnt«nraU A,<, A/, • • • la
which the oKcillatioiiH of x and y are <„ «)«•••, fp f^ ••• . Then tiM portloo o(
ihe curve due to the interval Ait may be inscribed In a reoUogto %% and
I>orti()n of the curve will lie wholly within a rectangle i§f%m **«TfTntTllf
thJH one. In thiH way may be obtained a itet of rectanglet which entlrtlj
the curve. The total area of these rectangles most exceed C For if all the
of all the rectangles be produced so as to rule the plane, the rectangles which go
to make up Au for this ruling nmst be contained within the original niftai^w,
and as A^>1^, the total area of the original rectangles is greater than C N«t
.sui)i>ose X {t) is of limited variation and is written as Zg + !!(() — N{t)^ the
ence of two nondecreasin^' functions. Then Stf ^ n(f,) -f JV(i|), that la, the
of the oscillations of x cannot exceed the total variation of z. On the <
:us y (t) is continuous, the divisions Ait could have been taken so small that % < f .
Hence
^. <^.. ^ 2^ 2e. . 2,,. < 41, 2^ e.^ 4,[n(ej) + iV(«,)].
The quantity may be made as small as desired, since it Is the product of a
quantity by i;. Hence U, = 0 and the quatlrature is possible.
It may be observed that if x{t) or y{t) or both are of limited TariaUon, a
all of the three curvilinear integrals
- Jydx, Jxdy, | Jxdy - ydz
may be defined, and that it should be expected that in this caae the ?atiie d the
integral or integrals would give the area of the curve. In fact if one desired to
deal only with rectifiable curves, it would be possible to take one or all of <
integrals as the dffinition of area, and thus to obviate the discussions of the
ent article. It seems, however, advisable at least to point out the problem of
cjuadrature in all its generality, especially as the treatment of the problem is teiy
siniilar to that usually adopted for double integrals (§ 182). From the preseiH
viewpoint, therefore, it would be a proposition for demonstration that the ennrl-
11 near integrals in the cases where they are applicable do give the Talue of the
area as here defined, but the demonstration will not be undertaken.
EXERCISES
1. For the continuous curve (27) prove the following properties
(a) Lines x = a, x = 6 may be drawn such that the cunre lies enureiy oetwprn
them, ha« at least one point on each line, and cuts every line x = (, a<|<ft|inat
least one point ; similarly for y.
(/3) From p = z cos a + y sin or, the normal equation of a line, prove the prop*
ositions like those of (cr) for lines parallel to any direction.
(7) If ({, n) is any point of the xy-plane, show that the distance of (|, f) from
the curve has a minimum and a maximum value.
314 INTEGRAL CALCULUS
(«) If m({, ii) and 3f({, i;) are the minimum and maximum distances of (f, -n)
from the curve, the functions m(f, v) and M{i, 17) are continuous functions of (^, tj).
Are the coordinates x(f, 17), y(e, 1?) of the points on the curve which are at mini-
mum (or maximum) distance from ({, v) continuous functions of (f , 17) ?
( e ) If t\ t'\ • • • , i<*>, • • • are an infinite set of values of t in the interval tf^^t^t^
and if t« is a point of condensation of the set, then x° = ip {t^), y^ = yf/ (t^) is a point
of condensation of the set of points (x', y'), (x", y"). • • •» (^^^^^ y^*^)i * • ' corre-
sponding to the set of values «',<"••., t(^\ • • • .
(i") Conversely to (e) show that if (x', y')* (a^", y")» • * * » (^^*^ 2/^*^), • • • are an
infinite set of points on the curve and have a point of condensation (x'', yO), then
the point (xO, y^) is also on the curve.
(17) From (f) show that if a line x = $ cuts the curve in a set of points y', y", • • • ,
then this suite of y's contains its upper and lower frontiers and has a maximum or
minimum.
2. Define and discuss rectifiable curves in space.
3. Are y = x^ sin - and y = Vx sin - rectifiable between x = 0, x = 1 ?
X X
4. If x(0 in (27) is of total variation n (t^) + N {t{), show that
f 'P{x, y)dx <M[n{t,) + l^{i,)l
where M is the maximum value of P (x, y) on the curve.
5. Consider the function ^(f, rj, t) = tan-i — which is the inclination of
^ — x{t)
the line joining a point (f , 17) not on the curve to a point (x, y) on the curve. With
the notations of Ex. 1 (5) show that
\^te\ = \0{^,V.t + At) - 0 (?, r,,t)\< ^^^ ,
m — 2 Mo
where 3 > | Ax | and 5>\Ay\ may be made as small as desired by taking M sufficiently
small and where it is assumed that m^O,
6. From Ex. 6 infer that 0 ({, 17, t) is of limited variation when t describes the
interval tQ^t^t^ defining the curve. Show that tf ({, 17, t) is continuous in (^, 77)
through any region for which m > 0.
7. Let the parameter t vary from t^ to t^ and suppose the curve (27) is closed so
that (x, y) returns to its initial value. Show that the initial and final values of
ff ({, 17, t) differ by an integral multiple of 2 v. Hence infer that this difference is
constant over any region for which m > 0. In particular show that the constant is
0 over all distant regions of the plane. It may be remarked that, by the study of
this change of tf as t describes the curve, a proof may be given of the theorem that
the closed continuous curve divides the plane into two regions, one interior, one
exterior.
8. Extend the last theorem of § 123 to rectifiable curves.
CHAPTER XII
ON MULTIPLE INTEGRALS
129. Double sums and double integrals. -
injitttT is so tliin and Hat that it can Im- coiisi., ■ :: .i ;
If any small portion of the body surrounding a given point P{x, y) be
considered, and if its mass be denoted by Am and V ' '
average (surface) density of the portion is the quoti<
actual density at the point P is defined as the limit of thi t
when A.l == 0, that is, .
The density may vary from point to point Now converaeljr snppow
that the density D(xj y) of the body is a known function of (jr, y) and
that it be required to find the total mass of the
body. Let the body be considered as divided
up into a large number of pieces each of which
is small in every direction^ and let A.4, Ixi the
area of any piece. If (^,-, i;,) be any point in
Ai4,., the density at that point is Z)(^f, 7^) and
the amount of matter in the piece is approxi-
mately i>(^i, 7]i)AAi provided the density be regarded as continuooa,
that is, as not varying much over so small an area. Then the sum
"xtendcd over all the pieces, is an approximation to the total
and may be sufficient for practical purposes if the pieoes be taken
tolerably small.
The process of dividing a body up into a large number of small pieoea
of which it is regarded as the sum is a device often resorted to ; for the
proj^erties of the small pieces may be known approximately, so that
the corresponding j)roperty for the whole body can be obtained approx-
imately by summation. Thus by definition the moment of ineKia of a
small ])article of matter relative to an axis is mr^f where m is the mass
of the i)article and r its distance from the axis. If therefore the
moment of inertia of a plane body with respect to an axis perpendicalar
815
816 INTEGRAL CALCULUS
to its plane were required, the body would be divided into a large
number of small portions as above. The mass of each portion would
be approximately D(^o i7,)A^< and the distance of the portion from
the axis might be considered as approximately the distance r,- from
the point where the axis cut the plane to the point (^,., 77.) in the por-
tion. The moment of inertia would be
or nearly this, where the sum is extended over all the pieces.
These sums may be called double sums because they extend over two
dimensions. To pass from the approximate to the actual values of the
mass or moment of inertia or whatever else might be desired, the
underlying idea of a division into parts and a subsequent summation
is kept, but there is added to this the idea of passing to a limit. Com-
pare §§16-17. Thus
.Jl!^^,.oXD(^<^vd^Ai and J^%X^(^-vdr.^^^i
would be taken as the total mass or inertia, where the sum over n
divisions is replaced by the limit of that sum as the number of
divisions becomes infinite and each becomes small in every direction.
The limits are indicated by a sign of integration, as
lim 2^ i> (^,, r,,)AAi =Cd (x, y) dA, lim ^ ^ (ii, vd ^i^^i = f^r^dA .
The use of the limit is of course dependent on the fact that the limit
is actually approached, and for practical purposes it is further depend-
ent on the invention of some way of evaluating the limit. Both these
questions have been treated when the sum is a simple sum (§§ 16-17,
28-30, 35) ; they must now be treated for the case of a double sum like
those above.
130. Consider again the problem of finding the mass and let Z),- be
used briefly for I>(ft, rji). Let Af^ be the maximum value of the density
in the piece Ai4< and let m,. be the minimum value. Then
TTiiAAi ^ DiAAi^ MiAAi.
In this way any approximate expression Z),A^4f for the mass is shut in
between two values, of which one is surely not greater than the true
mass and the other surely not less. Form the sums
extended over all the elements AA^. Now if the sums s and S approach
the same limit when A^f==0, the sum 2AA^4, which is constantly
ON MULTIPLE INTEGRALS
817
included between 8 and S must also approach that limit independeiitlv
of how the ])oint8 (^„ 17,) are chosen in the areas Ai4|.
That s and .S' do approach a common limit in the usual caae of a
continuous function />(ir, y) may be shown strikingly if the ttir&oe
z = D(Xf y) be drawn. The
term /),A/lf is then repre-
sented by the volume of a
small cylinder upon the base
A/1, and with an altitude equal
to the height of the surface
z = D(xj y) above some point
of A.4,.. The sum 2/).A/l,. of
all these cylinders will be aj)-
proximately the volume under
the surface z = D(x, y) and
over the total area A = 2AJ,-.
The term A/,- A. 4,. is represented
by the volume of a small cylin-
der ujK)n the base A/1,. and cir-
cumscribed about the surface ;
the term ?w,A.l„ by a cylinder
inscribed in the surface. When the number of elements ^A^ is in<
without limit so that each becomes indefinitely small, the three sums #,
Sy and 2Z>,A.l,- all approach as their limit the volume under the Burfaet
and over the area .4. Thus the notion of volume does for the
sum the same service as the notion of area for a simple sum.
Let the notion of the integral be applied to find the forwuUa /or tk» em
gravity of a plane lamina. Assume that the rectangular ooOrdtnatM of the
of gravity are (x, v)- Consider the body u divided into small areu Ail<. If(&t%)
is any point in the area A/1,, the approximate moment of
the approximate ma88D,A/l,- in that area with respect to
the line x = x is the product (f, — x)Z>,A4, of the man
by its distance from the line. The total exact moment
would therefore be
lim
^ (f. - X) D^AAi = fix -X)D(z,y)dA=0,
and must vanish if the center of gravity lies on the line
X = X as assumed. Then ^
fz)
fxD{x, y)dA -fxD{x, y)(L4 = 0 or CzDdA-^fD{*y i^dA,
These f onnal operations presuppose the facte that the difference of two fartegieli Is
the integral of the difference and that the integral of a ooneUnt I Umee a faaetkNi B
318 INTEGRAL CALCULUS
is the product of the constant by the integral of the function. It should be imme-
diately apparent that as these rules are applicable to sums, they must be applicable
to the limite of the sums. The equation may now be solved for x. Then
(xDdA fxdm CyBdA J ydm
fndA "^ jndA
where m stends for the mass of the body and dm for DdA, just as Am,- might replace
DiAAi ; the result for y may be written down from symmetry.
As another example let the kinetic energy of a lamina moving in its plane be cal-
culated. The use of vectors is advantageous. Let Iq be the
vector from a fixed origin to a point which is fixed in the
body, and let ii be the vector from this point to any other
point of the body so that
- = ---" f = W-^t - -T- + --
The kinetic energy is S ^ rjAm,- or better the integral of i vHm. Now
r? = v.-.Vi = Vo.Yo + Vii.vij + 2 Vo.vif = v^ + r^y + 2 Vo-Vi,-.
That Vi ,••▼!,• = r^iU^i where ru = |ri,| and w is the angular velocity of the body
about the point r©, follows from the fact that lit is a vector of constant length ru
and hence |dri,| = rudO^ where dO is the angle that ru turns through, and conse-
quently w = d$/dt. Next integrate over the body.
ji iMm = ji v^dm + Ti r^w^dm -\- JYo'^idm
^iv^M+iu'^Jr^dm + YQ.jYidm; (2)
for v^ and u^ are constants relative to the integration over the body. Note that
To* Cyydm = 0 if Vo = 0 or if Cy^dm = f — T^i^m = — ii^dm = 0.
But Tq = 0 holds only when the point r^ is at rest, and ( r^dm = 0 is the condition
that r^ be the center of gravity. In the last case
T=fi t^m = i v^M + i w2 j^ i^ fr^dm.
A« / is the integral which has been called the moment of inertia relative to an axis
through the iK)int T^^ perpendicular to the plane of the body, the kinetic energy is
■een to be the sum of ^ Mv'i, which would be the kinetic energy if all the mass were
concentrated at the center of gravity, and of i Iw^, which is the kinetic energy of
rotation about the center of gravity ; in case r^ indicated a point at rest (even if
only instantaneously as in §39) the whole kinetic energy would reduce to the
kinetic energy of rotation i lu^. In case r^ indicated neither the center of gravity
nor a point at rest, the thinl term in (2) would not vanish and the expression for
the kinetic energy would be more complicated owing to the presence of this term.
1
fiiMhtranif^T.* ...m
fS;
♦i*
If
i "•*
"al
\L « r
s
ON MULTIPLE INTEGRALS 819
131. To evaluate the double integral in ease the regiim is a rwUmmU
parallel to the axes of coordinateSf let the division be made into tmaU
rectangles by di-awing lines parallel to the
axes. Let there Ix^ vi e(iual divisions on one
side and n on the other. There will then be
7nn small pieces. It will be convenient to in-
tr(xlu(;e a double index and denote by A.-l^ the
area of the rectangle in the ith column andyth
row. Let (^y, i;^) be any point, say the mid-
dle point in the area A/1^ = Aj-iAy,-. Then the sum may be written
-I- i),jAjrjA»/a 4- D^^^ijt H -f- t^^A^m^Ht
+
4- A»Ax,Ay, + D^^Ax^y, + • • • + i^^Ao^.Ay..
Now the terms in the first row are the sum of the contribotioiit to
2,^ of the rectangles in the first row, and so on. But
(A^Aa-, 4- D,jAx, 4- • • 4- n^.Ax^)liyj = Ay,^ ^(^" nf)^i
i
and A//, 2^ Z>(^,, ry)AXi = \f 'd(x, ry)ibt + cl^y^.
That is to say, by taking m sufficiently large so that the individual
increments Aa*, are sufficiently small, the sum can be made to differ
from the integral by as little as desired because the integral is by
deiinition the limit of the sum. In fact
if c U> the maximum variation of D(Xj y) over one of the little reotanglet.
After thus summing up according to rows, sura up the rows. Then
X '^'J^'^^i = PH^^ Vi)dxAy, 4- f 'd(x, i7^c/j-Ay,
4- • • + r 'd(x^ v^ffr^V. 4- X,
|X| = |C,Ay^ 4- CAVa -^ • ■ • +C,AyJ ^ «(x - x^^Ay = c(x - xjijf - yj
If r'7>(x,y)rfx = ^(y),
then X ^^if^'h = <l>(Vx)^!li + <k(v^^!lt + • • 4- ♦(if.) Ay, + X
4»{y)dy 4- « 4- X, «. X small.
320
INTEGRAL CALCULUS
(3)
Henoe ♦ limX D^A^^^ = / DdA = I f D(x, y)dxdy.
It is seen that the double integral is equal to the result obtained by
first integrating with respect to x^ regarding y as a parameter, and then,
after substituting the limits, integrating with respect to y. If the sum-
mation had been first according to columns and second according to
rows, then by symmetry
DdA= I f D(x, y) dxdy = f I D(xyy) dydx. (3')
This is really nothing but an integration under the
sign (§ 120).
If the region over which the summation is extended
is not a rectangle parallel to the axes, the method
could still be applied. But after summing or rather
integrating according to rows, the limits would not
be constants as x^ and ic^, but would be those func-
tions x = ^o(y) and x = 4>^{y) of y which represent the left-hand and
right-hand curves which bound the region. Thus
D (Xj y) dxdy.
(3")
dx XiX
(3'")
And if the summation or integration had been first
with respect to columns, the limits would not have
been the constants y^ and y^, but the functions
y = ^Jix) and y = ^^(x) which represent the lower
and upper bounding curves of the region. Thus
DdA=: f I D{x,y)dydx.
The order of the integrations cannot be inverted without making the
corresponding changes in the limits, the first set of limits being such
functions (of the variable with regard to which the second integration is
to be performed) as to sum up according to strips reaching from one side
of the region to the other, and the second set of limits being constants
which determine the extreme limits of the second variable so as to sum
up all the strips. Although the results (3") and (3'") are equal, it fre-
quently happens that one of them is decidedly easier to evaluate than the
other. Moreover, it has clearly been assumed that a line parallel to the
• The rosult may also be obtained as in Ex. 8 below.
ON MULTIPLE INTEGRALS 821
axis of the first integration cuts the bounding curve in only two pointe •
if this condition is not fulfilled, the area must be divided into subaraM
for which it is fulfilled, and the results of integrating over these
areas must be added algebraically to find the complete value.
To apply these rules for evaluating a double Integral, wwMJ^tf the proliUm of
finding the moment of inertia of a rectangle of constant dendty with n&pti to
one vertex. Here
I = fl>r*dA = Df{x* + v»)dA = dJ" Y*(x« + |r*)di^
If the problem had been to find the moment of inertia of an elUpae of nnifonB
density with respect to the center, then
/ = D f(x^ ^y^)dA = Df'' f V^5?(«« + ^did^
/» + o /• + - Va' - *•
a
Either of these forms might be evaluated, but the moment of inertia of the whole
ellipse is clearly four times that of a quadrant, and hence the simpler remits
4/0 t/O
(x2 + y«)dxdy
Jq t/o 4
It is highly advisable to make use of symmetry, wherever poaible, to rednee tbt
region over which the integration is extended.
132. With regard to the more earful consideration af the UmKi inmbmd te tkt
dfifinition of a double integral a few observations will be sufficient. Cooiider tbt
sums .S and s and let 3f,A.4,- l)e any term of the first and ffiiLAt the t^rrmm^iA^y^
term of the second. Suppose the area A^l,- divided into two parts LAu and LAu^
and let 3/i,, Ma be the maxima in the parts and mu^ n^i the minima. Then dnoa
the maxinmin in the whole area A.-l, cannot be lew than that in either part^ and
the minimum in the whole cannot be greater than that in either part. It followi
that mu ^ rm^ ma ^ m^, Mu ^ Miy Mu ^ Mi^ and
viiAAi ^ tnnAAu + m^iAAti^ MnAAu -f Mti^Att ^ Jdt^Ao
Hence when one of the pieces A^, is subdivided the sum S cannot laemae nor too
sum 5 decrease. Then continued inequalities may be written ■•
mA ^ ^nuAAi ^ ]^D(f,, m)AAt ^^MiAAi S MA.
If then the original divisions AAi be subdivided indefinitely, both 5 and a wUl
approach limits (§§ 21-22) ; and if those limits art* the Mune, themn ZDt^At wVA
ai)pr();ich that connnon limit as its limit independently of how the polnu (|<, %)
are chosen in the areas A/1,.
322 INTEGRAL CALCULUS
It has not been shown, however, that the limits of S and s are independent of
the method of division and subdivision of the whole area. Consider therefore not
only the sums S and a due to some particular mode of subdivision, but consider all
such sums due to all possible modes of subdivision. As the sums S are limited
below by mA they must have a lower frontier X, and as the sums s are limited
above by MA they must have an upper frontier I. It must be shown that I ^ L.
To see this consider any pair of sums S and s corresponding to one division and
any other pair of sums S' and s" corresponding to another method of division ; also
the sums S" and «" corresponding to the division obtained by combining, that is,
by superposing the two methods. Now
It therefore is seen that any 5 is greater than any s, whether these sums correspond
to the same or to different methods of subdivision. Now if X < i, some B> would
have to be less than some s ; for as X is the frontier for the sums 5, there must be
some such sums which differ by as little as desired from X ; and in like manner
there must be some sums s which differ by as little as desired from I. Hence as no
S can be less than any s, the supposition X < Hs untrue and L^l.
Now if for any method of division the limit of the difference
Um {S-s) = lim V {Mi - rm) AAi = lim V O.A^f = 0
of the two sums corresponding to that method is zero, the frontiers X and I must be
the same and both S and s approach that common value as their limit ; and if the
difference S — s approaches zero for every method of division, the sums S and
8 will approach the same limit X = I for all methods of division, and the sum
ZDiAAi will approach that limit independently of the method of division as well
as independently of the selection of (f,-, in). This result follows from the fact that
L — l^S — 8, S — L ^ S —5, I— s^S — s, and hence if the limit of S — s is
zero, then X = I and S and s must approach the limit L = I. One case, which
covers those arising in practice, in which these results are true is that in which
D(x, y) is continuous over the area A except perhaps upon a finite number of
curves, each of which may be inclosed in a strip of area as small as desired and
upon which I) (x, y) remains finite though it be discontinuous. For let the curves
over which X> (z, y) is discontinuous be inclosed in strips of total area a. The con-
tribution of these areas to the difference S— s cannot exceed {M— m)a. Apart
from these areas, the function X)(x, y) is continuous, and it is possible to take the
divisions AAi so small that the oscillation of the function over any one of them
is leas than an assigned number e. Hence the contribution to S — s is less than
9 {A — a) for the remaining undeleted regions. The total value of S — s is there-
fore leas than {M — m)a ■\- e{A — a) and can certainly be made as small as desired.
The proof of the existence and uniqueness of the limit of 2D,A^,- is therefore
obtained in case I) is continuous over the region A except for points along a finite
number of curves where it may be discontinuous provided it remains finite.
Throughout the discussion the term " area " has been applied ; this is justified by the
previous work (§ 128). Instead of dividing the area A into elements A^, one may
rule the area with lines parallel to the axes, as done in § 128, and consider the sums
XMAzAv, ZmAxAy, 2/>A/Ay, where the first sum is extended over all the rectan-
l|[l«s which He within or upon the curve, where the second sum is extended over
all the rectanglen within the curve, and where the last extends over all rectangles
ON MULTIPLE INTEGRALS 8SS
within tlie curve and over an arbitrary number of tboie opon It. lo a errlAla
senHO thi8 method in simpler, in that the area then falls out M the iatif»l of iIm
special function which reduces to 1 within Uie curve and to 0 outride tin
and to eitlier upon the curve. The reader who deaires to follow ihUt mrtlxMl i
may do ko for himself. It is not within the range of thia book lo do noie fa tiM
way of rijgorous analysis than to treat the simpler questioni and to indkate I
need of corre8ix)n(iin^ treatment for other questiona.
The justitication for the metho<l of evaluatiuf^ a definite double intigial aa gii
above offers some diiticulties in case the function D(x, y) la dlaeoatlaooaa.
proof of the rule may be obtained by a careful conalderatlon of the intcfiatlon of
a function defined by an integral containing a parameter. Consider
It was seen (§ 118) that 4t{v) is a continuous function of y If D{x, y) la a con-
tinuous function of (x, y). Suppose that 2)(x, y) were diacontinuouai bi
finite, on a finite number of curves each of which la cut by a line parallel to
X-axis in only a finite number of points. Form A^ as before. Cut out the
intervals in which discontinuities may occur. As the number of such Inteirala la
finite and as each can be taken a^ short as desired, their total contribution to ^(y)
or 0(1/ + A^) can be made as small as desired. For the remaining portions of the
interval x^ ^ x ^ x^ the previous reasoning applies. Hence the diflerenee 64 caa
still be ma<le as small as desired and 0 {y) is continuous. If D(x, y) be dlsoootlaoooa
along a line y = p parallel to the x-axis, then 0 (y) might not be defined and adgbt
have a discontinuity for the value y = fi. But there can be only a finite naa»>
ber of such values if I>(x, y) satisfies the conditions imposed upon it in r»imld«l1nt
the double integral above. Hence 0 (y) would still be Int^rable from y^ to yj.
f'f ''D (X, y) dxdy exists
and
m(x, - XoXi/i - Vo)^ r "' r''^(-c, y)dxdy ^ 3f(x, -«,)(y, - y^
under tlie conditions imposed for the double integral.
Now let the rectangle x^ ^ x ^ x,, y^, ^ y ^ y, be divided up aa befon. Than
mo Ax. Ayy ^ j '' J I> (x, y) dxdy ^ Jf(/ A^xA^.
and 2; ly^'C^^^^' y^^V =X''C"<'' »>'^-
Now if the number of divisions is multiplied indefinitely, the limit to
J"' J'*D(x, y)djdy = Wm^ma^Aff = Um J) J»^</A/«a =/^('' V)^-
Thus the previous nde for the rectangle is proved with proper allowance for poa>
sible discontinuities. In case the area A did not form a rectangle, a rectaagla
could be described about it and the funcUon /)(x, y) could be daflnad for the
whole recungle as follows : For points within A the value of D(«, y) to already
824 INTEGRAL CALCULUS
defined, for points of the rectangle outside of A take D(x, y) = 0. The discon-
tinuities across the boundary of A which are thus introduced are of the sort
allowable for either integral in (4), and the integration when applied to the rec-
tangle would then clearly give merely the integral over A. The limits could then
be adjusted so that
f' f'Dix, y)dxdy = f' f^'^^^Cx, y)dxdy = Cl){x, y)dA.
The rule for evaluating the double integral by repeated integration is therefore
proved.
EXERCISES
1. The sura of the moments of inertia of a plane lamina about two perpendicular
lines in its plane is equal to the moment of inertia about an axis perpendicular to
the plane and passing through their point of intersection.
2. The moment of inertia of a plane lamina about any point is equal to the sum
of the moment of inertia about the center of gravity and the product of the total
mass by the square of the distance of the point from the center of gravity.
3. If upon every line issuing from a point 0 of a lamina there is laid off a dis-
tance OP such that OP is inversely proportional to the square root of the moment of
inertia of the lamina about the line OP, the locus of P is an ellipse with center at 0.
4. Find the moments of inertia of these uniform laminas :
(a) segment of a circle about the center of the circle,
(/S) rectangle about the center and about either side,
(7) parabolic segment bounded by the latus rectum about the vertex or diameter,
( 8 ) right triangle about the right-angled vertex and about the hypotenuse.
5. Find by double integration the following areas :
{a) quadrantal segment of the ellipse, (j3) between y^ = x* and y = x,
(7) between 3^/2 = 26 X and 5x2 = 9?/,
( J ) between x^ -\- y^ — 2x = 0, x^ -f- y^ _ 2 y = 0,
( e ) between 2/^ = 4 ax -f- 4 a^, y2 _ _ 4 53. ^ 4 52^
(f) within (2/ _ X - 2)2 = 4 - x2,
(17) between x2 = 4 ay, y{x^ + i a^) = 8 a^,
{$) 2/2 = ax, x2 -H 2/2 _ 2 ox = 0.
6. Find the center of gravity of the areas in Ex. 6 (a), (/3), (7), (5), and
(a) quadrant of a*2/* = a2x* - x«, (fi) quadrant of xf -f yt = a^,
(7) between xi = 2/^ -}- ai, x -|- 2/ = a, (5) segment of a circle.
7. Find the volumes under the surfaces and over the areas given :
(a) sphere z = Va^ — x^ — y^ and square inscribed in x^ -\- y^ = a*,
ifi) sphere z = Vg^ - x^ -~y^ and circle x^ -\- y^ - ax = 0,
(7) cylinder z = Via^-y^ and circle x^ + 2/* - 2 ox = 0,
(a) paraboloid z = kxy and rectangle O^x^a, 0^2/ = &,
( « ) paraboloid z — kxy and circle x2 + 2/2 — 2 ox — 2 aj/ = 0,
it) plane x/a ■\- y/b -|- 2/c = 1 and triangle xy (x/a + y/b - 1) = 0,
(i») paralK)lold « = 1 - x2/4 - 2/2/9 above the plane z = 0,
(0) paraboloid « = (x -|- y)« and circle x« 4- 2/2 = a«.
ON MULTIPLE INTEGRALS 825
8. Instead of chooeing ((f, ry) as particular pointa, namely the f*^A4h tif4irta of
the rectangleK and evaluating ZI>((^ ry) Ax^Aj/> subject to erron X, « whieli vmiUi te
the limit, EKHuine the function D(2, t^) continuoua and rawlve the do«ihle InlMiil
into a double huui by repeated uae of the Theorem of the Mean, aa
^^^^ =X''^^^' ^^^ =X ^<^'' ^^^' ^« property cboeen,
f\(v)dy=^i>in,)Ayj = ^\'^D{i{, n/)Aac,]Ajfy = yXl(6. %)A^^.
9. Consider the generalization of Osgood^a Theorem (|86) to apply to doubU
integrals and sums, namely : If a^ are inflniteaimala such that
where f^f is uniformly an infinitesimal, then
lim 2 irij = Jd(x, y)dA =/'*/''-D(«, y)did^^
Discuss the statement and the result in detail in view of f 84.
10. Mark the region of the xi^-plane over which the integration extendi:*
<''>/«7o'^'"^' w/'X"-^"'^' <^>/.X'"^'
11. The density of a rectangle varies as the square of the distance from ont
vertex. Find the moment of inertia about that vertex, and about a dde through
the vertex.
12. Find the mass and center of gravity in Ex. 11.
13. Show that the moments of momentum (§80) of a lamina about the oxtgjbi
and about the point at the extremity of the vector r,, satisfy
frxydm = r^x fydm + fr'xTdm,
or the difference between the moments of momentum about P and Q b the moment
about P of the total momentum considered as applied at Q.
14. Show that the formulas (1) for the center of gravity redace to
fxyDdx J"iyyZ>dx ^'*x(y,- y^Ite
X = -^ , y = -^ or « = -^2 (
f^yDdx fyDdx f'^wi^n)!^
piVx-Vo)
Ddx
* Exercises involving polar coordinates may be postponed until § 194 Is
the student is already somewhat familiar with the subject.
326 INTEGRAL CALCULUS
when D(x, y) reduces to a function D(x), it being understood that for the firsi
two the area is bounded by x = 0, x = a, y =/(x), y = 0, and for the second two
by X = Xoi * = ^1' y\ = A (^)' ^0 = /o(^)-
15. A rectangular hole is cut through a sphere, the axis of the hole being a
diameter of the sphere. Find the volume cut out. Discuss the problem by double
Integration and also as a solid with parallel bases.
16. Show that the moment of momentum of a plane lamina about a fixed point
or about the instantaneous center is Iw, where w is the angular velocity and I the
moment of inertia. Is this true for the center of gravity (not necessarily fixed) ?
Is ii true for other points of the lamina ?
17. Invert the order of integration in Ex. 10 and in / | Ddydx.
18. In these integrals cut down the region over which the integral must be
extended to the smallest possible by using symmetry, and evaluate if possible :
(a) the integral of Ex. 17 with D = y^ — 2x^y,
(^) the integral of Ex. 17 with D~(x-2 Vsfy or X> = (x - 2 Vs) y^,
(7) the integral of Ex. 10(e) with D = r (1 + cos 0) or D = sin ^ cos <f>.
19. The curve y =/(x) between x = a and x = 6 is constantly increasing.
Express the volume obtained by revolving the curve about the x-axis aK
TT [/(a)]*(^ — a) plus a double integral, in rectangular and in polar coordinates.
20. Express the area of the cardioid r = a (1 — cos </>) by means of double inte-
gration in rectangular coordinates with the limits for both orders of integration.
133. Triple integrals and change of variable. In the extension from
double to triple and higher integrals there is little to cause difficulty.
For the discussion of the triple integral the same foundation of mass
and density may be made fundamental. If JD(x, y, z) is the density of
a body at any point, the mass of a small volume of the body surround-
ing the point (^„ i;,-, ^,) will be approximately i>(^f, rji, ^t) AF,-, and will
surely lie between the limits il/.AF,- and m,AF,-, where Mi and m,- are
the maximum and minimum values of the density in the element of
volume AF,. The total mass of the body would be taken as
lim y /)(!,, ^,, C,) AF, = fD(x, y, z)dV, (6)
where the sum is extended over the whole body. That the limit of the
sum exists and is independent of the method of choice of the points
iiii Vti (i) *"^ o^ ^^® method of division of the total volume into elements
AF<, provided D(x, y, z) is continuous and the elements AF,- approach
zero in such a manner that they become small in every direction, is
tolerably apparent
ON MULTIPLE LSTB0BAL8
827
The evaluation of the triple integral by repeated or iterated tiitegm>
tion is the iiiunediate generalization of the method uaed for the doable
integral. If the region over whieh the integration takes place b a reiv
tangular parallelepiped with its edges parallel to the axes, tbe inlefftal it
JD(x,i/,z)dV=jjj D(x, V, r w/j-./v*it. (5*)
The integration with resjMict to x adds up the .1 :i
the eolunin upon the biise di/dZf the integration v. .:. n
adds these columns together into a lamina of thi< :
integration with respeet to ;; finally adds
together the laininas and obtains the mass
in the entire parallelepiped. This could
he done in other orders ; in faet the inte-
gration might he performed lirst with re-
gard to any of the three variables, second
with either of the others, and finally with
the liust. There are, therefore, six e(juiva-
lent methods of integration.
If the region over whieh the integration
is desired is not a rectangular parallele-
pi})ed, the only modification which must be introduced is to adjust the
limits in the successive integrations so as to cover the entire region.
Thus if the first integration is with respect to x and the region is
lx)unded by a sm*face x = xj;^ (//, z) on the side nearer the yar-plana and
by a surface x = ^^ (y, z) on the remoter side, the integration
X.
x^^^,^(M,z)
^(^7 Vy z)dxdydz = n(y, K)dydM
1^0 <"• «>
will add up the ma&s in elements of the column which has the
section dydz and is intercepted between the two surfaces. The problem
of adding up the columns is merely one in double integration over the
region of the yz-\A\x\\e upon which they stand ; this region is the pro-
jection of the given volume upon the y«-plane. Tbe value of the
integral is then
//»«» /»»-*,(«) /•«! /•♦!<•> /•♦!<«.»)
A/F= I / nrfyrf«= I / / Ddjcdydx. (5")
Here again the intt»grations may be performed in any order, prorided
the limits of the integrals are carefully a<ljusted to correspond to that
order. The method may best be learned by eiample.
828
INTEGRAL CALCULUS
Find the mass, center of gravity, and moment of inertia about the axes of the
volume of the cylinder x'^ + y^ — 2ax = 0 which lies in the first octant and under
paraboloid z* + y* = az, if the density be assumed constant. The integrals to eval-
uate are : r r r
j xdm j ydm I zdm
m
I. =» /i>(y* + 2^)dF, ly = D J(x2 + z2)dF, h = 2)/(x2 + y^)dV,
The consideration of how the figure looks shows that the limits for z are z = 0 and
« = (x» + y*)/a if the first integration be with respect to z ; then the double integral
in X and y has to be evaluated over a semi-
circle, and the first integration is more simple
if made with respect to y with limits y = 0
and y = V2 ax — x=^, and final limits x = 0
and x = 2aforx. If the attempt were made
to integrate first with respect to y, there
would be difficulty because a line parallel to
ihe y-axis will give different limits according
as it cuts both the paraboloid and cylinder or
the xz-plane and cylinder ; the total integral
would be the sum of two integrals. There
would be a similar difficulty with respect
to an initial integration by x. The order of
Integration should therefore be z, y, x.
ix:=2a
,y/zax-3? /.(a-!-
m
I I dzdydx = D /
= ^ J'^Ls V2 OX - x2 + -(2ax - x'^)(\dx
= Da?r\{\- cos ef sin2 tf + i sin* ^1
■V'2ax-
= 0
x2 +
dydx
a{l
ta /.Vaox-;
)2sin2tf + |sin*^|r?<?
ira^D
\ V2 OiC - x2 =
[ dx = a sin 6d6
cosO)
a sin 6
dydx
Jr'ia /t^aox — X- ^(x' + y')/a p2a /^"Wiax — x^x^ + XV^
I I xdzdydx = -D I / —
= ^ r'^^Fx' V2 OX - x2 + -x(2aa; - x^)(]dx = ira^D.
Hence x = 4 a/8. The computation of the other integrals may be left as an exercise.
134. Sometimes the region over which a multiple integral is to be
evaluated is such that the evaluation is relatively simple in one kind
of cottrdinates but entirely impracticable in another kind. In addition
to the rectangular coordinates the most useful systems are polar coor-
dinates in the plane (for double integrals) and polar and cylindrical
coordinates in space (for triple integrals). It has been seen (§ 40) that
the element of area or of volume in these cases is
dA = rdrd4>, rf K = r* sin BdrdOd^i, dV =^ rdrd<t>dz, (7)
ON MULTIPLE INTEGRALS
$29
> xcept for infiiiitesiinals of higher order. These qnantitiet may be
iil)stituted in the double or triple integral and the evaluation maj be
made by successive integration. The proof that the substitution eaa
Ih) made is entirely similar to that given in §f 34-35. The proof tluU
the inU>giul may still he evaluated by soooessive integration, with a
pro^x^r choice of the limits so as to cover the region, is cooiained in
the statement that the formal work of evaluating a multiple integial
))y rei)catcd integmtion is independent of what the coOrdinatee aotiiallj
represent, for the reason that they could be interpreUnl if desired as
representing rectangular coordinates.
Find the area of the part of one loop of the lemniscste r* = Sa*eost^ whldl k
exterior to the circle r = a; also the center of gravity and the monmit of Ineftia raU-
tivti tu tlie origin under the asHuniption of consUnt densitj. Here ths intsgnds ai«
Az^CdA, Ax=fxdA, Ay^CydA, I^dCt^A, m= DA,
The integrations may be performed first witli respect
to r so as to add up the elements in the little radial
sectors, and then with regard to ^ so as to add the
secttus ; or first with regard to 0 so as to combine the
eleinenU* of the little circular strips, and then with re-
gard to r so as to add up the strips. Thus
(a^H'l
(Vla,o)
A = 2f* C ^~'*rdni0= r*(2a«cos20-a«)d0 = Q>/8-^Wr^ ...«»a-,
Ax
aVaeoa'^
rcos
ip.rdrd4^ = - r*(2 V2<i«co«l 2#- o»)oos#d^
_2 , rI[2V2(l-2 8inV)^d8in0-coB^d^] = ^a« = .888a*.
Hence x = 3 7ra/(l2 Vs - 4 ir) = 1.15 a. The sym-
metry of the figure shows that y = 0. The calcula^
tion of / may be left as an exercise.
Given a sphere of which the density varies as the
distance from some point of the surface ; required the
ma«8 and the center of gravity. If j>olar coordinates
Willi the origin at the given point and the polar axis
along the diameter through that point be EMumed,
the equation of the sphere reduces to r = 2acottf
where a is the radius. The center of gravity from
reasons of symmetry will fall on the diameter. To
cover the volume of the sphere r must vary from r = 0
at the origin to r = 2acoHtf ujwn the sphere. The
polar angle must range from ^ = 0 to tf = J r, and the
longitudinal angle from 0 = 0 to 0 = 2 v. Then
330 INTEGRAL CALCULUS
m= r" n r"°"kr.,^Bmedrdedi.,
I r = 2 a COS 0
»w= f f' r At. r COS «9.r2 Sin <9drd(9d0,
m= f^' f^ ^ ka* C08* 0 Bin 6ded4> =: f -ka*d</> =
m2= r" r^ ??J^co8^e sin ed0d<p= f
J^^oJ0=o 6 "^0
35 35
The center of gravity is therefore z = 8 a/7.
Sometimes it is necessary to make a change of variable
or x = 4> («, V, w), y = il;(u, V, w), z = <o (u, V, w) (8)
in a double or a triple integral. The element of area or of volume has
been seen to be (§ 63, and Ex. 7, p. 135)
dAJ\ji^^^dudv or dV=\ji'^'^^\dudvdw. (8')
Hence f^C^, ?/)^^ = f^C*^, '/')|«^(^)|^^^^ (8";
and Cd{x, y, z)dV= C D{^, ,/,, a,)|/(^^)|^i.tZi;(;i.;.
It should be noted that the Jacobian may be either positive or negativ(
but should not vanish ; the difference between the case of positive and
the case of negative values is of the same nature as the difference
between an area or volume and the reflection of the area or volume.
As the elements of area or volume are considered as positive when
the increments of the variables are positive, the absolute value of the
Jacobian is taken.
EXERCISES
1. Show that (6) are the formulas for the center of gravity of a solid body.
2. Show that Jar =f{y^-\- z^) dm, ly =f{x^ + z^) dm, h = f{x^ + y^)dm are the
ftjnnulas for the moment of inertia of a solid about the axes.
3. Prove that the difference between the moments of inertia of a solid about
any line and about a parallel line through the center of gravity is the product of the
maM of the body by the square of the perpendicular distance between the lines.
4. Find the moment of inertia of a body about a line through the origin in the
direction determined by the cosines I, m, n, and show that if a distance OP be laid
off along thlH line Inversely proportional to the square root of the moment of
inertia, the Iocub of P is an ellipsoid with O as center.
ON MULTIPLE INTEGRALS SSI
5. Find the momenU of Inertia of tbete aolkU of uniform dmuiity t
(a) rectangular parallelepiped abe, about the edge a,
(/3) elliprnjid xVa* + vV^ + t*/c* = 1, about the t-«Ki%
(7) circular cylinder, about a perpendicular blaecior of Ita azU«
( S ) wedge cut from the cylinder x«+y* = r*by«=± mx, about ita edfe.
6. Find the volume of the solida of Ex. 6 (fi), (<), and of the :
(a) tri rectangular tetrahedron between xyt = 0 and x/a + y/b + t/e = 1,
(/3) Koliil bounded by the surfaces y* + «• = 4ar, y* = oj, x = So,
(7) solid common to the two equal perpendicular cylinderax'-fy's a', x*-f «*sa*
(a) octant of g + (?j + y = 1, (.) ocunt of (?) + (?) + (?)•= ,.
7. Find the center of gravity In Ex. 6 (a), Ex. 6 (a), (/J), («), (•), denidty uniform.
8. Find the area in these cases : (a) between r = a sin S^ and r s 1 a.
(p) between r* = 2 a> cos 2 0 and r = si a, (7) between r = a dn ^ and r = 6 o« #.
(«) r» = 20^0082^, r 0080 = i Via, (e) r = a(l + coe^), r = a.
9. Find the moments of inertia about the pole for the caaes in Ex« 8, demdty
uniform.
10. Assuming uniform density, find the center of gravity of the area of one loop :
(a) r» = 2a2cos2 0, (/9) r = a(l - co8 0), (7)r = a8ln20,
(a) r = a sin' I <(> (small loop), (e) circular sector of angle 9 a.
11. Find the moments of inertia of the areas in Ex. 10 (a), (/}), (y) aboai the
initial line.
12. If the density of a sphere decreases uniformly from D% at the center to D,
at the surface, find the mass and the moment of Inertia about a diameter.
13. Find the total volume of :
(or) (x2 + y2 + 22)2 = axyz, (^ (x« + y« + xV = 27a«XKt.
14. A spherical sector is bounded by a cone of revolution ; find the eMiar of
gravity and the moment of inertia about the axis of revolution if the dOMUf
varies as the nth power of the distance from the center.
15. If a cylinder of liquid rotates about the axla, the shape of the surface Is a
paraboloid of revolution. Find the kinetic energy.
16. Compute//^, j/^^JLlV / /?!J^\ and hence verify (7).
17. Sketch the region of integration and the curves 11 = const., • = const. ;
hence show:
(a) fT' 'f{x,y)dxdy= C C /(u - wr, ue)iMhidr, if « = y + x, r = ■*,
Jo X-o -^Vl + u l + u/(l + ii)« !•••«
(^) or = rr' f^—Audv^ r-fl^'f^-^^
^'^ Jo J«-0 (1 + M)* ''• •'— i (l + «)'
332 INTEGRAL CALCULUS
18. Find the volume of the cylinder r = 2acos<f> between the cone z = r and
the plane z = 0.
19. Same as Ex.18 for cylinder r* = 2a*co8 20; and find the moment of
inertia about r = 0 if the density variea as the distance from r = 0.
20. Assuming the law of the inverse square of the distance, show that the
attraction of a homogeneous sphere at a point outside the sphere is as though all
the maas were concentrated at the center.
21. Find the attraction of a right circular cone for a particle at the vertex.
22. Find the attraction of (a) a solid cylinder, (p) a cylindrical shell upon a
point on its axis ; assume homogeneity.
23. Find the potentials, along the axes only, in Ex. 22. The potential may be
defined as ^r-^dm or as the integral of the force.
24. Obtain the formulas for the center of gravity of a sectorial area as
x = — I -r»cos0d0, y = -7 I -r^sia,pd<f,,
and explain how they could be derived from the fact that the center of gravity of
a uniform triangle is at the intersection of the medians.
25. Find the total illumination upon a circle of radius a, owing to a light at a
distance h above the center. The illumination varies inversely as the square of the
distance and directly as the cosine of the angle between the ray and the normal
to the surface.
26. Write the limits for the examples worked in §§ 133 and 134 when the inte-
grations are performed in various other orders.
27. A theorem of Pappus. If a closed plane curve be revolved about an axis
which does not cut it, the volume generated is equal to the product of the area of
the curve by the distance traversed by the center of gravity of the area. Prove
either analytically or by infinitesimal analysis. Apply to various figures in which
two of the three quantities, volume, area, position of center of gravity, are known,
to find the third. Compare Ex. 3, p. 346.
135. Average values and higher integrals. The value of some special
interpretation of integrals and other mathematical entities lies in the
ooncreteness and suggestiveness which would be lacking in a purely
analytical handling of the subject. For the simple integral f f(x)dx
the curve y =/(a-) was plotted and the integral was interpreted as
an area; it would have been possible to remain in one dimension by
interpreting /(x) as the density of a rod and the integral as the mass.
In the oaae of the double integral jf(x, y)dA the conception of den-
sity and mass of a lamina was made fundamental ; as was pointed out,
it is possible to go into three dimensions and plot the surface z =/(«, y)
ON MULTIPLE INTEGRALS SS8
and interpret the integral as a yolume. In the traatment of the tripk
integral j f(x, y, x)dV the density and mass of a body in apaoe wera
made fundamental ; here it would not be possible to plot « ^/(r, y, c)
aa there are only three dimensions available for plotting.
Another important interpretation of an integral is found m in.- tx»n-
ception of average value. If q^y ?,»••• i 9. are n numbers, the aYertge of
the numbers is the quotient of their sum by n.
g=?-+y'+--+y-=Sb. (9)
n n ^ '
If a set of numbers is formed of w^ numbers q^, and w^ numbers
(/,,, • ■ -, and w^ numbers q^j so that the total number of the numbers
is w^ 4- w'jj -f • • • -|- n\y the average is
w'l 4- u\i H h u\ ^Wt ^ '
The coefficients w^, w'^j- -j^^wj or any set oi numbers which are pro-
portional to them, are called the weights of y^, y^, •••, y,. These defi-
nitions of average will not apply to finding the average of an infinite
numl)er of numbers because the denominator n would not be an arith-
metical number. Hence it would not be possible to apply the definition
to finding the average of a function f{x) in an interval x^'^x'^ x^.
A slight change in the point of view will, however, lead to a defi-
nition for the average value of a function. Suppose that the interval
a-Q ^ a- ^ a-j is divided into a number of intervals Ajr„ and that it be
imagined that the number of values of y = f{x) in the interval Ax,
is proportional to the length of the interval. Then the quantities
Aa-,- would be taken as the weights of the values /(^,) and the average
would be ^x,
/ /(')<*»
.^2^(fO, „,better y ^-^ (10)
i. "^
by passing to the limit as the Aa*/8 approach zero. Then
r 'f(x)dx
y-=%— — - or rV(x)r/a:=(x.-x^y.
^1 — ^0 Jx.
(lO*)
111 like manner if z =/(«, y) be a function of two variables or
u =f{'r, //, z) a function of three variables, the averages over an
334 INTEGRAL CALCULUS
or volume would be defined by the integrals
Cf{x,y)dA ff(x,y,z)dV
z = ^ and y^^"^—^ (10")
CdA = A jdV = V
It should be particularly noticed that the value of the average is de-
fined with reference to the variables of which the function averaged is a
function ; a change of variable will in general bring about a change in
the value of the average. For
if y =/(^), W) = - — - f V(^) d^ ;
but if y= f(<f>(t)), W) = 7-^- f /(<^(0) dt ;
and there is no reason for assuming that these very different expres-
sions have the same numerical value. Thus let
y = x% O^ic^l, x = smt, O^I^^tt,
1 r^ 1 - 1 r^ 1
y{x) = -\ xHx = -, y(t)=—j sinHdt = -'
The average values of x and y over a plane area are
x = -jjxdA, y = jjydA,
when the weights are taken proportional to the elements of area ; but
if the area be occupied by a lamina and the weights be assigned as
proportional to the elements of mass, then
= — I xdm, y = — I ydm.
and the average values of x and y are the coordinates of the center of
gravity. These two averages cannot be expected to be equal unless the
density is constant. The first would be called an area-average of x and
y; the second, a mass-average of x and y. The mass average of the
square of the distance from a point to the different points of a lamina
would be 1 /»
^=k' = — iMm=I/M, (11)
and is defined as the radius of gyration of the lamina about that point ;
it is the quotient of the moment of inertia by the mass.
ON MULTIPLE INTEGRALS M6
Ar a problem in averages consider the determination of the aversfe tsIos of a
proper fraction ; aim the average value of a proper fraction subject lo ths f*rtH-
tion that it be one of two proper fractions of which the sum shall be Isai •^-' ^
e(iual to 1. Let x be the proper fraction. Then in the first case
1 •/o S
III the second case let y be the other fraction so that x + y 2 !• Now if (x, y) bs
taken an coordinates in a plane, the range is over a trian^e, the nnmbsr o(f potets
(X, y) in the element dxdy would naturally be taken as proportional to the aiwi of
the cltiiunt. :ind the average of x over the r^on would be
Now if X were one of four proper fractions whose sum was not greater than 1, the
problem would be to average x over all sets of values (x, y, s, «) subject to the
relation x + 2/ + z + u^l. From the analogy with the above problems, the mult
would be
__.. ZxAxAyAzAu _ Ju=:o Jx^o Jw^o «/x-o ^ '^^"*
""- '"^ ZAxAyAzAu - r^ f'" V'-"- r""— 'dxrfydafci *
t/MzO vzsO t/ir—O t/xaiO
The evaluation of the quadruple integral gives x = 1/6.
136. The foregoing problem and other problems which may arise
lead to the consideration of integrals of greater multiplicity than three.
It will be sufl&cient to mention the case of a quadruple integral. In the
first place let the four variables be
x^^x^x^, y^^y^y^, z^^z^z^, k^^mSm^, (12)
included in intervals with constant limits. This is analogous to the
case of a rectangle or rectangular parallelepiped for double or triple
integrals. The range of values of a*, y, «, u in (12) may be spoken of
us a rectangular volume in four dimensions, if it be desired to use geo-
metrical as well as analytical analogy. Then the product A2:«Ay^^Wi
would l3e an element of the region. If
a-,. ^ $i ^ Xi 4- Aar<, • • -, t/, ^ 0^ ^ m< -f Am,,
the point (^,., 77,, d, $,) would be said to lie in the element of the fegkm.
The formation of a quadruple sum
could l)e carried out in a manner similar to that of double and triple
sums, and the sum could readily be shown to have a limit when
336 INTEGRAL CALCULUS
Aa-,-, Ay,, A«f, Am, approach zero, provided / is continuous. The limit of
this sum could be evaluated by iterated integration
p'l rvi /•*, >^«i
limV/AiTiAy.A^.Aw, = I / / / f{x,y,^iU)dudzdydx
*Jx^ *Jvo *^'o ^\
where the order of the integrations is immaterial.
It is possible to define regions other than by means of inequalities
such as arose above. Consider
F(Xy y, z, u)=0 and F(xj y, «, u) ^ 0,
where it may be assumed that when three of the four variables are
given the solution of F = 0 gives not more than two values for the
fourth. The values of x, y, «, u which make F <0 are separated from
those which make F > 0 by the values which make F = 0. If the sign
of F is so chosen that large values of x, y, z, u make F positive, the
values which give F > 0 will be said to be outside the region and those
which give F < 0 will be said to be inside the region. The value of the
integral of /(x, y, z, u) over the region F^ 0 could be found as
I / / / f{x,y,z,u)dudzdydx,
where u = <o^(x, y, z) and u = a>^(a;, y, z) are the two solutions of F = 0
for u in terms of x, y, «, and where the triple integral remaining after
the first integration must be evaluated over the range of all possible
values for (x, y, «). By first solving for one of the other variables, the
integrations could be arranged in another order with properly changed
limits.
If a change of variable is effected such as
« = 0(2^,1^, «',«'), y = i^{x\v',z\u'), z = x{x\y',z%u'), u = uix^y^z'.u') (13)
the integrals In the new and old variables are related by
fffffi'^^ y, 2, y) dxdydzdu =ffff /{<(>, f , X, «) U (J' ^: ^: I) \dxWdz'du\ (14)
The result may be accepted as a fact in view of its analogy with the results (8) for
the simpler cases. A proof, however, may be given which will serve equally well
as another way of establishing those results, — a way which does not depend on the
somewhat loose treatment of infinitesimals and may therefore be considered as
more satisfactory. In the first place note that from the relation (38) of p. 134
inyoWing Jacobians, and from its generalization to several variables, it appears
that If the change (14) is possible for each of two transformations, it is possible
for the succession of the two. Now for the simple transfonnation
« = «', v = v\ zz=iz\ u = w {x\ y% z\ uO = w (jc, y, z, u% (ISO
ON MULTIPLE INTECiRALS SS7
which involves only one variablef J — l^/Wy and
J/(x, V, z, u)d.u =//(x, y, «, uOJ — ldu' =//(x', iT, f*, ^ M*»'.
and each side may be integrated with respect to ac, y, f. Hence (14) to tme in thto
. iM . For the transformation
which involves only three variables, J ( f * *!* ^! ** J =j( ^ ^l'\ and
///•^^•'' ^' '^' «)^^i'^ =///-^^^' ^' X' ii)|/|dr'cly'df'.
and each side may be integrated with respect to u. The nilif therefore hoMe la
this case. It remains therefore merely to show that any transfomuUkm (IS) nej
be resolved into the succession of two such as (IS'), (W). Let
Xi = X', Vx = V\ 2i = «'» "i = « («', y't «',«') = •• (X,, y„ «,. mO-
Solve the equation Uj = «(Xj, y^ Zj, u') for m' = t#, (r^, y,, tj, m,) and write
x = 0(Xi, y^Zj, «,), y = ^ (Xi, yi, Zp «,), « = x (Xp Vi^ «i. -i), « = «,.
Now by virtue of the value of <•»,, this is of the type (18"), and the •abetiiuUoo of
^v Vv ^v ^i ^" *^ gives the original transformation.
EXERCISES
1. Determine the average values of these functions over the Intenrato:
(a) x^.O^x^ 10, (/3) sinx, 0 ^ X s J »,
(7) x", 0 ^ X ^ n, (3) coa^^r, 0 s x s J «•.
2. Determine the average values as indicated :
(a) onlinate in a semicircle x^ -^^ v^ = a*, y > 0, with z aa yariable,
(/3) onlinate in a semicircle, with the arc as variable,
(7) ordinate in semiellipse x = aco8 0, y = 68in0, with ^ aeTarUble,
(5) focal radius of ellipse, with equiangular spacing about foow,
(e ) focal radius of ellipse, with etiual spacing along the major azto,
(f) chord of a circle (with the most natural aasiimption).
3 . Find the average height of so much of these surfaces as Hee aboTe the cr-plaae :
(a) x2 + y* + z'» = a2, (fi) z = a*-p^z^-q^y (y) i« = 4 - *• - y«.
4. If a man's heijjht is the average height of a conical tent, on how mneh ol the
floor space can he stand erect ?
5. Obtain the average values of the following:
(a) distance of a point in a sqiiare from the center, (/J) ditto frwn Teitaz,
(7) distance of a point in a circle f n)m the center, (*) ditto for ephere,
( f ) distance of a point in a sphere from a fixed point on the surface.
6. From the S.W. corner of a township persons stjurt in rftodom directloM
between N. and E. to walk across the township. What to their arenge walk ?
Which has it ?
338 INTEGRAL CALCULUS
7. On each of the two legs of a right triangle a point is selected and the line
joining thera is drawn. Show that the average of the area of the square on this
line is \ the square on the hypotenuse of the triangle.
8. A line joins two points on opposite sides of a square of side a. What is the
ratio of the average square on the line to the given square ?
9. Find the average value of the sum of the squares of two proper fractions.
What are the results for three and for four fractions ?
10. If the sum of n proper fractions cannot exceed 1, show that the average
value of any one of the fractions is l/(n + 1).
11. The average value of the product of k proper fractions is 2-*.
12. Two points are selected at random within a circle. Find the ratio of the
average area of the circle described on the line joining them as diameter to the
area of the circle.
13. Show that J = r^ sin^ ff sin <f> for the transformation
X = r cos ^, y = r sin ^ cos <f>, z = rsinO sin 0 cos i/', u = r sin 6 sin 0 sin ^,
and prove that all values of x, y, z, u defined by x^ -\- y^ + z^ + u^ ^ a^ are covered
by the range O^r^a, O^^^tt, 0^0^ tt, 0^^^2ir. What range will
cover all positive values of x, y, z, u?
14. The sum of the squares of two proper fractions cannot exceed 1. Find the
average value of one of the fractions.
15. The same as Ex. 14 where three or four fractions are involved.
16. Note that the solution of u^ = w(Xj, y^, Zj, u') for m' = Wi(Xj, y^, z^, u^)
requires that du/du^ shall not vanish. Show that the hypothesis that J does not van-
ish in the region, is sufficient to show that at and in the neighborhood of each point
(x, y, z, u) there must be at least one of the 16 derivatives of 0, f , x, w by x, y, z, u
which does not vanish ; and thus complete the proof of the text that in case J ^0
and the 16 derivatives exist and are continuous the change of variable is as given.
17. The intensity of light varies inversely as the square of the distance. Find
the average intensity of illumination in a hemispherical dome lighted by a lamp
at the top.
18. If the data be as in Ex. 12, p. 881, find the average density.
137. Surfaces and surface integrals. Consider a surface which has
at each point a tangent plane that changes contin-
uously from point to point of the surface. Consider
also the projection of the surface upon a plane, say
the acy-plane, and assume that a line perpendicular
to the plane cuts the surface in only one point. y
Over any element dA of the projection there will /
be a small portion of the surface. If this small
portion were plane and if its normal made an angle y with the «-axis,
the area of the surface (p. 167) would be to its projection as 1 is to
Y
WdA
ON MULTIPLE INTEGRALS 8S9
cos y and would be sec yd A, The value of oo8 y may be i«ad from (9)
on ])age 9C>. This suggests that the quantity
../»,w. =//[..(£)•. (1)1'^ <";
be taken as the dejinition of the area of the wurfaee^ where the doobk
integral is extended over the projection of the surface; and this defi-
nition will be adopted. Tliis definition is really dependent on the
particular })lane upon which the surface is projected ; tliat the value ol
the area of the surface would turn out to be the same no matter what
))lane was used for i)rojection is tolerably apparent, but will be proved
later.
Let the area cut out of a hemisphere by a cylinder upon the radias of Um
hemisphere as diameter be evaluated. Here (or by geometry directly)
J L z^ z'^i Jx = o^» = o Va« - *« - ^
This integral may be evaluated directly, but it is better to tnuiafonn It to polar
co-ordinates in the plane. Then
.S = 2 f r'**"*-^^=nird0 = 2 r a«(l-8in^)d# = (»-«)a«.
It is clear that the half area which lies in the first octant could be projected opoa
the xr-plane aiul thus evaluated. The region over which the Integration would
extend is that between x^ + z* = «* and the pn)jection
z^ -^ ax = d^ of tlie curve of intersection of the sphere
and cylinder. The projection could also be made on the
yz-plane. If the area of the cylinder between r = 0 and
the sphere were desired, projection on 2 = 0 would be
useless, projection on x = 0 would be inv(>lve<l owing to
the overlapping of the projection on itself, but projection
on y = 0 would be entirely feasible.
To show that the definition of area does not depend,
except apparently, upon the plane of projection consider
any second plane which makes an angle 0 with the first. Let the line of
tion be the y-axis ; then from a figure the new coordinate x' Is
j' = xcostf + zsin^, 1/ = y, and ji-li^ = — = coa#+ — ihif,
(X, y) ar «f
~Jj coay~JJ ' (X', y) CO87 J J coBy{coB0 -^ pdUk0)
It remains to show that the denominator coe 7(006^ + pdn^ = coe7'. Referred
to the original axes the direction ooftinee of the normal are — p: — '/ ^ *»** "^
340 INTEGKAL CALCULUS
the z'-Axis are — sin tf : 0 : cos B. The cosine of the angle between these lines is
therefore
, t)8in^+ 0+ co8^ psin^ + cos^ "
0087'=^- — — — = - = cos 7 (cos ^ -\- psin$).
Hence the new form of the area is the integral of secy'dA' and equals the old form.
The integrand dS = sec ydA is called the element of surface. There
are other forms such as dS = sec (r, n) r^ sin 6d6d<}>, where (r, n) is the
angle between the radius vector and the normal; but they are used
comparatively little. The possession of an expression for the element
of sui-face affords a means of computing averages over surfaces. For if
u = u(Xf y, z) be any function of (x, y, z), and z =f(x, y) any surface,
the integral
^=^jn (x, y, z) dS = -jj u (x, y, f) -y/l + p^ -^ qHxdy (16)
will be the average of u over the surface S. Thus the average height
of a hemisphere is (for the sui-face average)
=2^/"^^=2i^xr"!^^^2^
29ra2
2 1.
ira^ = - ;
whereas the average height over the diametral plane would be 2/3.
This illusti-ates again the fact that the value of an average depends
on the assumption made as to the weights.
138. If a surface z =f(Xj y) be divided into elements A5,., and the
function u (x, y, z) be formed for any point (^,., ly,-, t,>) of the element,
and the sum 2w,A.S\- be extended over all the elements, the limit of
the Bum as the elements become small in every direction is defined
as the surface integral of the function over the surface and may be
evaluated as
lim2jtt(ft, m, C..)A5, = Cu{x 2/, z)dS
=ff'^ l^> y^ f(^y y)l Vi+/;^+/;^ dxdy. (it)
That the sum approaches a limit independently of how (^,., rji, ^,) is
chosen in ^S^ and how A5,. approaches zero follows from the fact that
the element i/(^,., ,;,., ^^)a.S.. of the sum differs uniformly from the
integrand of the double integral by an infinitesimal of higher order,
provided « (x, y, z) be assumed continuous in (x, y, z) for points near
the surface and VT+T^+y^' l)e continuous in (x, y) over the surface.
For many purposes it is more convenient to take as the normal
form of the integrand of a surface integral, instead of udS, the
ON MULTIPLE INTEGRALS
S41
prrxluct It cos y</.s' of a function R (a-, y, x) by the ootine of Um itw
cliiiiition of tlie surfcoce to the x-axis by the element dS of the ■ttrfiuse.
Then tlie integral may Ije evaluated over either aide
of the surface ; for R (xj y, x) has a definite value
on the surface, dS is a positive quantity, but 006y
is positive or negative according as the normal is
drawn on the up]X5r or lower side of the surfaoe.
Tlie value of the integral over the surface will be
^
/
mdA
I n (jr, y, z) cos yds = \\Rdxdy or - ijRdxdp (18)
according as the evaluation is made over the upper or lower tide. If
the function R (x, y, x) is continuous over the surface, these integrands
will be iinite even when the surface becomes perpendicular to the
j*y-plane, which might not be the case with
an integrand of the form u (x, y, z) dS.
An integral of this sort may be evaluated
over a closed surface. Let it be assumed
that the surface is cut by a line parallel to
the ;5;-axis in a finite number of points, and
for convenience let that number be two. Let
the normal to the surface be taken con-
stantly as the exterior normal (some take
tlie interior normal with a resulting change
of sign in some formulas), so that for the
upi^er part of the surface cos y > 0 and for
the lower part cos y < 0. Let z =/j(-'*, y)
and z =ff^(xj y) be the upper and lower values of x on the sorfsoe. Then
the exterior integral over the closed surface will have the form
Jr cos yds =JJr [x, y,/, (x, y)] dxdy -JJr [x. y,/,(x, y)]dxdy, (IS")
where the double integrals are extended over the area of the projeotian
of the surface on the a-y-plane.
From this form of the surface intv^Mal ovrr :i cli^^srd surf.Kf
it api)ears that a surface integral over a dosed surfarr may U* *'x-
pressed as a volume integral over the volume inoloeed by the sorfaoe.*
* Certain restrictions upon the functions and deri\'aUTM« aa regards tiMtr
infinite ami the like, nuist lioM it)M>n and within the aorfaoe. It will ba qolta
if the functions and derivatives remain tiuite and oootinaoiia, bat soeh eztrana
are by no means necessary.
342 INTEGRAL CALCULUS
For by the rule for integration,
/ I / ^-^dzdxdy = R(x,y, z) dxdy.
Hence I -R cos ydS =z \ — dV
•^° -^ (19)
if the symbol O be used to designate a closed surface, and if the double
integral on the left of (19) be understood to stand for either side of
the equality (18'). In a similar manner
Cp cos adS = jjPdydz = jjj ^ dxdydz = j ydV,
fQcosfidS= (I Qdxdz= 1 1 j y dydxdz = I j- dV.
r r /dp do dR\
Then \ {Pco%a-\- Qqo^ p-\- Rco^y)dS = \ (^ + ^+^)^^
Jo J \^ y ^1 ^^^^
or J I {Pdydz + Qdzdx + Rdxdy) =111 [Y "^ ^ + jz) ^^dydz
follows immediately by merely adding the three equalities. Any one of
these equalities (19), (20) is sometimes called Gauss's Fonimla^ some-
times Green's Lemma, sometimes the divergence formula owing to the
interpretation below.
The interpretation of Gauss's Formula (20) by vectors is important.
From the viewpoint of vectors the element of surface is a vector dS
directed along the exterior normal to the surface (§ 76). Construct the
vector function
F(a:, y, z) = iP(x, 2/, z) + ]Q(x, y, z) -f ^R(x, y, z).
Let dS = (i cos a H- j cos )3 + k cos y) dS = idS^ + ]dSy + kr/^^,
where rf5,, dS^, dS, are the projections of dS on the coordinate planes.
Then P cos adS + Q cos fidS -}- R cos ydS = F.f/S
and
ji (Pdydz -h Qdxdz + Rdxdy) = Tf.^/S,
where dS^, dS^, dS, have been replaced by the elements dydz, dxdz, dxdy,
which would be used to evaluate the integrals in rectuigular coordinates,
ON MULTIPLE INTEGRALS UZ
without at all implying that the projections dS,, dS^, dS, Mtt aotuftllT
re(;tangular. The combination of partial deriyatiTet
dp BQ BH ,. „ ^
where V.F is the symbolic scalar product of V and P (Ex. 9 below), b
called the divergence of F. Hence (20) becomes
fdiv FdV = JV.TUV= TF.r/S. (20^
Now the function F (x, y, z) Ih such that at each point (x, y, z) of qwoe a Twior
is ilehned. Such a function is Been in the velocity in a moving fluid nich a« air or
water. The picture of a scalar function u (z, y, z) was by means of the mrfseas
u z^ const. ; the picture of a vector function F (jr, y, z) may be found in the ijHsai
of curves tangent to the vector, the stream lines in the fluid
if F be the velocity. For the immediate purposes it is better
to consider the function F(j', y, z) as the llux Dr, the prod-
uct of the density in the fluid by the velocity. With this
interpretation the rate at which the fluid flows through an
element of surface dS is Dy»dS = F»(/S. For in the time
dt the fluid will advance along a stream line by the amount
ydt and the volume of the cylindrical volume of fluid which advances through
surface will be Y»dSdt. Hence S/>y*(iS will be the rate of diminution of Um i
of fluid within the closed surface.
As the amount of fluid in an element of volume dF is IMV, the rate of (
of the fluid in the element of volume is — dD/dt where dZ>/d( is the rate of
of the density I) at a point within the element. The total rate of diminution of
amount of fluid within the whole volume is therefore — ZdD/HdV, Hmms, by
virtue of the principle of the indestructibility of matter,
Tf^S = Tdt^S = - f^dV. (MT)
Jq Jq J 01
Now if Vx, Vy, Vt be the components of y so that P = Dp,, Q = /^ B = Dib m
the components of F, a comparison of (21), (20'), {20") shows that the integnls of
— dl)/ct and div F are always equal, and hence the integrands,
tt ~ d£ ty dt dt dy H *
are 6qual ; that is, the sum P^ + Q^ + i?,' represents the rate of diminutkMi of
density when iP + jQ + k/J is the flux vector; this combinatioo is called ths
divergence of the vector, no matter what the vector F really leprassota
139. Not only may a surface integral be stepped up to a Toliuiie
integral, but a line integral around a closed curve may be stepped up
into a surface integral over a surface which spans the curve. To begin
844
INTEGRAL CALCULUS
with the simple case of a line integral in a plane, note that by the
same reasoning as above
x^'" -ff- % <^^' X'^'" =xrs '^''
flP(x, i,)dx+ Q(x, y)dy-\ =jf{£ " '^J^'^'J-
(22)
This is sometimes called Green's Lemma for the plane in distinction
to the general Green's Lemma for space. The oppo-
site signs must be taken to preserve the direction
of the line integral about the contour. This result
may be used to establish the rule for transforming a
double integral by the change of variable ic = <^ (w, v),
y =z kjfiuy v). For
A=Jxdi/ = ±Jx^du + x-£dv
° -*//[s(-l)-l('£)]-
J J \du dv dv duj
(The double signs have to be introduced at first to allow for the case
where J is negative.) The element of area dA = \J\dudv is therefore
established.
To obtain the formula for the conversion of a
line integral in space to a surface integral, let
P(x, y, z) be given and let z —fixj y) be a surface
spanning the closed curve O. Then by virtue of
X =^f(x, y), the function P(xy y, z) = P^(xy y) and
where O' denotes the projection of O on the xy-plane. Now the final
double integral may be transformed by the introduction of the cosines
of the normal direction to « =/(«, y).
cos^:co8y = -y:l, dxdij = cos yds, qdxdij = - cm pdS = - dxdz.
ON MULTIPLE INTEGRALS S46
If this result and those obtained by permuting the letters be
X
{Pdx + Qdy -h iidx)
-//[(|-B)^^'-(S-g)«.^(g-g)H-<»>
This is known as Stokat^s FomiuUi and is of eepeoial importanoe in
}i}'(lroniechanics and the theory of electromagnetism. Note Mmt the
line integral is carried around the riin of the surface in the diieotkm
which ap})ears positive to one standing u]>on tliat side of the titrfMe
over which the surface integral is extended.
Again the vector interpretation of the result is valoaUe. Let
F(x, y, z) = iP{x, y, z) + jQ(x, y, «)-|- k7?(ar, y, «),
— ' raf-fc>'(£-£)-'(S-5)- <")
Then jF.rfr = fcurl F.rfS = fv^F.dS, (28')
where V^F is the symbolic vector product of V and F (Ex. 9, below),
is the form of Stokes's Formula ; that is, the line integral of a veotor
around a closed curve is equal to the surface integral of the curl of the
vector, as defined by (24), around any surface which spans the onnre.
If the line integral is zero about every closed curve, the surface inte-
gi*al must vanish over every surface. It follows that curl F = 0. For
if the vector curl F failed to vanish at any point, a small plane sur-
face dS })erpendicular to the vector might be taken at that point and
the intogi-al over the surface would be approximately |curl F\dS and
would fail to vanish, — thus contradicting the hypothesis. Now the
vanishing of the vector curl F requires the vanishing
of each of its components. Thus may be derived the condition thai
Pdx + Qf/y + Rdz be an exact differential.
If F be interpreted as the velocity t in a fluid, the integnJ
rT«dr sifv^ + lyiy + Ogdi
of the component of the velocity along a curve, whether open or doted, li
the circulation of the fluid along the curve; it might be more natural to
346 INTEGRAL CALCULUS
the integral of the flux Dw along the curve as the circulation, but this is not
the convention. Now if the velocity be that due to rotation with the angular veloc-
ity a about a line through the origin, the circulation in a closed curve is readily
computed. For
T = «xr, fy-dr = faxT»dr = fa.rxdr = a. frxdr = 2 a.A.
The circulation is therefore the product of twice the angular velocity and the area
of the surface inclosed by the curve. If the circuit be taken indefinitely small, the
integral is 2 a.dS and a comparison with (23') shows that curl v = 2 a; that is, the
curl of the velocity due to rotation about an axis is twice the angular velocity and
is constant in magnitude and direction all over space. The general motion of a
fluid is not one of uniform rotation about any axis ; in fact if a small element of
fluid be considered and an interval of time 8t be allowed to elapse, the element
will have moved into a new position, will have been somewhat deformed owing to
the motion of the fluid, and will have been somewhat rotated. The vector curl v,
as defined in (24), may be shown to give twice the instantaneous angular velocity
of the element at each point of space.
EXERCISES
1. Find the areas of the following surfaces :
(a) cylinder x^ -\- y^ — ax = 0 included by the sphere x^ -\- y^ -^ z^ = a^,
(/S) x/a + y/b + z/c = 1 in first octant, (7) x^-{-y^ + z^ = a?- above r = a cos n0,
(a) sphere x^ 4. 2,2 ^ ^^ = a^ above a square |x| ^ 6, |y | ^ 6, 6 < ^ V2 a,
(c) z = xy over a;* + 2/2 _ ^2^ (^) 2 az = a;^ — y^ over r^ = a^ cos0,
(1;) z* + (x cos a: + y sin a)^ = a^ in first octant, (0) z = xy over r^ = cos 2 0,
( 1 ) cylinder x^ + y2 _ g2 intercepted by equal cylinder y^ ■}- z^ = a^.
2. Compute the following superficial averages:
(a) latitude of places north of the equator, Ans. S2j\°.
iP) ordinate in a right circular cone h^{x^ + y^) — a^{z — h)^ = 0,
(7) illumination of a hollow spherical surface by a light at a point of it,
(a) illumination of a hemispherical surface by a distant light,
(e) rectilinear distance of points north of equator from north pole.
3. A theorem of Pappus: If a closed or open plane curve be revolved about an
axis in its plane, the area of the surface generated is equal to the product of the
length of the curve by the distance described by the center of gravity of the curve.
The curve shall not cut the axis. Prove either analytically or by infinitesimal
analysis. Apply to various figures in which two of the three quantities, length of
curve, area of surface, position of center of gravity, are known, to find the third.
Compare Ex. 27, p. 332.
4. The surface integrals are to be evaluated over the closed surfaces by express-
ing them as volume integrals. Try also direct calculation :
(*») Jj(x*dydz + xydxdy + xzdxdz) over the spherical surface x* + y* -j- z« = a«,
<^) fj(x*dydz + y^dxdz + z^dxdy), cylindrical surface x'^ -1- j/« = a^, z=±h,
ON MULTIPLE INTEGRALS $47
(7) ff[i^* - yz)dvdz - 2xvdxdx + dxdy] OY«r Um eube 0 S s, y, i S c,
(8) ffxdydz = ffl/<ixdz = fftdxdy = kffixdydz + ydadf + idW^ » K,
(e) Calculate the line iiitc^'raU of Ex. 8, p. 207, around a doMd paib foraMd by
two patliK there given, by applying Green*a Lemma (tt) and eraluatliif the rmaHu
iw^ double integrals.
5. If x = 0j(u, »), y = 0,(u, r), t = 0,(m, o) are the parameCrie aqiuUioni of a
surface, the direction ratios of the nomuil are (aee Ex. 16, p. 186)
C08a:C08/S:C087 = Jj:J,:J, if Jf ^ J (l!l±l2.^Ltl\
Sliow P that the area of a surface may be written an
-- -x(^)- -SO'- -s^^.
itiui cUfl = £du« + 2 Fdudo + Ode*.
Show 2° that the surface integral of the first type becomes merdy
///(-c, y, z) 8ec7dzdy = ///(^,t 0„ 0,) ViSTO - F«diids,
and determine the integrand in the case ot the developable surface of Ex. 17, p. 148.
Show 3° that if x =/,({, v, f), V =/i{^, V, i), z =/,({, i?, f) i* » transformation of
space which transforms the above surface into a new surface ( = ^,(tt, v), f = ^((a, v),
■'('d)-('ir:)'(!r.)-(:Tf)'(.=r3*-'(n)^0-
Show 4*^ that the surface integral of the second type beoomea
ffRa.,y=ffRj[^ya.
whore the integration is now in terms of the new variables (, f, f in place of c, y, f.
Show 5° that when R = z tlie double integral above may be transformed by
Green's Lemma in such a manner as to establish the formula for change of variables
ill triple integrals.
6. Show that for vector surface integrals J l/dS zs JVUdV.
7. Solid angle as a surface integral. The area cut out from the unit sphere by a
cone with its vertex at the center of the sphere Is called the sottd tmgk w soblsnded
at the vertex of the cone. The solid angle may also be defined as the ratio of the
area cut out upon any sphere concentric with the vertex of the OOIM, Id the aqoara
of the radius of the sphere (compare the deflnition of the angle betw##n two lines
848 INTEGRAL CALCULUS
in radians). Sliow geometrically (compare Ex. 16, p. 297) that the infinitesimal solid
angle d« of the cone which joins the origin r = 0 to the periphery of the element dS
of a surface is dw = cos(r, n)dS/r^, where (r, n) is the angle between the radius
produced and the outward normal to the surface. Hence show
"J fS "^ J 1* J r^dn J dnr J r
where the integrals extend over a surface, is the solid angle subtended at the origin
by that surface. Infer further that
_rAlds = 4» or -f±ldS = 0 or -f^ldS^g
Jq dn T Jq dn r Jq dn r
according as the point r = 0 is within the closed surface or outside it or upon it
at a point where the tangent planes envelop a cone of solid angle 0 (usually 2 7r).
Note that the formula may be applied at any point ({, ?;, f) if
r2 = (e-x)2 + (r,-y)2 + (f-2:)2
where (x, y, z) is a point of the surface.
8. Gau88''8 Integral. Suppose that at r = 0 there is a particle of mass m
which attracts according to the Newtonian Law jP = m/r^. Show that the
potential is V=—m/r so that F= — VF. The induction or flux (see Ex. 19,
p. 308) of the force F outward across the element dS of a surface is by definition
— Fcos(F, n)dS = F'dS. Show that the total induction or flux of F across a
surface is the surface integral
fF-dS = - fdS-VV = - f—dS = m fdS-V - ;
J J J dn J r
and m = — r F.dS = — f dS.VF= nl f — -dS,
iir Jq Ait Jq 4^ir Jq dn r
where the surface integral extends over a surface surrounding a point r = 0, is the
formula for obtaining the mass m within the surface from the field of force F
which is set up by the mass. If there are several masses wij, tHj, • • • situated at
points (fp 1,1, r^), (fj, Vi, fB)> • • •» let
F = Fi + F2 + ---, F=Fi+F2 + ...,
Vi = -m [(?,- - XiY + (,,, - ViY + (fi - ZiYT ^
be the force and potential at (x, y, z) due to the masses. Show that
— i rF^S = -i- rdS.VF=- — V C — -dS=^'mi = M,
iw Jo iirJo ^ir ^ Jodnn ^
(26)
where 2 extends over all the masses and S' over all the masses within the surface
(none being on it), gives the total mass U within the surface. The integral (26)
which gives the mass within a surface as a surface integral is known as Gauss's
Integral. If the force were repulsive (as in electricity and magnetism) instead of
attracting (as in gravitation), the results would be F = m/r and
^ / F^8 = ^ rdS.VF=fi V r A !^dS =y'm, = U. (26')
4wJo \ir Jo ifT^Jodnrt ^ ^ '
ON MULTIPLE INTEGRALS S49
9. ^^^ = 'r: + J7- + kr-bethe operator deflnod on {Mifo I7f, ahow
by formal operation on P = Pi + QJ + Rk. Show forUier thmt
Vx(VxF) = V (V-F) - (V-V) F (write the C&rteiUui form).
Show that (V»V) U = V.(Vt7). If u is a con«Unt unit rector, abow
(u.V)F = ^co8a + ^coe/J + ??c«i7 = —
ex cy dt d§
i» the directional derivative of F in the direction n. Show ((ir*V) F = cfF.
10. Green's Formula (space). Let F(x, y, z) and 0(x, y, x) be two
so that VF and VG become two vector functions and FVO and GVF two
vector functions. Show
V.(FVG) = VF»VG + FV.VG, V,{GVF) = VF.VO + GV.V/',
dx\ dz/ dy\ dvJ dz\ dz/
&y dy dz dz WV"*" *•/"
dF 8G dF dO , dF dG
dx bx
and the similar expressions which are the Cartesian equivalents of the abore vector
forms. Apply Green's Lemma or Gauss's Formula to show
CFVG'dS - CvF»VGdV -^^ CFV.VGdV, ^
j*GVF»dS = fvF'VGdV -\- fcV.VFdV, (IT)
f{FVG - GVF)HiS =zf{FV.VG - GV.VF)dV, (JT^
The formulas (26), (26'), (26'') are known as Greenes FormuUu; in particular the flm
two are asymmetric and the third symmetric. The ordinary Caitesiaii fonnt d
(26) and (26") are given. The expression c^F/dx* + d^F/V + i^F/dtfi la ohm
written as AF for brevity ; the vector form is V»VF.
11. From the fact that the integral of F^r has opposite valaes when the
is traced in opposite directions, show that the integral of VxFover a doaed aurfMa
vanishes and that the integral of V'VxF over a volume vaniabes. lofer tbat
V.VxF = 0.
350 INTEGRAL CALCULUS
12. Reduce the integral of VxVU" over any (open) surface to the difference in
the values of U at two same points of the bounding curve. Hence infer VxVZJ = 0.
13. Comment on the remark that the line integral of a vector, integral of F.dr,
is around a curve and along it, whereas the surface integral of a vector, integral
of F^S, is over a surface but through it. Compare Ex. 7 with Ex. 16 of p. 297. In
particular give vector forms of the integrals in Ex. 16, p. 297, analogous to those of
Ex. 7 by using as the element of the curve a normal dn equal in length to dr,
instead of dr.
14. IfinF = Pi4- Qj + Kk, the functions P, Q depend only on x, y and the
function i? = 0, apply Gauss's Formula to a cylinder of unit height upon the
zy-plane to show that
fV'FdV=fF-dS becomes JT^— + — )dxdy = jF.dn,
where dn has the meaning given in Ex. 13. Show that numerically F«dn and Fxdr
are equal, and thus obtain Green's Lemma for the plane (22) as a special case of (20).
Derive Green's Formula (Ex. 10) for the plane.
15. If fF.dr = fG'dS, show that C{G - VxF).dS = 0. Hence infer that if
these relations hold for every surface and its bounding curve, then G = VxF.
Ampere's Law states that the integral of the magnetic force H about any circuit is
equal to 4 tt times the flux of the electric current C through the circuit, that is,
through any surface spanning the circuit. Faraday's Law states that the integral
of the electromotive force E around any circuit is the negative of the time rate
of flux of the magnetic induction B through the circuit. Phrase these laws as
integrals and convert into the form
4irC = curl H, — B = curl E.
16. By formal expansion prove V»(ExH) = H'VxE — E»VxH. Assume VxE= — H
and VxH = fe and establish Poynting's Theorem that
r(ExH).dS = - -^ ri(E.E+H.H)dF.
dtJ 2
17. The '* equation of continuity " for fluid motion is
dD . BDvx . dDv„ dDv^ ^ dD
1 1 H = 0 or \-
dt dx dv dz dt
\dx dy dzj '
where D is the density, y = iVx-\- j»y + kvz is the velocity, dD/dt is the rate of
change of the density at a point, and dD/dt is the rate of change of density as one
moves with the fluid (Ex. 14, p. 101). Explain the meaning of the equation in view
of the work of the text. Show that for fluids of constant density vv = 0.
18. If f denotes the acceleration of the particles of a fluid, and if F is the
external force acting per unit mass upon the elements of fluid, and if p denotes
the pressure in the fluid, show that the equation of motion for the fluid within any
surface may be written as
2^fIWr = VFDdF-VpdS or fiDdV^fvDdV-fpdS,
ON MULTIPLE INTEGRALS 861
where the Huniniations or int^piitioiui extend over t)ie Toluroe or iu boundliif fluw
face and the prewures (except thmte acting on the boundiog muimet lawmnt) wmj
be diKregarded. (See the first half of f 80.)
19. By the aid of Ex. 6 tranifonn the mrfAoe Integral in Ex. 18 and iad
fjAav=f(Dr-vp)dV or ^ = »-iv,
as the equations of motion for a fluid, where r is the Tecior to any particle. Prov«
20. If F is derivable from a potential, so tliat F = —VU^ and If the deiMity la a
function of the pressure, so that dp/D = dP, show that the equatiooaof moUoo ars
£l_vxVxv=-v(r7+P + ir«). or ^(T^r) =- d/i;+ P- i^)
after multiplication by dr. The first form is Helmholtz^s, the Moond Is RelTlii*a.
Show
l(v.dr) = - f Y^r=-\u-^P--y\ and fT* = coii*.
a,b.e dt dtJa,b,c L 2 J«.».e ^O
In particular explain that as the differentiation d/dt follows tlie particles In their
motion (in contrast to d/ct^ which is executed at a single point of qiace), tiM
integral must do so if the order of differentiation and int^n^o" is to be Inters
chan<^a.'able. Interpret the final equation as stating that the circulation in a corre
which moves with the fluid is constant.
22. Show that, apart from the proper restrictions as to continuity and dlfferm-
tiability, the necessary and sufficient condition that the surface InlegTal
ffPdydz + Qdzdx + Rdxdy = f pdx '^- ijdy -^ rdx
depends only on the curve bounding the surface U that P^ + Q^-^ B^ = 0. Show
further that in this case the surface integral reduces to the line integral given abore,
provided p, g, r are such functions that r^ — q'^ = P, pi — r^ = (^« 9» ~ P» = ^
Show finally that these differential equations for p, q, r may be satisfied by
p=f'Qdz-fR(z,y,Zo)dy, q^-f'Pdx, r = 0;
and determine by inspection alternative values of p, g, r.
CHAPTER XIII
ON INFINITE INTEGRALS
140. Convergence and divergence. The definite integral, and hence
for theoretical purposes the indefinite integral, has been defined.
f f(x)dx, F(x)= f f{x)dx,
when the function f(x) is limited in the interval atoh, or a to ic ; the
proofs of various propositions have depended essentially on the fact
that the integrand remained finite over the finite interval of integration
(§§ 16-17, 28-30). Nevertheless problems which call for the determina-
tion of the area between a curve and its asymptote, say the area under
the witch or cissoid,
have arisen and have been treated as a matter of course.* The inte-
grals of this sort require some special attention.
When the integrand of a definite integral hecom.es infinite within or
at the extremities of the interval of integration, or when one or both of
the limits of integration become infinite, the integral is called an infinite
integral and is defined, not as the limit of a sum, but as the limit of an
integral with a variable limit, that is, as the limit of a function. Thus
I f{x)dx=\\m\F(x)=\ f(x)dx\, infinite upper limit,
I f(x) d^ = lim F(x) = i f(x) dx , integrand f(b) = oo.
These definitions may be illustrated by figures which show the connec-
tion with the idea of area between a curve and its asymptote. Similar
definitions would be given if the lower limit were — oo or if the inte-
grand became infinite at a; = a. If the integrand were infinite at some
intermediate jwint of the interval, the interval would be sulxiivided
into two intervals and the definition would be applied to each part.
• Hew and below the construction of figures is left to the reader.
862
ON INFINITE INTEGRALS S58
Now the behavior of F(x) as x approaobes a definite ralue or
infill it<^ may be of three distinct aorta ; for F(x) may approaeb a i
finite.' (}iiantity, or it may become infinite, or it may n«i>jHgf^ witbool
appioa(;hing any finite quantity or becoming definitely infinite. The
examples
i i^T47» = ±[i ?T4T«-^«*^-"2^J-2-'. -Hmit.
/ V ~ i^"i / V ^ ^^^ * r ^^'®^°*®* infinite, no limits
J I cos xdx = lim / cos xdx = sin x L osoiUates, no limit,
0 "^iJo J
illustrate the three modes of behavior in the case of an infinite npper
limit. In the first case, where the limit exists^ the infinite integmt ia
said to converge ; in the other two cases, where the limit does not exist,
the integi-al is said to diverge.
If the indefinite integral can be found as above, the question of the
convergence or divergence of an infinite integral may be determined
and the value of the integral may be obtained in the case of oonrergenoe.
If the indefinite integml cannot be found, it is of prime importance to
know whether the definite infinite integral converges or diverges; for
there is little use trying to compute the value of the integral if it does
not converge. As the infinite limits or the points where the integrand
becomes infinite are the essentials in the discussion of infinite integrals,
the integrals will be written with only one limit, as
J}(x)dx, J''f{x)dx, f/i')'^-
To discuss a more complicated combination, one would write
Jo Vi*logx Jo J( J\ Ji Vxlogx
and treat all four of the infinite integrals
Jr e-'dx r^ e-'dx P e-'dx C* e-'dx
0 V^logx J Vx*logx Jj V?logx J V5log*
Now by definition a function E(x) is called an J?-funotion in the
neighborhood of the value x = a when the function is continuous in
the neighborhood of a* = ff and approaches a limit which ia neither lero
nor iiifinitv (p. fi2>. T/ir hehavior of the. Infinite iniegraU of a
354 INTEGRAL CALCULUS
which does not change sign and of the product of that function by an
E-function are identical as far as convergence or divergence are concerned.
Consider the proof of this theorem in a special case, namely,
rf{x) dx, J fix) E (x) dx, (1)
where f{x) may be assumed to remain positive for large values of x
and E (x) approaches a positive limit as x becomes infinite. Then if K
be taken sufficiently large, both /(a;) and E(x) have become and will
remain positive and finite. By the Theorem of the Mean (Ex. 6, p. 29)
m rf(x)dx < C f(x)E(x)dx <M C f{x)dx,
x> K,
where m and M are the minimum and maximum values of E (x) between
K and oo. Now let x become infinite. As the integrands are positive,
the integrals must increase with x. Hence (p. 35)
' /(ar) 6?a; converges, I f(x)E(x)dx<MJ f(x)dx converges,
K J K J K
if I f{x) E ix) dx converges,
f(x) dx < — / f(x) E (x) dx converges ;
and divergence may be treated in the same way. Hence the integrals
(1) converge or diverge together. The same treatment could be given
for the case the integrand became infinite and for all the variety of
hypotheses which could arise under the theorem.
This theorem is one of the most useful and most easily applied for determining
the convergence or divergence of an infinite integral with an integrand which
does not change sign. Thus consider the case
r "^ =rr ^' i'^, e(x)=[ ^' 1^ r"- = --"
•^ (ox + x2)i -^ Lax + x^^J x2 ^ ' Lax + x^\ J x^ x
Here a simple rearrangement of the integrand throws it into the product of a func-
tion E{x), which approaches the limit 1 as x becomes infinite, and a function l/x^,
the integration of which is possible. Hence by the theorem the original integral
converges. This could have been seen by integrating the original integral ; but
the integration is not altogether short. Another case, in which the integration is
not possible, is
r^ dx _ r^ 1_
dx_
Vl-x* ^ Vl + x^ vTTx Vi^
E{x\= ^ r ^ =,
Vl 4-x« Vl + X •^ Vl - X
ON INFINITE INTEGRALS 866
Hero E{1) = \. The integral U again convergent. A caee of dIveifMM* wookl hm
141. The interpretation of a definite integral as an area will tuggeal
another form of test for convergence or divergence in case the int5-
j^rand does not change sign. Consider two functions /(x) and ^(x)
hoth of which are, say, positive for large values of x or in the neigh-
l)oiliood of a value of x for which they become infinite. J/ (he currt
,j=z^i,{x) remains above y =/(«), the integral of f{x) must converts if
the integral of\l/(x) converges, and the integral of^{x) must diverge if
the integral off{x) diverges. This may be proved from the definition.
For/(.T) < ^{x) and
r f(x)dx < i ^(x)dx or F(x) < ♦CjrV
Now as X approaches b or oo, the functions F(x) and ♦(x) In-t !i in. n-i'-^v
If ♦(a-) approacihes a limit, so must F(x) ; and if F(x) in«i. i^.^ with-
out limit, so must ♦(x).
As the relative behavior of /(a*) and ^(a?) is cons**' ;</// nr^tr
particular values of x or when x is very great, the < ^ may )«
expressed in terms of limits, namely : Jfflf(x) does not change sign and
if the ratio f{x)/^(x) approaches a finite limit (or zero), the integral of
f(x) will converge if the integral of \lf{x) converges; and {f the ratio
f(x)/\ff(x) approaches a finite limit (not zero) or becomes ui^finite, the
Inttgral off(x) will diverge if the integral oftlf(x) diverges. For in the
tirst case it is possible to take x so near its limit or so large, as the
case may be, that, the ratio f(x)/^ (x) shall be less than any assigned
number G greater than its limit; then the functions /(x) and G^(x)
satisfy the conditions established above, namely / < G^, and the inte-
i;ral of f(x) converges if that of \^(x) does. In like manner in the seoood
case it is possible to proceed so far that the ratio /(j')/^(x) shall have
l)ecome to remain greater than any assigned numljer g less than its
limit; then/> <7^, and the result above may be applied to show that
the integral of /(a-) diverges if that of ^(jr) does.
For an infinite upper limit a direct integration shows that
converges if * > 1,
/
dx -11
a^ A-lx*
or log X
diverges if Ar ^ L ^
Now if the teM function ^(x) be chosen as l/x*«x-*, the ratio
f(x)/<f>(x) becomes a^/(x), and ♦/ the limit of th^ product «•/(«)
856 INTEGRAL CALCULUS
and may be shown to be finite (or zero) as x becomes infinite for any
choice ofk greater than 1, the integral off(x) to infinity will converge;
but if tJis product approaches a finite limit {not zero) or becomes infinite
for any choice of k less than or equal to 1, the integral diverges. This
may be stated as : The integral oif(x) to infinity will converge iif(x)
is an infinitesimal of order higher than the first relative to 1/x as x
becomes infinite, but will diverge i^f(x) is an infinitesimal of the first
or lower order. In like manner
/
dx
{b^xf k-l(b-x)
k-]
or — log (b — x)
converges iik<l,
diverges if A;^l, '
and it may be stated that : The integral of f(x) to b will converge if
f(x) is an infinite of order less than the first relative to (b — x)~^ as x
approaches i, but will diverge if f(x) is an infinite of the first or higher
order. The proof is left as an exercise. See also Ex. 3 below.
/I 00
As an example, let the integral | x'^e-'^dx be tested for convergenoe or diver-
Jo
gence. If n > 0, the integrand never becomes infinite, and the only integral to
examine is that to infinity ; but if n < 0 the integral from 0 has also to be consid-
ered. Now the function e-^ for large values of x is an infinitesimal of infinite
order, that is, the limit of x* + »e- ^ is zero for any value of k and n. Hence the
integrand x^e-^ is an infinitesimal of order higher than the first and the integral
to infinity converges under all circumstances. For x = 0, the function e-^ is finite
and equal to 1 ; the order of the infinite x»e-^ will therefore be precisely the order
n. Hence the integral from 0 converges when n > — 1 and diverges when n ^ — 1.
Hence the function
r(a) = r*x«-ie-«dx, a > 0,
defined by the integral containing the parameter a, will be defined for all positive
values of the parameter, but not for negative values nor for 0.
Thus far tests have been established only for integrals in which the
integrand does not change sign. There is a general test, not particularly
useful for practical purposes, but highly useful in obtaining theoretical
results. It will be treated merely for the case of an infinite limit. Let
^(*) =y* f(x) dx, F(x") -F(x')= r f(x) dx, x\ x" > K. (4)
Now (Ex. 3, p. 44) the necessary and sufficient condition that F(x)
approaxsh a limit as x becomes infinite is that F(x") — F(x') shall
approach the limit 0 when x' and x", regarded as independent varia-
bles, become infinite; by the definition, then, this is the necessary
and sufficient condition that the integral of f(x) to infinity shall
converge. Furthermore
ON INFINITE INTEGRALS 867
if J \f(^)\^ eanverget, then f /{x)djc (S)
must converf/e aiul is suid to be abgolutely convergent. The proof of
iui(X)rtant theorem is contained in the above and iu
£ f(x)dxs£\f(x)\dx.
To see whether an integral is absolutely convergent, the
lished for the convergence of an integral with a positive integimnd
may be applied to the integral of the absolute value, or some obriout
direct method of compariaun may be employed; for example,
/" co&xdx r* Idx , . ^
liFir^^-J ^;q:^^h^c»^<^°vergee,
and it therefore appears that the integral on the left converges abiCK
lutely. When the convergence is not absolute, the question of con-
vergence may sometimes be settled by integration by parts. For
suppose that the integral may be written as
J/(x)rfx=JVw^(x)<te = [*(x)J^(x)</J-JVwJ'*(«)«i«'
by separating the integrand into two factors and integrating by perts.
Now if, when x becomes infinite, each of the right-hand tenns approaches
a limit, then
j^ f{x)dx = Inn U{x)j^{x)d^- \\mj\\x)j^(x)dxdx,
and the integral oif(x) to infinity converges.
. /••zooaxdx _ /••s|ootx|dlc
As an example consider the convergence of I — - — -^ • Here J —3 — -j-
does not appear to be convergent ; for, apart from the factor I ooex| which oedlUtee
between 0 and 1, the integrand is an inflnite«ini&l of only the tint order snd tbe
integral of such an integrand does not converge ; the original integral !• therefore
apparently not absolutely convergent. However, an Integratioii by parts gives
/^xcoszdx zsinxl' r'z* — a* _,
,, zsmz
lim
xsinz r'_£LliLcoezrfz<f -.
Now the integral on the right is seen to be convergent and, in fact* absolately
conver<;ont as x becomes infinite. The original integral therefore
a limit and be convergent ae x becomes infinite.
358 INTEGRAL CALCULUS
EXERCISES
1. EsUbllsh the convergence or divergence of these infinite integrals:
(3) r x*-*(l — x)^-kiz (to have an infinite integral, a must be less than 1),
/I ^ r** dx /** dx
x'-Hl-x)P-^dx, (f) / —==. (1,)/-—==,
''O Vox - X2 ''I xVx2-l
r* dx , V r^ a;dx , , /•^ x^-i
(X) f\ ^ ^ , ^ < 1, ^ = 1, (.) r\/iE^^, , < 1.
Jo V(l-x2)(l-ifc2x2) *^0\1-X^ '
2. Point out the peculiarities which make these integrals infinite integrals, and
test the integrals for convergence or divergence :
(a) fVlogiVdx, conv. if n>-l, div. if n^-1, (/3) f^^^dx,
Jo \ x/ Jo 1 — X
it) f (— log2;)"dx, (5) r^logsinxdx, (c) j xlogsinxdx,
"'J ^\ zh + x' ^ ' Jo (sinx + cosx)* ^ ' Jo \^x/ '
■. (k) I x^dx, (\) I logxtan — dx,
0 Vxlog(x + l) Jo ^Mo ^ 2 '
wrfr-:-- <')/:;-•'-• ^rf^-
. . /»»sin2x - . . /»ilogxdx . . /*• -fx-^V
Jo x^ Jo V 1 — x^ *^o
3. Point out the similarities and differences of the method of ^-functions and
of test functions. Compare also with the work of this section the remark that the
determination of the order of an infinitesimal or infinite is a problem in indeter-
minate forms (p. 63). State also whether it is necessary that /(x)/V' (x) or x*/(x)
should approach a limit, or whether it is sufficient that the quantity remain finite.
Distinguish "of order higher" (p. 356) from "of higher order" (p. 63); see Ex. 8, p. 66.
4. Discuss the convergence of these integrals and prove the convergence is
absolute in all cases where possible :
(«) /• '-^dx, (« /-cosxMx, (V) /- £^dx.
ON INFINITE INTEGRALS 869
(«) f x''-'e-'c<^^iMm(x8iu/<)dx, (X) f* •*»»•«■ g»^
5. If /^(2) and /,(z) are two limited funcUons int««rsble (In tiM tnm at
U 28-30) over tlie integral a ^ z ^ b, show that their product /(x) =/,(x)/(x)
is integrable over the interval. Note that in any interval ^, the rvUtioM
mi. mi, s m. ^ 3f, ^ Jf^Afaf and 3/,,3/,i - mum,< = Ifulftf - Mutrngf +
3f,,m-^, — mum^i = MnOqi -^ nhiOn hold. Show further Uiat
fy,{x)f,{x)dx = lim2)./',(f.)/,(e.)«<
= lim5^/,(f,)[^/,(x)dji-^' /,(x)dxj,
or fy{x)dx=f,{i,)fy,ix)dx + lim]^[/,({,) -/,((,.,)] fy{z)dx.
6. rA€ /Second Theorem of the Mean. If /(x) and ^(x) are two llmitMl -MTmtnn»
integrable in the interval a ^ x ^ 5, and if 0(x) is positive, DondecWMh^ sad
less than K, then
fy{x)/{x)dx = Kf''/{x)dx, a S ( s 6.
And, more generally, if 0(x) satisfies — oo<ii;^^(x)^ir<ao and te citliar
nondecreasing or nonincreasing throughout the interval, then
£<t>{x)f{x)dx = kfj/(x)dx + Kf%)dx,
a^i^b.
In the first case the proof follows from Ex. 5 by noting that the intflfnJ of
0 (x)/(2) may be regarded as the limit of the sum
<t>{i,)£f{x)dx + ^[0(£.) - <p{ii.i)]fy{x)dx + [A-- ♦((.)] j^/C)*^
where the restrictions on 0(x) make the coefficients of the integrmle all podtive or
zero, and where the sum may consequently be wriuen aa
m[*(^,) + 0(y - 0(ft) + • + 0(W - f{U-i) + X" - ^iU)] = 1^
if /i be a properly chosen mean value of the integrals which multiply tbOM eoeA>
fients ; as the integrals are of the form / /(x)dx where | = a, X|, • • •, a^ It foUowe
•'I
360 INTEGRAL CALCULUS
that fi must be of the same form where a ^ $ ^ 6. The second form of the theorem
follows by considering the function <f> — k or k — <^.
7. If ^(x) is a function varying always in the same sense and approaching a
finite limit as x becomes infinite, the integral / <l>{x)f{x)dx will converge if
f /(x)dx converges. Consider
f'''<f>{x)f(x)dx = <p(x^fjf{x)dx + 0(x")/'^/(x)dx.
8. If 0 (x) is a function varying always in the same sense and approaching 0 as
a limit when x = oo, and if the integral F{x) of /(x) remains finite when x = oo,
then the integral I <f> (x)/(x) dx is convergent. Consider
£% (x)/(x) dx = 0 (xO lF{i) - F(xO] + 0 (x'O [F(x'0- F{^)-\.
This test is very useful in practice ; for many integrals are of the form j <f> (x) sin xdx
where 0 (x) constantly decreases or increases toward the limit 0 when x = oo ; all
these integrals converge.
142. The evaluation of infinite integrals. After an infinite integral
has been proved to converge, the problem of calculating its value stijl
remains. No general method is to be had, and for each integral some
special device has to be discovered which will lead to the desired
result. This tnay frequently he accomplished hy choosing a function
F(z) of the complex variable z =x-\- iy and integrating the function
around some closed path in the z-plane. It is known that if the points
where F {z) = X (x^ y) -\- iY (x^ y) ceases to have a derivative F'(z),
that is, where X{x^ y) and r(cc, y) cease to have continuous first par-
tial derivatives satisfying the relations X'^ = Yy and X'^ = — K^, are cut
out of the plane, the integral of F{z) around
any closed path which does not include any of "^"^y td±J^
the excised points is zero (§ 124). It is some-
times possible to select such a function F{z)
and such a path of integration that part of
the integral of the complex function reduces
to the given infinite integral while the rest of
the integral of the complex function may be computed. Thus there
arises an equation which determines the value of the infinite integi-al.
CoMider the integral J* ?^dx which is known to converge. Now
dx=f 1 — dx=r —-f ^—dx
0 « Jo 2ix Jo 2ix Jo 2ix
■QgiasU at once that the function eU/z be examined. This function has a definite
d«rivativ« &t every point except « = 0, and the origin is therefore the only point
dz=»-\rdx
dz-^+idy dz^idy
^ dz^dx
0 A
ON INFINITE INTEGRALS S61
y» liich has to be cut out of the plane. The Integral of «<•/< aroand anj patli ^n^
as tiiat marked in the figure * i« therefore caro. Then If a la ■nail aad A la »*frr^
Jot Ja z ^Jo A-^iy '^Ja x-^itt
J*o e-u-f t~*^ /•♦••*•
But r'??<b=-r^f^dx=-r?:^ «d r"5:fd.«r-LL?*,
ilie tifHt by the ordinary rules of Integration and the aeooiid Iqr MaolaariJi**
Formula. Hence
— dz= +r - + four other Inlefimk.
O Z Ja Z J-a X
It will now be shown that by taking the rectangle aniBetently laiga and Ito
semicircle about tlie origin Kuf!iciently Huiall each of the foar intograla may be
nuule as sumll as desired. The method is to replace each integral by a laifMr oaa
which may be evaluated.
\Jo A + iy I Jo |^+<y|' ' Jo A A
These changes involve the facts that the integral of the absolute raloe Li aa great
as the absolute value of the integral and that e*^ - » = e'^e- », |<'^| = l,|^ + ^|>il,
e-y<l. For the relations je'-^lzrl and \A ■\-iy\>A^ the interpretation of the
quantities as vectors suffices (§§ 71-74) ; that the integral of the abaolate ralae la
as great as the absolute value of the integral follows from the same fact for a aiun
(p. 154). The absolute value of a fraction is enlargetl if that of lU numerator ia
enlarged or that of its denominator diminished. In a similar
\Ja z-\-iB'\^J-AB B IJa-A^iy^^A
Furthermore I C''-^\^ C \^\t^ = r'NI**^
I •/—a Z I J-a \Z\ •'0
/• + « d« _ /»o re»<td0
J-a z Jw re*'
Then 0= r£?dz= r^t^dx-^i + /e, | R| < J^ +t.-*4 + ^^
Jo z Ja X A a
where < is the greatest value of jij] on the semicircle. Now let the raotaagto b«
so chosen that A = Bei^; then | R| < 4 c~ i * + »f. By taking B watttlk
e~ i ^ may be made as small as desired ; and by taking the aemlolrBla
• It is also iMissible to integrate along a aemtelrele from J to - J, or to <
directly from ili to the origin and separate real from Imaginary parta.
in method may be left as exerrises.
362 INTEGRAL CALCULUS
small, c may be made as small as desired. This amounts to saying that, for A sufB-
ciently large and for a sufficiently small, i? is negligible. In other words, by taking
A large enough and a small enough | may be made to differ from - by
Ja Z 2
as little as desired. As the integral from zero to infinity converges and may be
regarded as the limit of the integral from a to ^ (is so defined, in fact), the integral
from zero to infinity must also differ from ^ ir by as little as desired. But if two
constants differ from each other by as little as desired, they must be equal. Hence
i:
— = ?• (6)
X 2 ^ '
As a second example consider what may be had by integrating e*'/{z^ + k^) over
an appropriate path. The denominator will vanish when z = ±ik and there are
two points to exclude in the 2-plane. Let the integral
be extended over the closed path as indicated. There is
no need of integrating back and forth along the double
line O a, because the function takes on the same values
and the integrals destroy each other. Along the large
semicircle z = Be*"^ and dz = Rie*'>d<f>. Moreover
.«irM^ = -io ^lT^=i ^TT^ by elementary rule.,
and 0 = C -^ dz = 2 f^^^SS^ g^ ^ T e^^^'"" BieOd<p r j^^dz_
Now I e*^**^ I = I 6*^(0080 + tain *) I = I g- ^ sin <^gi.R cos <(> I — g- JBsin*^^
Moreover | R^e^** + k^ \ cannot possibly exceed R^ — k^ and can equal it only when
0 = i IT. Hence
IX
n e*^^Rie^dtp\ ^ r^ Kg-iJein.^ ^| 2Je-if«in«
/o R^e^<'i> + k^ I =Jo R^-k'^ "^^ = ^0 i?2_A:2 ^^-
Now by Ex. 28, p. 11, sin 0 > 2 ^/tt. Hence the integral may be further increased
IX
2(fr
0 li^e^it^ + A:«
p ^e ^a<f>^ ^ (e-i^-l).
Moreover, f _^!dz_ ^ r _e^ _d^ ^ r (e^.XJ^^
Jaa'a Z^ + fc* »/«a'a Z + tA: Z - ifc J««'a \2ki / Z- ik
where iy is uniformly infinitesimal with the radius of the small circle. But
Jaa'a X-ik Jaa'a Z^ + k^ 2k *'
where | M S 2 w« If « is the largest value of \ri\. Hence finally
ON INFINITE INTEGRALS M|
By taking the small circle small enough aitd tlie largv divle large
two terniH may be made aa near zero as desired. Henoe
/.
_cos»_ _»•-*
0 «« + *« IT' <^
It may be noted that, by the work of § 126, f «^ IwiC^ ls«a^
Jaa'a t -^ ki t - ki 1«^^^
and not merely approximate, and remains exact for any closed curre about gmki
which does not include z =— ki. That it is approximate in the small dids folkms
innnediately from the continuity of e^/{z + M) =s e- V^M -f f and a dlraci InU-
^'ration about the circle.
As a third example of the method let f" ^ — dz be evaluated. This InUcnl
will converge if 0 < a < 1, because the infinity at the origin is then of onlsr Iss
than the first and the integrand is an infinitesi-
mal of order higher than the first for large values
of X. The function z«- V(l + z) becomes infinite
at z = 0 and z = — 1, and these points must be
excluded. The path marked in the figure is a
closed path which does not contain them. Now
here the integral back and forth along the line
aA cannot be neglected ; for the function has a
fractional or irrational power z«-i in the nu-
merator and is therefore not single valued. In
fact, when z is given, the function z*-* is deter-
mined as far as its absolute value is concerned, but its angle may taks oo any
addition of the form 2 7rk{a — 1) with k integral. Whatever value of the
is assumed at one point of the path, the values at the other points most
as to piece on continuously when the path is followed. Thus the values
line aA outward will differ by 2ir(a — 1) from those along ^a inward
the turn has been made about the origin and the angle of x has increased by Sv.
The double line be and c6, however, may be disregarded because no turn aboat the
origin is made in describing cdc. Hence, remembering that <^ = — 1,
J A 1 -I- re*'" Jabba 1 + X J<^€ 1 + f
A Y411-1 /*afm-\fA9wi pA fM^l
Now
J. l + r J A 1 + r J. 1 + r'
Mo \ + AeK \ Jo A-l\ I ^-1
MoN>al + r I |Jf»l + ae*< I Ja 1-a l-«
364 INTEGRAL CALCULUS
.*el + « J 1 + z
Hence O = (l-c«-'0r ^ dr + 2 7rie'«' + f , |r|<- 7 + — -
Ja 1 + r ^ — 11 — a
If il be taken suflBciently large and a sufficiently small, f may be made as small
M desired. Then by the same reasoning as before it follows that
0 = (1 — e*»«"*) r* dr + 2 iHe^^^, or 0 = — sin ira f dr + tt,
./o 1 + r t/o 1 + r
f __da; = -^ (8)
0 1 + « sin nrir
143. One integral of particular importance is I e'^^dx. The evalu-
ation may be made by a device which is rarely useful. Write
XA r r^ r^ 1^ r r^ r^ i^
e-'^dx^l I e-'^dxl (i-y'dy\^\ I \ e-^'-^'dxdy\ .
The passage from the product of two integrals to the double integral
may be made because neither the limits nor the integrands of either
integral depend on the variable in the other. Now transform to polar
coordinates and integrate over a quadrant of radius A.
f f e-^-y^dxdy^ f' f e-'-'rdrde -^ R = j7r(l - e-^')-h R,
c/O Jo Jo Jo ^
where R denotes the integral over the area between the quadrant and
square, an area less than ^ A^ over which e''''^ ^ e~^'. Then
A r*A
<iAh-^\
R<iAh-^\ f f e-^-y'dxdy-iir
Jo Jo
Now A may be taken so large that the double integral differs from \ ir
by as little as desired, and hence fo^ sufficiently large values of A the
simple integral will differ from \ Vtt by as little as desired. Hence *
/ e~^dx = ^ Vtt.
(9)
• It should be noticed that the proof just given does not require the theory of infinite
double Integrals nor of change of variable ; the whole proof consists merely in finding
a number i V^ from which the integral may be shown to differ by as little as desired.
Thi« was also true of the proofs in § 142 ; no theory had to be developed and no limiting
prooeMM were used. In fact the evaluations that have been performed show of them-
selves that the Infinite Integrals converge. For when it has been shown that an integral
with a Urge enough upper limit and a small enough lower limit can be made to differ
from a cerUin c^msUnt hy as little as desired, it has thereby been proved that that
lotegral from rero to Infinity must converge to the value of that constant.
ON INFINITE INTEGRALS S«5
When some infinite integrals have been evaluated, oihen may be
obtained from them by various oi)erations, such as integimtion bj parte
and change of variable. It should, however, be borne in mind thai the
I ules for operating with definite integrals were established onlj for
tinite integrals and must be reestablished for infinite integiak. From
the direct application of the definition it follows that the integral of
II function times a constant is the product of the constant by the
integral of the function, and that the sum of the integrab of
functions taken between the same limits is the integral of the
of the functions. But it cannot be inferred conversely that an integral
may be resolved into a sum as
r'[/(')+ *(*)]<*« = /*/(')<*« + /*'♦(')««»
when one of the limits is infinite or one of the functions beoooiee
infinite in the interval. For, the fact that the integral on the left
converges is no guarantee that either integral upon the right will
(;onverge ; all that can be stated is that if one of the integrals an the
riffht convergesy the other willj and the equation will be tme. The
same remark applies to integration by parts.
If in the process of taking the limit which is required in the defi-
nition of infinite integrals, two of the three terms in the equation
approach limits, the third will approach a limits and the equation will
be true for the infinite integrals.
The formula for the change of variable is
f f(.x)dx= /[♦«]*'(0<".
where it is assumed that the derivative ^'(^ is continuous and does
not vanish in the interval from t to T (although either of theil
ditions may be violated at the extremities of the interval). As
two quantities are equal, they will approach equal limits, provM*^
they approach limits at all, when the limit
f{x)dx= f[i^m^'(t)dt
required in the definition of an infinite integral is taken, where one of
the four limits a, b, t^, t^ is infinite or one of the integrands
366 INTEGRAL CALCULUS
infinite at the extremity of the interval. The formula for the change
of variable is therefore applicable to infinite integrals. It should be
noted that the proof applies only to infinite limits and infinite values
of the integrand at the extremities of the interval of integration ; in
case the integrand becomes infinite within the interval, the change of
variable should be examined in each subinterval just as the question
of convergence was examined.
/* * sin X TT
As an example of the change of variable consider f dx = — and take x = ax\
Jo X 2
Jf*-«8inax' , , /' + *8ina'x' , /•-* sin ara;' , , r^'=«sinQ:a!' ,
— ; — ax'= I — ; — ax or = / — ; — aa;' = — i dx\
x-O X Jx' = 0 X t/x' = 0 X Jx' = 0 X'
according as or is positive or negative. Hence the results
r?!lL^dx = +^ if a>0 and -"- if a < 0. (10)
Jo _ X 2 2 ^ '
Sometimes changes of variable or integrations by parts will lead back to a given
integral in such a way that its value may be found. For instance take
- 0 -
/= r^logsinxdx = — f logcosydy= f^logcosydy, y=i- — x.
*fo Jw Jo 2
2
Then 21= P(logsina; + logcosx)dx = C^log^^^^dx
«/o Jo 2
- IT
= - C log sin xdx-^ log 2= f^logsinxcte- -log2.
2 «/o 2 »/o 2
n
Hence I = T ^ log sin xdx = — - log 2. (11)
Here the first change was y = \'jr — x. The new integral and the original one
were then added together (the variable indicated under the sign of a definite inte-
gral is immaterial, p. 26), and the sum led back to the original integral by virtue
of the substitution y = 1x and the fact that the curve y = log sin x is symmetrical
with respect to x = J tt. This gave an equation which could be solved for I.
EXERCISES
1. Integrate / ' , m for the case of (7), to show f"^^'"^ dx = - c-*.
«* + ** Jo x2 + ifca 2
2. By direct integration show that C e- (a-fcO*d« converges to (a - bi)-\ when
a > 0 and the integral is extended along the line y = 0. Thus prove the relations
XV"coete<te = -jA_. _£V».i„ted. = -jA_. „>o.
Along what linni Imuing from tlie origin would the given integral converge ?
ON INFINITE INTEGRALS 867
3. Show f'^^Zl^=iLfL5!lE. To\nttgniB9hooiMm^itmthBUmmi9i
Jo (1 + X)* dn air —••••• •hipi»w
expansion ««-» = [- 1 + 1 + «]«-» = (- l)*-»[l ^. (I -«)(!+ t) + ^(| ^ g)j^
t; Ninall.
4. Intf^'rato e-«* aroiuul a circular sector with vertex at x - 0 ainl b<#u!Hl««l by
the real axis and a line inclined to it at an angle of ^ w. Hence fthuw
ei " r*(co8r» - < sin r^ dr = C^e-^dz = — .
*/o Jo S
r*co8x«dx = r*8inx«dx = i .^.
5. Integrate e- ** around a rectangle y = 0, y = B, z = i-^, and diow
r e-''co8 2axdx = jVire--', f e-'^ainaaciis asO.
6. Integrate z* -^c- «, 0 < or, along a sector of angle q<\w\o ahow
secag \ x« - »e- *•<>•» cos (x sin 5) dx
= c8caq i x*-ic-*««««8in(x«in7)dx= r x*-*«-^te.
7. Establish the following results by the proper change of variable :
, , r^cosffx , ire-** ^ ,^, /••x*-VLc «<>■"' ^ ^^
(7) re-<^^dx = ^V^. (a) /"e— -Ld« » Jl,
Jo 2 a Jo -y/jj ^a
e-**^co8tedx = , a > 0, (f) / "_ = Vir,
0 2 a Jo V— logx
. . /'"C08X, /""slnx, /ir ,^ r^loaxdz », ^
Jo Vx "^o Vx ^^ •'0 Vl — X* *
8. By integration by parts or other devices show the following :
^ xlog8inxdx = --iraiog2, {fi) j -^-<«» = r-»
0 2 Jo X* »
, . /* • sin X cos arx - v., , . w ,. . , aki^i^i
(7) / dx = - if -l<a<l, or- If dr= i 1, orOlf|a|>l,
Jo X 2 4
(Or(a + l) = ar(cr)ifr(a')=jr*x— le-'dx, <^> X'f+cS^i'^ T*
(^) flog (x + -^ -^ = w log a, by virtue of x = tan r
Jo \ x/ 1 + ac
368 INTEGRAL CALCULUS
9. Suppose f*f{x) — . where a > 0, converges. Then if p > 0 , g > 0,
Ja X
Jo X aiOL*'a ^ "^P" ^ *'9« ^ -I
Show r-/(t")-/to^) d, = Hm rnx) ^ =/(0) log » .
Jo X aiO Jpa iC p
Hence («) f" ""P^-""^^dx = 0, (fi) f^H^Ox = log£.
^ ' Jo X Jq X p
J^ixp-i — a:«-i , , (7 ,., /»« cosx — cosax - ,
^- ^l-dx = log^, 8 / ;; dx = loga.
0 logx p Jo X
10. If /(x) and/'(x) are continuous, show by integration by parts that
lim f fix) sin fcxdx = 0. Hence prove lim f f{x) — — dx = ^/(O).
Apply Ex. 6, p. 359, to prove these formulas under general hypotheses.
Jf% h sin lex
f{x) dx = 0 if b > a > 0. Hence note that
a X
lim lim rV(x)— — dxTi lim lim f^f{x)^^^dx, unless /(O) = 0.
Xrsao a»0 Ja X a = Oifc = oc Jo X
144. Functions defined by infinite integrals. If the integrand of an
integral contains a parameter (§ 118), the integral defines a function of
the parameter for every value of the parameter for which it converges.
The continuity and the differentiability and integrability of the func-
tion have to be treated. Consider first the case of an infinite limit
f(x, a)dx= I /(x, a)dx + R (cc, a), 72 = | /(a;, a) dx.
%J a *Jx
If this integral is to converge for a given value a = a^, it is necessary that
the remainder R (xj a^ can be made as small as desired by taking x large
enough, and shall remain so for all larger values of x. In like manner if
the integrand becomes infinite for the value a; = 5, the condition that
fix, a)dx= j f(x, a)dx -\-R(x, a), R= I f(x, a)dx
converge is that R (Xy aj can be made as small as desired by taking x
near enough to /->, and shall remain so for nearer values.
Now for different values of a, the least values of x which will make
I /? (aj, or) I s «, when c is assigned, will probably differ. The infinite inte-
grals are said to converge uniformly for a range of values of a such as
ON INFINITE INTEGRALS 869
tx^^a^ a^ when it is possible to take x so large (or « to near b) that
I It (x, a)\< € holds (and continues to hold for all larger Tallies, or valoea
nearer b) siniultaneously for all values of a in the raoge a^Saftc,.
The most useful test for uniform convergence is oontained in the
theorem : If a jmsUivefuncti/m ^(x) e4in be found tueh thai
f
4> {x) dx converges and ^ (x) S |/(ac, a) |
for all large values ofx and for all values of a in ths interval a, S a S a »
the integral of f(x^ a) to infinity converges uniformly (and absnluUly)
for the range of values in a. The proof is contained in the relation
f(x,a)dx\^j i,(x)dx<€,
whi('h holds for all values of a in the range. There is dearly a Bimikr
theorem for the case of an infinite integrand. See also Ex. 18 below.
Fundamental theorems are : ♦ Over any interval a^^ a ^ a^ where
an infinite integral converges uniformly the integral defines a
tinuous function of a. This function may be integrated over any
interval where the convergence is uniform by integrating with respect
to a under the sign of integration with respect toz. The function may
be differentiated at any point a^ of the interval a^^aS a^hy differ-
entiating with respect to a under the sign of integration with respect
to x provided the integral obtained by this differentiation converges
uniformly for values of a in the neighborhood of a^. Proofs of these
theorems are given immediately below, t
To prove that the function is continuous if the convergenoe \a uniform leC
\{'(a)=j^*/(?, a)dx=£'f{x,a)dx + R(x,a), a^Safiffi,
yp (a + Aa) = /"/(«, a + Aa)dx + R(x, a + Aa).
\Ai^\^\f\f{x, a + Anr) -/(x, or)] dx | + | R (x, a + Atf)| + |B(*, a)\,
• It is of course assumed that/(x, or) I« continuous In (ar, a) for all raloM of « aad a
under consideration, and in the theorem on differentiation It U further aMuniad Itet
/^ (x, a) is continuous.
t It should be noticed, however, that although the coadltlona whkb hav« Um
\m\wsed are sufficient to esUiblish the theorenjs, they are noi mcamarj/; tbat la. It aay
hiippon that the function will he continuous and that \u deriratlTe and lalefral aay W
(.btained by operatinj; under the sign although the convergence to »ot nolforw. totfcit
(Hse a special investigation would have to be undertaken ; and If no proeeM far |««fytaC
tlje continuity, integmtion, or «lifferentiation could be dertoed. It mlgitt b» aaeawMy Is
the case of an integral (K'curring in s<ime a])plication to aMume that the fonMl ^^^^***
to the right result if the result looked reasonable from the point of view of >^ I
under discussion, — the chance of getting an erroneoua reettll would b«
370 INTEGRAL CALCULUS
Now letx be taken so large that |R|<e for all a's and for all larger values of x
— the condition of uniformity. Then the finite integral (§118)
f'f{x, a)dx is continuous in a and hence J [f{x, a + Aa) —f{x, a)] dx
can be made less than e by taking Aor small enough. Hence | A^|<3€; that is, by
taking Aa small enough the quantity | A^ | may be made less than any assigned
number St. The continuity is therefore proved.
To prove the integrability under the sign a like use is made of the condition of
uniformity and of the earlier proof for a finite integral (§ 120).
r''V(a)dar = f' f'fix, a)dxda + f'Rdx = ^ ('"'/(x, a)dadx + f.
Now let X become infinite. The quantity f can approach no other limit than 0 ;
for by taking x large enough B < e and | f | < c (nr, — or^) independently of a. Hence
as X becomes infinite, the integral converges to the constant expression on the
left and „ n^ r'^x
I \p{a)dcc= I f{x,a)dadx.
Moreover if the integration be to a variable limit for a, then
^{a)= f''rf^{a)da= f* f/Cx, a)dadx= f^F{x, a)dx.
Also f'^ F{x,a)dx\=\ f" f''f{x^a)dadx\-\ f f^fix, a)dxda <e(a: -«(,).
Hence it appears that the remainder for the new integral is less than e {a^ — oTq)
for all values of a ; the convergence is therefore uniform and a second integration
may be performed if desired. Thus if an ivfinite integral converges uniformly^ it may
he integrated as many tim^s as desired under the sign. It should be noticed that the
proof fails to cover the case of integration to an infinite upper limit for a.
For the case of differentiation it is necessary to show that
Xoo /too
/^ (x, a^) dx = <f>' (or^ ) . Consider / /^ (x, a)dx = <a{a).
As the infinite integral is assumed to converge uniformly by the statement of the
theorem, it is possible to integrate with respect to a under the sign. Then
J\{a)da = f^ £fa{^^ a)dadx = JJ [/(x, a) -/(x, a^)]dx = <t>{oc) - ip{a^).
The integral on the left may be differentiated with respect to a, and hence
<f>{a) must be differentiable. The differentiation gives u>{a) = 4>'{a) and hence
'^{^i) = ^'{^$)- The theorem is therefore proved. This theorem and the two
above could be proved in analogous ways in the case of an infinite integral due
to the fact that the integrand /(x, a) became infinite at the ends of (or within)
the interval of integration with respect to x ; the proofs need not be given here.
145. The method of integrating or differentiating under the sign of
integration may be applied to evaluate infinite integrals when the condi-
tions of uniformity are properly satisfied, in precisely the same manner as
the method was previously applied to the case of finite integrals where
ON INFINITE INTEGRALS ^j
the question of the uniformity of convergence did not ariie (f 1 119-120).
The examples given below will serve to illustrate how the method woritt
and in particular to show how readily the test for uniformity may be
applied in some cases. Some of the examples are purposely chosen idea.
tical with some which have previously been treated by other methods.
Consider first an integral which may be found by dlract lnt«gnuJoo, namdj,
f e-«'co8tedg= ^ . Compare r"e-«»d« = l.
The integrand e-<» is a positive quantity greater than or equal to r^timbm
for all values of h. Hence, by the general test, the first integral ngaided •■ a
functiun of b converges uniformly for all values of b, defines a continuous fuae-
tion, and may be integrated between any limits, say from 0 to 6. Then
J I e-"coBbxdxdb= f f e-^cmbacdbdz
0 Jo Jo Jo
/•• ,8inte_, /•* adb 6
Jo X Jo a* + 6* a
Integrate again, f f V«'-l^dMx = f V-l^Hf^^dx
Jo Jo z Jo X*
= 6 tan-» - - ^ log (a« + 6»).
1 — cos to J , /**1 — costo
Compare f%-»xl^:^dx and f
Jo X* Jo
dx.
Now as the second integral lias a positive integrand which is never leas than the Inta-
grand of the first for any positive value of a, the first integral oonveiiges uniformly
for all positive values of a including 0, is a qontinuous function of a, and the value
of the integral for a = 0 may be found by setting a equal to 0 in the integrand. Then
The change of the variable to x' = ^ x and an integration by parts give respeetivdy
Jr»*sin2to, ir .. , /'•sin to. r w . ^ . ^
— — (Zx = -6, / dx=+- or --, as 6>0 or 6<0l
0 z* 2 Jo z 2 2
This last result might be obtained formally by Uking the limit
1- r* -,8l"to , /'•sinto. ^ ,6 . »
hm / e-«« dx= I dz = Un->-=±-
a = oJo Z JoZ OS
ifter the first integration ; but such a process would be unjustifiable without fint
iiowin<; that the integral was a continuous function of a for small positive values of a
indforO. In this case jz-^ e- "'sin to|^|z-i sin z|, but as the integral of jx^'dnla)
iloes not converge, the test for uniformity fails to apply. Hence the limit would not
be justified witliout special investigation. Here the limit does give the right result,
but a Kimple case where the integral uf the limit is not the limit of the IntsgnJ Is
,. /'•sin to, ,j / tX w /••„ sin to. /••O. ^
Inn / dz = lim(db-) = db - ^ I Hm dx I -dx^Q,
6^0 Jo z 6Jbo\ 2/ S Jo »A0 X Jo X
872 INTEGRAL CALCULUS
As a second example consider the evaluation of j e \ xj dx. Differentiate.
To justify the differentiation this last integral must be shown to converge uni-
formly. In the first place note that the integrand does not become infinite at the
origin, although one of its factors does. Hence the integral is infinite only by vii-
tue of its infinite limit. Suppose a ^ 0 ; then for large values of x
r\ */ [l j^e2ae-2^ and j e-^dx converges
(§143).
Hence the convergence is uniform when a ^ 0, and the differentiation is justified.
But, by the change of variable x' = — a/x, when a > 0,
Jo x^ Jq Jo
Hence the derivative above found is zero ; <t>' {a) = 0 and
<f>(a)= f e~ \~^) dx = const. = / e-^^dx = ^ Vtt ;
Jo Jq
for the integral converges uniformly when a ^ 0 and its constant value may be
obtained by setting a = 0. As the convergence is uniform for any range of values
of a, the function is everywhere continuous and equal to | Vir.
/"*
As a third example calculate the integral <p{b) = | e-^'^^costedx. Now
Jo
— =1 — xe-«^^8in6xdx = e-«*^sin6x f e- «''^ cos tedx.
db Jo 2a2|_ Jo 2a^Jo
The second step is obtained by integration by parts. The previous differentiation
is justified by the fact that the integral of xe- «*^, which is greater than the inte-
grand of the derived integral, converges. The differential equation may be solved.
-^ = - -^ 0, 0 = Ce 4««, 0(0) = / e--'^dx = f-.
db 2a^ Jq 2 a
Hence 0(6) = 0(O)e 4a' = / e-«*^cos&xdx =
Jo 2 a
In determining the constant (7, the function 0(!>) is assumed continuous, as the
integral for 0 (6) obviously converges uniformly for all values of 6.
146. The question of the integration under the sign is naturally
connected with the question of infinite double integrals. The double
integral I /(x, y) dA over an area A is said to be an infinite integi-al
if tliat area extends out indefinitely in any direction or if the function
fix, y) becomes infinite at any point of the area. The definition of
ON INFINITE INTEGKAL8 «TS
convergence is analogous to that given before in the mat of infimlo
simple integrals. If the area A is infinite, it is replaced by a fiaila
:iiea A' which is allowed to expand so as to cover more and mora of
! he area A. If the function /{x^ y) becomes infinite at a point or along
I line in the area A , the area A is replaced by an area A* from which tbo
singularities oi /{x^ y) are excluded, and again the area yl' is allowed to
«x{)and and approach coincidence with A. If then the double integral
extended over .4' approaches a definite limit which is indapoiideilt of
how .1 ' approaches A , the double integral is said to converge. As
jjf(x, y) dxdy = jj I / (^)| /(♦, ^; dud.,
where x = ^(*/, r), y = ^(?<, r), is the rule for the change of Tariable
and is applicable to A\ it is clear that if either side of the equality
a})proaches a limit which is inde})endent of how A* approaches ^4^ the
other side must approach the same limit.
The theory of infinite double integrals presents numerous difficultieay
the solution of which is beyond the scope of this work. It will be siiA*
cient to ]K)int out in a simple case the questions that arise, and then
state without ])roof a theorem which covers the cases which arise in
practice. Suppose the region of integration is a complete quadrant so
that the limits for x and y are 0 and oc. The first question is, If the
double integral converges, may it be evaluated by successive inti^Lrm-
tion as
Cf{x,y)dA=r rf(x,y)dydx=f' f /{x, y)didyT
And conversely, if one of the iterated integrals converges so that it may
be evaluated, does -the other one, and does the double integral, converge
to the same value ? A part of this question also arises in the case of a
function defined by an infinite integral. For let
</»(^)= r/(^,y)^y and r%(x)rfx=r r /(x,^)^^*,
it being assumed that ^ {x) converges except possibly for certain ndnea
of jr, and that the integral of <^ (x) from 0 to oo converges. The question
arises, May the integral of ^ {x) be evaluated by integration under the
sign ? The proofs given in § 144 for uniformly convergent integrals inte-
grated over a finite region do not apply to this case of an infinite inte-
gral. In any ])articular given integnd special methods may possibly be
devised to justify for that case the desired transformations, l*ut most
cases are covered by a theorem due to de la Vall^e-Poussin : If lAs
374 INTEGRAL CALCULUS
function f(xy y) does not change sign and is continuous except over a finite
number of lines parallel to the a^es of x and y, then the three integrals
Cf(x,y)dA, r r f(x,y)dydx, f f f(x,y)dxdy, (12)
cannot lead to different determinate results ; that is, if any two of them
lead to definite results, those results are equal* The chief use of the
theorem is to establish the equality of the two iterated integrals when
each is known to converge; the application requires no test for uni-
formity and is very simple.
As an example of the use of the theorem consider the evaluation of
Jo Jo
Multiply by e-*' and integrate from 0 to oo with respect to a.
Jo Jo Jo Jo
Now the integrand of the iterated integral is positive and the integral, being equal
to /*, has a definite value. If the order of integrations is changed, the integral
r* r*«e-«'(i+^)dadx= r''_^^ = ltan-ioo = -
Jo Jo Jo l + x2 2 2 4
is seen also to lead to a definite value. Hence the values I^ and \ ir are equal.
EXERCISES
1. Note that the two integrands are continuous functions of (x, a) in the whole
region O^ar<oo, 0^x<oo and that for each value of a the integrals converge.
Establish the forms given to the remainders and from them show that it is not pos-
sible to take X so large that for all values of a the relation \R{x, a:) | < c is satisfied,
but may be satisfied for all or's such that 0 < «(, ^ a. Hence infer that the conver-
gence is nonuniform about or = 0, but uniform elsewhere. Note that the functions
defined are not continuous at or = 0, but are continuous for all other values.
(a) J ae-«^dx, E{x,a)=C ae- «^ dx = e- o"^ — \,
ta\ r* sin ax, „, . r'*>B\nax. /^*sinx ,
(/3) I dx, R (x, a) = / dx = I dx.
Jo X Jx X Jax X
2. Repeat in detail the proofs relative to continuity, integration, and differ-
entiation In case the integral is infinite owing to an infinite integrand at x = 6.
• The theorem may be generalized by allowing /(«, y) to be discontinuous over a
finite number of curves each of which is cut in only a finite limited number of points
by lines parallel to the axis. Moreover, the function may clearly be allowed to change
sign to a certain extent, as in the case where / > 0 when x>a, and / < 0 when 0 < x < a,
etc., where the integral over the whole region may be resolved into the sum of a finite
number of integraU. Finally, If the integrals are absolutely convergent and the integrals
o'l/(»t y)\ lead to definite reaults, so will the integrals of/(x, y).
ON INFINITE INTEGRAX8 875
3. Show that differentiation under tba lign Ui allowabto la Um follovlaf ^
ami hence derive the resulut that are given :
^0 2\^ Jo f>a*^l
_rl-8...(>ii~|)
w*ooi<nr
r*x-dx = -i-, n>-l, r*x-(-logx)-dx = ?1^
/ :; dx = -- ' 0<a<l, I — - — ~S-dx = —
Jo 1 + z sin air Jo 1 •!> x cod* ov — 1
4. Establish the right to integrate and hence evaluate tbaae:
J* oe /* * C " '^ ff~^ ft
e-«dz, 0< «() ^ a, / dz = log - , ft, a iB <>•«
0 Jo X a
/ z«dx, - 1< a^ < a, / dx = log — — , 6, a S a,.
Jo Jo log z i>+ 1
r* /•• e"*"— «"*■ 1 6* ^ ■•*
I e-*' cos mzdz, 0 < ao ^ a, I cos mzdz = - log -—^^ — -.
Jo ' *• ' Jo z 2^a«+««
J* * /• «o g— ax _ g—ft* ft (
e-«»sinmzdx, 0<ao^a, i sin mzdlz = tAn-* tmn"*-
0 Jo z mi
r*e-«*^dz= — ^, 0<ao^a, r*^"^- e"i»dz = (ft- «)Vir.
Jo 2a Jo
5. Evaluate: (a) f*e-"^^^dx=tMH-^^,
^ ' Jo X a
/•« ,1 — COBOZ , , rr— ; , , /•• ,»ln2<r»
I e-' dz = logvl + nr*, (7) f f^ «,
Jo z Jo
6. If 0 < a < 6, obtain from f "e-^dx = - \- and juKtifv the relatloiM:
Jo 2 \ J"
/ — -dr = — zz f \ e- '^sin rdzdr = — = \ f e"'^tinrdrdz
Ja -y/f y/ir*''^ •'^ Vw'* *^*
2 r. /'•e-'^z^dz . - /••e-»-^z«dr
= — — sin a I Bin ft I -— — -r-
^l Jo \-\-i* Jo l + z«
--..ooeft/ - . , It
/•'^sinr^ /i^ 2r, /»«g->^x«dz . „._r*^"^1
376 INTEGRAL CALCULtJS
., . /•'•cosr^ fir 2r n'^ e-^^x^dx . r* e-^dx\
Similarly, I dr=-v/ cosr | sinr | .
^* Jo Vr ^2 tL Jo 1 + x* Jo 1 + x* J
7. Given that = 2 f* ae-«'(i+^)da, show that
1 + X* Jo
—^ — -dx = -(1 + e-'») and / -dx = -e-'«, m > 0.
0 l + x^ 2^ ' Jo H-x2 2
J^ * X sin crx
' g dx, by integration by parts and also by substi-
X 1 "T X
tuting x' for ax, in such a form that the uniform convergence for a such that
0 < aQ ^ a is shown. Hence from Ex. 7 prove
X«xsinax , ir ^ ^ ^ ,, ,.„ ^. ^. ,
— dx = — e- «, a > 0 (by differentiation).
Show that this integral does not satisfy the test for uniformity given in the text ;
also that for a = 0 the convergence is not uniform and that the integral is also
discontinuous.
9. If /(x, or, /3) is continuous in (x, a, /3) f or 0 ^ x < oo and for all points (a, /9)
/»00
of a region in the cr/S-plane, and if the integral <f>{a, /3) = | /(x, a, /3)dx con-
Jo
verges uniformly for said values of (a, /3), show that <f> (a, /3) is continuous in (a, /3).
Show further that if /^ (x, or, /3) and /^ (x, a, /S) are continuous and their integrals
converge uniformly for said values of (or, /3), then
jT /;(x, or, ^)dx = <f>^, X*"^^^^' ^' '^^^ "^ ^^'
and 0^, 0^ are continuous in (or, /3). The proof in the text holds almost verbatim.
10. If /(x, 7)=/(x, a + i/3) is a function of x and the complex variable
7 = a + t^ which is continuous in (x, a, /3), that is, in (x, 7) over a region of the
7-plane, etc., as in Ex. 9, and if /^(x, 7) satisfies the same conditions, show that
Jr»oo
/(x, 7) dx defines an analytic function of 7 in said region.
0
11. Show that J e-y^dx, 7 = a + i/3, a ^ otq > 0, defines an analytic func-
tion of 7 over the whole 7-plane to the right of the vertical a = a^. Hence infer
'Jo 2 \7 2 \a + i/3
a ^ OTo > 0.
Jo "^ 2 \2 «a 4. fl2
Jo 2 >2 a* + i3a
ON INFINITE INTEGRALS ITT
12. Integrate / -er^^xcMfizHz of Ex. 11 bj paru with zvmB^^m4m
to show that the convergence U uniform at a = 0. Henea find / mw/mKi,
13. Yrom J^Jco»xHz=f^*coB{Z'¥a)*dxzz^=J*\n{fi'^a^4M,wtlk
/• + • /•■♦■•
the re8ult8 f coe 2> sin 2 nrxdx = J sin x* tin S aaedx s 0 dua Co Um fact ckai
sin z is an odd function, establiHii tiie relations
i cos a:'' cos 2 oxdx = —-cos/- — a* j. f stox^oosSoadbB b — ^dn/- — a*].
14. Calculate: (a) f e- «^ cosh iccdx, (fi) f^ze- «*oci«tedi,
and (together) (7) J^* cos (^ ± ^) dx, (8) jT "sin ^^ i ^J d«,
15. In continuation of Exs. 10-11, p. 868, prove at least formally the rtlitlow 1
k-»*/-a X I k»m»wJ-n X
f^f''f{x)co8kxdxdk=f''f''/{x)coBkxdkdx=f'/(x)^^^^
- r* f''/{x)coakxdxdk= lim i f /{x)^^dx=i/{%
- r* r*/(a:)co8fcEdxdA:=/(0), - f* r*/(z)coslc(x- OtottB/(0.
The last form is known as Fourier's Integral ; it represents a funcUoo /(O at a
•louble infinite integral containing a parameter. Wherever poasible. Justify tha
steps after placing suflBcient restrictions on/(x).
16. From r*e-»i'dy = - prove T*^! — Z-5I_ dat = log - • Prorealn
Jo . X Jo X a
Jz^-^e-'dx f z*»-Je-'dx
0 Jo ^
= 2 r*ra"+««-«e-'^dr« f •sin««->^ooa«— >^d^
17. Treat the integrals (12) by polar coordinates and show that
f/{x, v)dA=£^f^'"/{rcM^,r^nf)rdrd^
will converge if |/| < r-*-* as r becomes Infinite. If/(x,|f)
origin, but |/| < r-* + *, the integral converges as r approaches tero.
these results to triple integrals and polar coordinates In space ; the only
is that 2 becomes 3.
18. As in Exs. 1, 8, 12, uniformity of conveiigenoe may often be leeied direaft^,
without the test of page 309 ; treat the integrand r^€"*dnbtttd pegeSTl,'
that test failed.
CHAPTER XIV
SPECIAL FUNCTIONS DEFINED BY INTEGRALS
147. The Gamma and Beta functions. The two integrals
f a;"-ie-^c?x, B(m, «)= / x'^-\l - xy-^dx (1)
0 Jo
converge when n > 0 and m > 0, and hence define functions of the
parameters n or n and m for all positive values, zero not included.
Other forms may be obtained by changes of variable. Thus
V(n) = 2J f^-'e-y'dy, by x = y\ (2)
rW=jr(log^J dy, by e-' = y, (3)
B{m,n)=jr-\^-yT-^dy = B{n,m), by x = l-y, (4)
BK^)=X"(ia^^ by . = ^, (5)
TT
B (m, n) = 2 r ^in^^-i*^ cos2«-i<^^<^, by a; = sin^ <^. (6)
If the original form of V(n) be integrated by parts, then
r(n) = r a»-ie-'c?a; = - ic»e-^ + - f a;"e-^<?a; = - r(n -h 1).
Jo ^ Jo ^Jo n ^ ^
The resulting relation r(7i + 1) = nV{n) shows that the values of the
r-function for n -h 1 may be obtained from those for n, and that con-
sequently the values of the function will all be determined if the values
over a unit interval are known. Furthermore
r(n + 1) = nV{n) = n{n- l)V(n - 1)
= n(7i - 1) . . . (/I - k)V{n - k) ^^
is found by successive reduction, where k is any integer less than n.
If in particular n is an integer and k = n — l, then
r(n H- 1) = n(n - 1) . . . 2 . 1 . r(l) = n ! r(l) = n ! ; (8)
878
FUNCTIONS DEFINED BY INTEGRALS S79
since when n = 1 a direct integration shows that r(l) m L Thof/br Ail^
gral values ofn the T-furutlon is the factorial ; and for other than inlmtl
values it may l)e regarded as a sort of generalization of the ibctorkL
Both the r- and B-functions are continuous for all valnee of the
parameters greater than, but not including, zero. To prore thie It it
sufHcient to show that the convergence is uniform. Let n be any taIim
in the interval ^ <n^^n^ N\ then
The two integrals converge and the general test for uniformitj (f 144)
therefore applies ; the application at the lower limit is not neoeeaary
except when n < 1. Similar tests apply to B(m, n). Integratkm with
respect to the parameter may therefore be carried under the sign. The
derivatives d''V(n) C*
"^ Jo ^"'""(^^'^)*'^ W
may also be had by differentiating under the sign ; for these deriTed
integrals may likewise be shown to converge uniformly.
By multiplying two T-functions expressed as in (2), treating the
product as an iterated or double integral extended over a whole quad*
rant, and evaluating by transformation to polar codrdinates (all of
which is justifiable by § 146, since the integrands are poeitiTe and
the processes lead to a determinate result), the B-f unction may be
♦'xpressed in terms of the T-f unction.
r(n)r{m) = i:f x^'^-'e-^'dx Py*— V»^rfy=4 f'Tx^'-y^W^^ibBi^
*/0 %Jo 4/0 */o
= 4 r r2'' + «'"-ie-^rfr r%in«'"-^<^cos*"-»^^ = r(n + m)B(fisii).
Jo Jo
Hence B (m, n) = ^W^W = B (n, m). (10)
^ ' ^ r(m4-n) ^'^ / ^ /
The result is symmetric in m and n, as must be the case
as the B-function has been seen by (4) to be symmetric
That r (i) = Vtt follows from (9) of § 143 after setting » - ( in (3);
it may also be deduced from a relation of importance which is obtained
from (10) and (6), and from (8) of § 142, namely, if » < 1,
r(n)r(i-n) „, ^ , r*y"'' ^ y
= 1 ^ = B(n,l-n)=J^ iTy^''
r(l)
sin
or r(n)r(l-n)=-: (Ji>
380 INTEGRAL CALCULUS
As it was seen that all values of T(n) could be had from those in a
unit interval, say from 0 t/O 1, the relation (11) shows that the inter-
val may be further reduced to ^ ^ n ^ 1 and that the values for the
interval 0 < 1 — w < ^ may then be found.
148. By suitable changes of variable a great many integrals may
be reduced to B- and T-integrals and thus expressed in terms of
r-functions. Many of these types are given in the exercises below;
a few of the most important ones will be taken up here. By y = ax,
J' x'*-\a — xy-^dx = a'« + '-i / y^-\l — yy-Hy = a"+"-iB(m, n)
0 t/o
or r a:--i(a-x)«-i<^a; = a™ + ''-»^;^?^^i^^, a > 0. (12)
Jo r(m -h n) ^ ^
Next let it be required to evaluate the triple integral
1=111 x^-^y^-^z^-^dxdydzj x -{-y + z^lj
over the volume bounded by the coordinate planes and x -{- y -{- z = 1,
that is, over all positive values of x, y, z such that x-\-y ^z-^\. Then
j x^-Y^'^^'^^dzdydx
Jo
""»/"/ x'-'y»-'(l -X- yydydx.
By (12) J">-'(1 - X - y)'dy = "^^^^l^'Hf^ (1 - xf^'.
Then ° / = ryr(. + i)/-^,_^ ^^ ^.^
nT(m + n-\-l)J^ ^ ^
^ V{m)T{n + 1) T{l)T{m + n + l)
wr(m-f w-hl) T{l + m-\-n-\-l)'
This result may be simplified by (7) and by cancellation. Then
There are simple modifications and generalizations of these results which are
■ometimes useful. For instance if it were desired to evaluate / over the range
of positive values such that x/a + y/b + z/c ^ A, the change x = ah^, y = 6/117,
t = ehf gives
/ = a'6^V + - + » jyj(/-i,«-ifn-idf(i^df, f + , + f ^ 1,
•'•'•' r(i + m+n+l) ' a^b^c-
FUNCTIONS DEFINED BY 1NTEGKAL8 S81
The value of this integral extended over the Uunina hetweeo two parmlkl pliMi
determined by the values h and A + dA for the oooftant k would Iw
r(/ + m + n) *
Hence if the integrand contained a function /(A), the redoction would bt
///x.-v-U-./ (? + ? + £) drdvd.
if the integration be extended over all values x/a + y/6 -f g/e S /'.
Another modification is to the case of the integral extended over a vohmi
wliicii i8 the octant of the surface {x/a)p + (y/5)« + {z/ey = A. The iwtuelloo to
fff^''^^~^~'t''''^^f^ i + f + rai,
pqr
is made by (A = (?)", t^A = (0', fA = 0^, d« = ? A>{>"\ . . • .
r r r z ' - V" ~ *2 •* ~ ^dxdydz
a^lF^c*
r(i + - + ? + ,\
\1» g r /
A* f^^
This integral is of importance because the bounding surface here oocnrriiig It of a
type tolerably familiar and frequently arising ; it includes the ellipsoid, the snrfaes
JJ^ + y^ + z^ = cL^t the surface xl + yf + zf = a\. By taking < = m s n s 1 the
volumes of the octants are expressed in terms of the T-functlon ; by ttildQf flnt
/ = 3, m = n = 1, and then m = 3, / = n = 1, and adding the results, the inimiHi
of inertia about the z-axis are found.
Altliough the case of a triple integral has been treated, the results for a doabit
integral or a quadruple integral or integral of higher multiplicity are mado obvloas.
For example,
rrx'-iy--idxdy = a'6«A' + "-i;M<^. * + ?«*,
J J r(/ + m + l) ah
x\p . /y\«
//---4(r-m'^-«'
./M^^''
382 INTEGRAL CALCULUS
rrCf^^-iyi-iz^-it^-^dxdydzdt =
pqrs
'MMhC^^
\p Or r 8 I
-I 2 -« 1
149. If the product (11) be formed for each of - > - > • • •> j and
^ n n n
the results be multiplied and reduced by Ex. 19 below, then
The logarithms may be taken and the result be divided by n.
Now if n be allowed to become infinite, the sum on the left is that
formed in computing an integral if dx = I/ti. Hence
lim V log r (a;,.) Ax = C log T (x) dx = log V2^. (15)
Then f log T(a + x)dx = a (log a - 1) + log V27r (15')
may be evaluated by differentiating under the sign (Ex. 12 (^), p. 288).
By the use of differentiation and integration under the sign, the
expressions for the first and second logarithmic derivatives of T(n)
apd for log T (n) itself may be found as definite integrals. By (9)
and the expression of Ex. 4 (a), p. 375, for log jc,
x*'-h-''\ogxdx= j a;"-^^-^ I dadx.
If the iterated integral be regarded as a double integral, the order of
the integrations may be inverted ; for the integrand maintains a posi-
tive sign in the region l<ic< oo, 0<a<oo, and a negative sign in
the region 0<a;<l, 0<a<oo, and the integral from 0 to oo in a:
may be considered as the sum of the integrals from 0 to 1 and from
1 to 00, — to each of which the inversion is applicable (§ 146) because
the integrand does not change sign and the results (to be obtained)
are definite. Then by Ex. l(a:).
v^
FUNCTIONS DEFINED BY INTEGRALS
" ?g-£'-^<")-r(--<rb)T- m
This value may be simplified by subtracting from it Uie |i«**i<^|ftf
value - y = r'(l)/f(l)= r(l) found for n = 1. Then
r(n) r(l) r(n)^^ j, Vl + a (l + a//«
The change of 1 + a to 1/a or to «• gives
FW^^=Jo "T^^'^^Jo 137^-^ 07)
Differentiate: y-^logr(n)=/ r rj*'^- (W)
To find log r (n) integrate (16) from n s 1 to n =s n. Then
"'-'•'-Xl'->--"":K'.f'-]f. <■-
since r(l) = 1 and log r(l) = 0. As r(2) = 1,
and log r(») = / -;— — "5 - ^ ^ ^ ^ — 77-7— r
by subtracting from (19) the quantity (» - 1) log r(2) = 0. Finallj
iogr(«)=£j95f-(n-iK]^ rm
if 1 4- rr be changed to e'". The details of the redocttons and the jiuU-
tication of the differentiation and integration will be left as exerdses.
An approximate expression or, better, an tuympMic exprmaiamf
I that is, an expression with small percentage error, may be found for
T{n + 1) when n is Uirge, Choose the form (2) and note that the inte-
grand rf^^e'^ rises from 0 to a maximum at the point y* « 11 -f | and
falls away again to 0. ^Make the change of variable y = Va + ^t where
a = 71 4- ^, so iis to bring the origin under the maiimnm. Then
r(n H- 1) = 2 r ( Vcr -f- wy'e-'-^^^'^dw,
^ Tin + 1) = 2«^.-J*^a*-K-;7:)-^— ',/«,.
Now 2alog^l4- 7^)-2Viir^0, - Vi < ir < co.
884 INTEGRAL CALCULUS
The integrand is therefore always less than e~ "'*, except when t^; = 0
and the integrand becomes 1. Moreover, as w increases, the inte-
grand falls off very rapidly, and the chief part of the value of the
integral may be obtained by integrating between rather narrow
limits for w^ say from — 3 to 4- 3. As a is large by hypothesis,
the value of log(l + w/^/a) may be obtained for small values of w
from Maclaurin's Formula. Then
T(n -h 1) = 2 0-^6-" r e-^'^'^i-'^di
is an approximate form for V(n-\- 1), where the quantity c is about
§ w/ Va and where the limits ± c of the integral are small relative to Va.
But as the integrand falls off so rapidly, there will be little error made
in extending the limits to oo after dropping c. Hence approximately
r(w + 1) = 2 afe-" I e-^'^^dw = V^o^e-",
«y — 00
or T{n + 1) = V2^(n + i)«+i e-(" + i)(l + ,;), (20)
where iy is a small quantity approaching 0 as w becomes infinite.
EXERCISES
1. Establish the following formulas by changes of variable.
or) r(n) = a« r*x"-ie-«^dx, a:>0, (/3) P sin«xcte = is /^ + i, iV
7) B(n, n) = 2i-2»»B(n, i)by(6), (5) r^ic'«-i(l - x2)«-i(to = J B(^m, n),
a;m-i(i_a.)n-i^^ B(m,n) ^ 1 ^ (m) r (n) ^^^^^ ^ ^ ^
(z + a)"« + »» a»>(l + a)'» a»(l + a)»» r (m + n) ' x + a 1 + a'
1 z"«-i(i_a;)"-idx _ V{m)T (n) . , 6y
a{\-y)-\-hy
Jo [ax + 6(1 — a;)]'« + « a'«6«r(m + n)*
1,) r^g"'-Hl-g)"-^dx^ B(m, n) /»i xMx ^V^r(^n-h^)
./o (6 + cx)'» + « 6"(6 + c)»»* ^ Wo Vr^x2 2 r(in + l)*
•/o ' n V ' n J ^Uo Vr=^ n r(n-i + i)
2. From r (1) = 1 and r (J) = Vv make a table of the values for every integer
and half integer from 0 to 6 and plot the curve y = r (x) from them.
8. By the aid of (10) and Ex. 1 (7) prove the relations
v/irr(2a) = 2a«-ir(a)r(a + i), V^r(n) = 2»-ir(in)r(J n + I).
4. Given that r (1.76) = 0.9191, add to the table of Ex. 2 the values of r (n) for
every quarter from 0 to 8 and add the points to the plot.
FUNCTIONS DEFINED BY INTEGRALS $86
5. With the aid of the T-f unction proTe Umm reluloot (IM Ki. 1) i
[a) r^Hin.xdx=pco^xd» = lli±i:i<!!Lzil!: or «^ • '(«»-t)
M«U«T«aorodd.
(ti) C' ^"^ ^l»8.6...(2n-l)y /^ia;>.4mg_ 146..?
''o Vr^:i5 2.4.e...Mn 2* ^^' J« VTT^ " l.S.*...(X,
(3)X"-^'^^^^^^'dx = l^. (.)jrV(a.-xt)ld,o!^,
(f) Find r*—:^= to four decimate, M Find f' ^ .
6. Find the areas of the quadrants of these conres :
(or) xi + yJ = ai, (^) xl + y^ = a!. (>) x« + yi s 1,
( «) xVa« + yV^ = 1, (<) the evolute (ax)l + (6y)i = («• - 6»)l.
7. Find centers of gravity and inomenUi of Inertia about the axes in Ex. 6.
8. Find volumes, centers of gravity, and momenta of inertia of Um oetaaiaoC
(a) xi + yi + ^i = ai, (/3) xi + yi + zt = of. (>) x« + y« + xi s 1.
9. (a) The sum of four proper fractions does not exceed unity ; find the ETenig*
valiu> of their product, (p) The same if tho sum of the squarM doM Boi titwd
unity. (7) Wliat are tlie results in the case of k proper fractiooaf
10. Average e-**^-''*^ under the supposition ox* + 6y* ^ H^.
11. Evaluate the definite integral (15') by differentiation under the iign.
12. From (18) and 1 < — ^^—— < 1 + or show that the magnitude of D* lof r(a)
is about 1/n for large values of n.
13. From Ex. 12, and Ex. 23, p. 76, show that the error in taking
logr(n + ^) for J^"^'logr(x)dx is about _-J_ log F (a + i) .
14. Show that r" logr(x)dx= r log r (n + x) dx and hanoa oonpara (Ift*)*
(20), and Ex. 13 to show that the small quantity if Is about (24 a + 12)- >.
15. Use a four-place table to find the logarithms of 61 and 10!. Find Um
logarithms of the approximate values by (20), and determine the pereentaga errors.
16. Assume n = 11 in (17) and evaluate the first integral. Take the IqgariUNiie
derivative of (20) to find an approximate expression for r'(n)/r(a), and la
ular compute the value for n = 11. Combine the results to find y = 0.678, By
accurate methods it may be shown that Euler's Constant 7 = 0.677,216,686. • ••
17. Integrate (190 ^rom n to n + 1 to find a definite Integral for (W). Sabuacft
1 r^ «^ — e* dec „ g, ,
the integrals and add - log n = I — • Hence find
2 J— • 2 a
logr(n)-n(logn-l)-logV^+lloga==J^^^-i + 5js- — .
386 INTEGRAL CALCULUS
18. ObUin Stirling' n approzimation, T (n + 1) = V2imn»e-», either by compar-
ing it with the one already found or by applying the method of the text, with th»
substitution x = n + VTny, to the original form (1) of r (n + 1).
*=i"-i . kir . IT . 2ir . (n— l)ir n ,
19. The relation TT sm — = sin - sm sm -i '— = — — may be
jfcJi n n n n 2«-i
obtained from the roots of unity (§ 72) ; f or z« - 1 = (x - 1) TT V-c - e * / ,
l~iri . _,v^
n = lim? i= TT U-e - J, TT -^ = l^^Z7-. = ^T-. '
a-ilX — 1 k = \ ii = i 2i (2i)«-i 2'»-i
150. The error function. Suppose that measurements to determine
the magnitude of a certain object be made, and let w^, m^, • • • , m„ be a
set of n determinations each made independently of the other and each
worthy of the same weight. Then the quantities
which are the differences between the observed values and the assumed
value m, are the errors committed ; their sum is
9'i + 72 -J 1- (/n = (^1 + ^2 ^ f- ^«) - ^'^^•
It will be taken as a fundamental axiom that on the average the errors
in excess, the positive errors, and the errors in defect, the negative
errors, are evenly balanced so that their sum is zero. In other words it
will be assumed that the mean value
nm = m^ + m^ + • • • + m„ or m — - {in^ ■\- 'm.^-\- - • - -\- m^ (21)
is the most probable .value for m as determined from m^, m^, • • • , m„.
Note that the average value in is that which makes the sum of the
squares of the errors a minimum ; hence the term " least squares."
Before any observations have been taken, the chance that any par-
ticular error q should be made is 0, and the chance that an error lie
within infinitesimal limits, say between q and q -f dq, is infinitesimal ;
let the chance be assumed to be a function of the size of the error, and
write <^ {q) dq as the chance that an error lie between q and q -\- dq. It
may be seen that (^ (<y) may be expected to decrease as q increases ; for,
under the reasonable hypothesis that an observer is not so likely to be
far wrong as to be somewhere near right, the chance of making an
error between 8.0 and 8.1 would be less than that of making an error
between 1.0 and 1.1. The function <^(7) is called the error function.
It will be said that the chance of making an error qi is <^ (y,) ; to put it
more precisely, this means simply that <^ (y,) rfy is the chance of making
an error which lies between y< and y^ -f- dq.
FUNCTIONS DEFINED BY INTEGRALS SST
It is a fundamental principle of the theory of ohiAea »htt the
chance that several independent events take place is the pradool of
the cliances for each separate event The probability, thai, that tha
errors q^i q^t'"* 9m^ made is the product
<t>('/x) H^^ ■ • <^(7.) = *K - «) ♦(«», --!)••• ♦(•s - «•). (23)
The fundamental axiom (21) is that this probability is a mairimiym
when 7/1 is the arithmetic mean of the measurements « , m »•••,«.;
for the errors, measured from the mean value, are on the whola less
than if measured from some other value.* If the probability is a masi-
mum, so is its logarithm : and the di'i-ivutive of the logarithm of (22)
with respect to m is
it»'(mi-m) ^ <^'(m,-m) ^ ^ ^'('*. ~ <*) „ p
<f>(m^ — m) ^(//i, — m) ^(m, ~m)
wlien y^ -\- f/.^-\ h q^ = (m^ — m) + (m, — m) H + (w, — »i) — a
It remains to determine <f> from these relations.
For brevity let F(q) be the function F = ^'/^ which is the ratio
of <^'(«/) to <^(y). Then the conditions become
^(Yi)+^(^/-i) + --- + ^W = 0 when y, + 5r, + ... + y.-0.
In particular if there are only two observations, then
^X^/i) + ^(y,) = 0 and q, + q^=-0 or y, = -y,.
Then F(q,) + ^(-'Ii) = ^ o** ^i^ 9)='- ^q)-
Next if there are three observations, the results are
Hence F(q;) -f- F(q^ = - ^(y^ = F(- y^ = ^(7, + 7^.
Now from F(a?) + F(y) = F(a; + y)
the function F may be determined (Ex. 9, p. 45) as F(x) -• Cx. Then
and <^(y) = a*^+'=(7e*^.
This determination of ^ contains two arbitrary constants which may
be further determined. In the first place, note that C is negattve, for
if> (q) decreases as q increases. Let ^ C « — ib*. In the second place, the
• The derivation of the expression for ^ b phytleal mtlwr ^'^ '"g'^'^'^^''*'
ment. The real justiticatlon or proof of the vaUdl^ of th* •xpcwrioa uhtillii !• * ^•••
teriori and depeuds on the experience that in practice errora do follow tko !•» O*^-
388 INTEGRAL CALCULUS
error q must lie within the interval — oo < 2- < + oo which comprises
all possible values. Hence
f ^<t>(q)dq = l, G C e-'^^'^dq = 1.
»/ — 00 */ — ao
(23)
For the chance that an error lie between q and q + dq is <t}dq, and if
an interval a^q^bhe given, the chance of an error in it is
ft b ^b
2) *^ (•?) ^2' or, better, lim ^<t>(q)dq= j <t> (q) dq,
and finally the chance that — oo < ^^ < + <» represents a certainty and
is denoted by 1. The integral (23) may be evaluated (§ 143). Then
G ^Jirlk = 1 and G = k/^fir. Hence *
<^(?) = -^^-*'^'- (24)
"VTr
The remaining constant k is essential ; it measures the accuracy of
the observer. If k is large, the function <^ {q) falls very rapidly from
the large value A;/ Vtt for g- = 0 to very small values, and it appears
that the observer is far more likely to make a sm^ll error than a large
one ; but if /: is small, the function <j> falls very slowly from its value
k/^Jir for 5' = 0 and denotes that the observer is almost as likely to
make reasonably large errors as small ones.
151. If only the numerical value be considered, the probability that
the error lie numerically between q and q -\- dq is
2k , 2k r^
e-^i*dq, and —/= \ e-^^'i^dq
£
^. ,^^, .... ^
is the chance that an error be numerically less than ^. Now
2k r^ 2 r*f
is a function defined by an integral with a variable upper limit, and the
problem of computing the value of the function for any given value of ^
reduces to the problem of computing the integral. The integrand may
be expanded by Maclaurin's Formula
aj» . a;» x' x' ~" ^^^^
/■
e-^dx = x-'^-h~-^ + ^-R^ R<
3 ' 10 42 ■ 216 ' ^ 1320
• The reader may now verify the fact that, with ^ as in (24), the product (22) is a
maxiroam If the lum of the squares of the errors is a minimum as demanded by (21).
FUNCTIONS DEFINED BY INTEGRALS
For small values of x this series is satisfactory ; for z ^ } it will be
jiccurute to five decimals.
The probable error is the technical term used to denote thai error |
which makes ^(Q = ^; that is, the error such that the chance of a
mailer error is \ and the dm nee of a larger error is also \, Thk it
iound by solving for x the equation
The first term alone indicates that the root is near x v .45, and a trial
with the first three terms in the series indicates the root ae between
./■ = .47 and x = .48. With such a close approximation it it etaj to Aa
the root to four places as
a; = A:^ = 0.4769 or ^ = 0.4769 *">. («7)
That the probable error should depend on k is obvious.
For large values of a; = A:^ the method of expansion by Maclanrin't
Formula is a very poor one for calculating ^(^); too many terms are
required. It is therefore important to obtain an expantiom ofeoriimf
to descending powers of x. Now
The limits may be substituted in the first term and the method of in-
tegration by parts may be applied again. Thus
e-'*/ 1 1.3\ 1.3.5 r- e-^dx
" 2xy 2x«"^2V; 2* J, a* '
and so on indefinitely. It should be noticed, however, that the t^rm
T = 2^; ^ ^ diverges as fi» 00.
In fact although the denominator is multiplied by 2ar* at eaoh elep^ the
numerator is multiplied by 2 n — 1, and hence after the integratioiia by
parts have been applied so many times that » > as* the terms in tha
parenthesis begin to increase. It is worse than nselees to oany tba
integrations further. The integral which remains is (Ex. 5, p. 29)
390 INTEGRAL CALCULUS
■£
1.3. 5. ..(2714-1) r* e-^'dx 1-3. 5.-(2 71-1) _^
Thus the integral is less than the last term of the parenthesis, and it
is possible to write the asymptotic series
(28)
1
0.5643
kyf^
= k '
1
V^ =
0.7071
k
with the assurance that the value obtained hy using the series will differ
front the true value hy less than the last term which is used in the series.
This kind of series is of frequent occurrence.
In addition to the probable error, the average mtmerical error and the
mean square error, that is, the average of the square of the error, are
important. In finding the averages the probability <f> (q) dq may be taken
as the weight ; in fact the probability is in a certain sense the simplest
weight because the 'sum of the weights, that is, the sum of the prob-
abilities, is 1 if an average over the whole range of possible values is
desired. For the average numerical error and mean square error
•^^ - -• /oq^
Y' = ~ q'e-^^'dq
•VTrJo
It is seen that the average error is greater than the probable error, and
that the square root of the mean square error is still larger. In the
case of a given set of n observations the averages may actually be
computed as
» k-y/TT |?|Vir
p_g? + g| + -- + ?.'_ 1 ,_1
* » _ 2A.^_^ ^V2
Moreover, 7r\qf = 2qK
It cannot be expected that the two values of k thus found will be pre-
cisely equal or that the last relation will be exactly fulfilled ; but so
well does the theory of errors represent what actually arises in prac-
tice that unless the two values of k are nearly equal and the relation
nearly satisfied there are fair reasons for suspecting that the observa-
tions are not bona fide.
168. Consider the question of the application of these theories to
the errors made in rifle practice on a target. Here there are two
FUNCTIONS DEFINED BY INTEGRALS S91
errors, one due to the fact tliat the shots may fall to the right or lofl
of the central vertical, the other to their falling above or below Um
(;enti-ul horizontal. In other words, each of the ooOidinalea (^ p) ol
the position of a shot will be regarded as subject to the law of enott
independently of the other. Then
Vtr Vir w ^
will be the probabilities that a shot fall in the vertical strip befeweta
X and X -f dx^ in the horizontal strip between y and y -f </y, or in the
small rectangle common to the two strips. Moreover it will be mnnMid
tliat the accuracy is the same with respect to horizontal and refftieal
deviations, so that k — k'.
These assumptions may appear too special to be reasonable. In particoUr it
ini>,'ht seem as tliongh the accuracies in the two directions would be very dlbreut,
i)wii)<; tu the possibility that the marksman's aim should tremble more to the rifhl
and left than up and down, or vice versa, so that k^k". In this case the shots woold
not tend to lie at equal distances in all directions from the center of the taifeC,
but would dispose themselves in an elliptical fashion. Moreover ss the ^»«*«*^Ty it
(lone from the ri^tht shoulder it might seem as though there would besooi
line through the center of the target along which the accuracy would be
a line perpendicular to it along which the accuracy would be greatest, so
disposition of the shots would not only be elliptical but Inclined. To
general assumption the probability would be taken as
Ge- ^^ - 2AXV - i' V(tcdy, with G f^* /*^*^ - « a«» - ^'Vdjoiy = 1
las the condition that the shots lie somewhere. See the exercises below.
With the special assumptions, it is best to transform to polar ooOr-
dinates. The important quantities to determine are the average distanee
of the shots from the center, the mean square distanoei the probable
distance, and the most probable distance. It is necessary to distinguish
carefully between the probable distance, which is by definition the dis-
tance such that liulf the shots fall nearer the center and half fall farther
away, and the most probable distance, which by definition is that die-
tance which occurs most frequently, that is, the distance of the nng
between r and r -j- dr in which most shots fall.
The probability that the shot, lirs in the element nird^ is
- e- ^^rant4>, and 2 ;fe*e-*«'*nfr,
IT
)btained by integrating with respect to ^, is the probability tbat the
jhot lies in the ring from r to r -^ dr. The mo»t probable distance r^ it
892 INTEGRAL CALCULUS
that wnich makes this a maximum, that is,
The mean distance and the mean square distance are respectively
k^ k
The probable distance r^ is found by solving the equation
-X'
(30')
2 1
VW2 0.8326 ,^^„,
2 kh-^^rdr = 1 - e-'^'l , r^ = — ^ = — ^ (30")
Hence ^p < ^f < ^ < v ^.
The chief importance of these considerations lies in the fact that,
owing to Maxwell's assimiption, analogous considerations may be applied
to the velocities of the molecules of a gas. Let u, v, w be the compo-
nent velocities of a molecule in three perpendicular directions so that
V = (u^ -\- v^ -\- w^^ is the actual velocity. The assumption is made that
the individual components u, v, w obey the law of errors. The proba-
bility that the components lie between the respective limits u and n ■\- duy
V and V -{• dVjW and w + dw i^
*' e- *v - i«v' - i^i^dudvdw, and — ^ e- *«f* 72 sin dd VdOd^*
IT
V^ ' ttVtt
is the corresponding expression in polar coordinates. There will then
be a most probable, a probable, a mean, and a mean square velocity.
Of these, the last corresponds to the mean kinetic energy and is subject
to measurement.
EXERCISES
1. If Ac = 0.04476, find to three places the probability of an error $ < 12.
2. Compute f e-*'dx to three places for (a) x = 0.2, (/3) x = 0.8.
«/o
8. State how many terms of (28) should be taken to obtain the best value for
the Integral to x = 2 and obtain that value.
4. How accurately will (28) determine f e~^dx — \ Vv? Compute.
«/o
6. Obtain these asy-.ptotic expansions and extend them to find the general law.
Show ttiat the error introduced by omitting the integral is less than the last term
retained in the series. Show further that the general term diverges when n be-
comes infinite.
FUNCTIONS DEFINED BY 1NTE0BAL8 aM
6. (a) Find the value of tlie average of any odd power Sm^f 1 ol tbt wror;
(fi) also for the average of any even power ; (>) alao for any power.
7. The observations 195, 226», 190, 210, 206, 180», 170«, IW, SCO, tlQ, 210, M»»,
175*, 192 were obtained for deflections of a galvanometer. Conpote k twm Um
mean error and mean square error and compare the reMilta. Sappoie tiM otaerv*.
tionn marked *, which show great deviations, were dJaouded ; eoaqNite k by ibe
two methods and note whether the agreement la ao good.
8. Find the average value of the product qq' of two errom trtwtwl at nuidoai
:iii(l the average of the product |9|>|g^| of numerical valoea.
9. Sliow that the various velocities f or a gaa are F, s - , ^i^ ,
2 _ 1.1284 /yi_ Va _ 1.2247 * *
~ Vxk~ ^ * V2k~ *
10. Fur oxygen (at QPC. and 76 cm. Hg.) the square root of th* mtin iqaarv
velocity is 462.2 meters per second. Find k and show that only about It or 14
molecules to the thousand are moving as slow as 100 m./eec. What qiead la mtm
probable *?
11. Under the general assumption of ellipticity and inclination in
butiun of the shots show that the area of the ellipae Hx* -^iXzy -f k^ = H I0
irHik^k^ -\^)~K and the probability may be written Oe- «r(A«4r^ - X«)" \dB,
12. From Ex. 11 establish the relations (a) 0 = - y/W^ - X«,
2(JkSJk^_X») '' 2(*«*^-X«) " ' 2(I*I^-X«)
13. Find Hp,H^ = 0.698, 3, H* in the above problem.
14. Take 20 measurements of some object. Determine Ic by the two
and compare the results. Test other points of the theory.
153. Bessel functions. The use of a definite integral to define
tions wliich satisfy a given differential equation may be illnttnfeed bj
the treatment of xy" + (2 n + l)y' + xy = 0, which at the tame tfaae
will afford a new investigation of some functions which have pre-
viously been briefly discussed (§§ 107-108). To obtain a tolotaon of
this equation, or of any equation, in the form of a definite integral, aoine
special type of integrand is assumed in part and the ramiindur of tha
394 INTEGRAL CALCULUS
integrand and the limits for the integral are then determined so that
the equation is satisfied. In this case try the form
yW
= fe^'Tdt, y' = C ite^'Tdt, y" = j - fe^Tdty
where r is a function of t, and the derivatives are found by differen-
tiating under the sign. Integrate y and y" by parts and substitute in
the equation. Then
(1 - ^ Te'**] - Ce'^*[T'(l -f)-\-(2n- l)tT;\dt = 0,
where the bracket after the first term means that the difference of the
values for the upper and lower limit of the integral are to be taken ;
these limits and the form of T remain to be determined so that the
expression shall really be zero.
The integral may be made to vanish by so choosing T that the
bracket vanishes ; this calls for the integration of a simple differential
equation. The result then is
T = (1 - fy -\ (1 - t^y + V^'] = 0.
The integral vanishes, and the integrated term will vanish provided
t = ± 1 or e""^ = 0. li X be assumed to be real and positive, the expo-
nential will approach 0 when t = 1 -{- iK and K becomes infinite. Hence
y(x)=C e'-\l-ty-^dt and ^(x)=C ' e'^Xl-fyht (31)
are solutions of the differential equation. In the first the integral is an
infinite integral when n < -\- ^ and fails to converge when n ^ — \.
The solution is therefore defined only when ti > — ^. The second in-
tegral is always an infinite integral because one limit is infinite. The
examination of the integrals for uniformity is found below.
Consider j e"<(l — t^)'*~idt with n < J so that the integral is infinite.
f e^'(l-t2)"-ide= r {l-t^y-lco8xtdt-\-i f (1 - <2)«-isinx<ctt.
From considerations of symmetry the second integral vanishes. Then
1/ ^**^<^ - <V~i(tt| = I J ^\l - <2)*-i coaxtdtl ^f^\^ - t^y^dt.
This last Integral with a positive integrand converges when n> — \, and hence the
Ifiven integral converges uniformly for all values of x and defines a continuous
f anction. The successive differentiations under the sign give the results
FUNCTIONS DEFINED BY INTEGRALS 906
These integrals also converge uniformly, and henoe the diflowittetiocMi wn JMtf.
fiable. The second integral (81) may be written with < = 1 4- te, ■«
This integral converges for all values of x > 0 and n > > |. Haaot Um given lat*-
^ral converges uniformly for all values of z ^ x^ > 0, and dallMt a oootlaaDW
function ; when z = 0 it is rea<lily seen that the integral dWeigas and i*««iM noi
define a continuous function. It is easy to Justify th« diflerantialkNW as bafoiw.
The first form of the solution may be expanded in
= 2 j (l-^""-^ cos xtdt (82)
-X'<'-'^-'('-f*Tr-iT-'iT)"''"<i"<'-
The expansion may be carried to as many terms as desired. Esoh of
the terms separately may be integrated by B- or r-funotions.
r(2 k 4- i)r(;i + ^• + 1) 2**r(;fc -j. i)r(n + * + 1) '
is then taken as the definition of the special function /,{x), wliere the
expansion may be ciirried as far as desired, with the coeffioieiit $ for
the last term. If n is an integer, the F-functions may be written as
factorials.
154. The second solution of the differential equation, namely
z(x) = y^(x) + i>,(^) =J'^*' - 2.'-(l - f»)- »c^
where the coefficient — 2 has been inserted for conrenienoey is for
])urposes more useful than the first It is complex, and, as the equation
is real and x is tiiken as real, it affords two solutions, namely its real pait
and its pure imaginary pjirt, each of which mnst satisfy the equation. As
!/{x) converges for x = 0 and x(x) diverges for xaO, so that jfj(jr) oc
(«•)
Q
396 INTEGRAL CALCULUS
yj(x) diverges, it follows that y (x) and y^(x) or y (x) and y^(x) must be
independent ; and as the equation can have but two independent solu-
tions, one of the pairs of solutions must constitute a com-
plete solution. It will now be shown that y^(x) = y(x) ^
and that Ay{x) -\- By^{x) is therefore the complete solu-
tion of xy" -\-{2n+l)y' + xy = 0.
Consider the line integral around the contour 0, 1 — «,
1 -f ci, 1-1-00 1, 00 1, 0, or OPQRS. As the integrand has a
continuous derivative at every point on or within the
contour, the integral is zero (§ 124). The integrals along ^ ^
the little quadrant PQ and the unit line 725 at infinity may be made as
small as desired by taking the quadrant small enough and the line far
enough away. The integral along SO is pure imaginary, namely, with
f -2 6*^(1 - ef-^dt = 2if e— ♦(! + uy-^du,
J so Jo
The integral along OP is complex, namely
f -2e'^(l-fy-^dt
= _ 2 / (1 - ff'^Go^xtdt -2i\ (1 - ^"-i mnxtdt.
Jo Jo
Hence 0=^-2 f (1- t^-^ cos xtdt-2i f (1- ty-hinxtdt A- 1^
Jo Jo
+ r _ 2e^(l - t-^^-^dt 4- ^2 -f 2i r e— (1 + u^-^duy
Jq Jo
where f^ and ^^ are small. Equate real and imaginary parts to zero
separately after taking the limit.
2 J (1 - e)''-ho8xtdt = y(x) = /^ P""*'- 2e-'(l - t^''-idt = y^(x),
2 r (1 - e)""-^ Bin xtdt - 2 r*e-"(l -f- u^-^du
= jf '^-2e'-*(l-ty-^dt = y^(x).
The signs ^ and J are used to denote respectively real and imaginary
parts. The identity of y(x) and y^(x) is established and the new solu-
tion y^(x) is found as a difference of two integrals.
FUNCTIONS DEFINED BV INTEGKAL8 Wl
It 18 now possible to obtain the important expuuion of tha inliHwii
!/(x) and y,(j!) in descending powers of x. For
J -2e'«(l-<^-»rf<=jf'_2i««— (««_2i«)-»rf,, r-l+ta.
Since z ^ 0, the transformation ux = via pemuMiUe and girw
2"*i(~{)'*ie<'x—ire-'v-i(l+^-ld«
The expansion by the binomial theorem may be carried as far at de*
sired; but as the integration is subsequently to be performed, the
values of v must be allowed a range from 0 to oo and the nie of
Taylor's Formula with a remainder is required — the leriea would not
converge. The result of the integration is
z(x) = 2- + V'-ir(n+ i>'C'"(""'^)^][f>(x) + iQCx)], (34)
^^ 2!(2a:)« ^ 4!(2a:)*
Take real and imaginary parts and divide by 2*x"" Virr(n + J). Then
"'^'^ = ^l^[«("> ""^ ("-(»+ 1) i) + ''<'> "'" ('-("+ 1) 5)]
are two independent Bessel functions which satisfy the eqnatkm (36)
of § 107. If ji-\- i is an integer, P and Q terminate and the tolatioiia
are expressed in tiums of elementary functions (§108); but if a -f )
is not an integer, P and Q are merely asymptotic expressions which do
not terminate of themselves, but must be cut short with a remainder
term l)ecause of their tendency to diverge after a certain point; for
tolerably large values of x and small values of a the values of /.(x)
and K^{x) may, however, be computed with great aoooracj by wiof
the first few terms of P and Q.
398 INTEGRAL CALCULUS
Ihe Integration to find P and Q offers no particular difficulty.
f %- V-i+*dt = T{n-{-\ + k) = (n-\-k- ^)(n + A;- f)-. (n + i)r(n + i).
The factors previous to r (n + J) combine with n— i, n— |,---,n — A; + i, which
occur in the Jkth term of the binomial expansion and give the numerators of the
terms in P and Q. The remainder term must, however, be discussed. The integral
form (p. 57) will be used.
Let it be supposed that the expansion has been carried so far that n — k—^<0.
Then (1 + ri/2x)"~*~ i is numerically greatest when v = 0 and is then equal to 1.
Hence
' *'^Jo (fc-l)! (2x)* k\ (2a;)*
|(n._lV..(.._(?illi)!\|
and |XV.--i.^.|<l^ ^' X^^ ' '^^h'i-
It therefore appears that when fc > n — ^ the error made in neglecting the remain-
der is less than the last term kept, and for the maximum accuracy the series for
P •\- iQ should be broken off between the least term and the term just following.
EXERCISES
1. Solve xy" + (2 n + 1) y' — xy = 0 by trying Te^' as integrand.
aC (l-<2)«-V«d< + B r~ («2-l)"-^e^'d«, x>0, n>-h
2. Expand the first solution in Ex. 1 into series ; compare with y{ix) above.
3. Try r(l - tx)^ on x(l - x)y'' + [7 _ (a + |3 + l)x]y' -a^ = 0.
OneeoXxitioniB f t»-\l-t)y-P-\l-~tx)-^dt, /3 > 0, 7 > /3, |x|<l.
4. Expand the solution in Ex. 3 into the series, called hypergeometric,
L 1-7 1 • 2 7 (7 + 1)
g(a-H)(a + 2)/3(/3 + l)(/3 + 2)^, "I
1.2.37(7 + l)(7 + 2) J'
5. Establish these results for BessePs J-functions :
^ I sin2«
2«Virr(n + J)*'o
(a) J,(x) = — I sin2« 0 cos (x cos 0) d0, n > — \,
2«Virr(n4 "
1 x»
(/»)
•^•(x) = -r-r r 8in2»0cos(xcos0)(f0, n = 0, 1, 2, 3
IT o • 0' • • (* n — 1) »/o
FUNCTIONS DEFINED BY INTEGRALS 999
1 r'
6. Show - / C08 (n^ - z Kin ^)d^ Mtiiflaf
7. Find tho equation of the second order MttiAfledbj f (l«C*)*'lrfaaML
8. Show Jo(2x^ = l-«« + -^- — + -i? —^
* (a!)« (8!)« (4I)« (6!)*
9. Compute J,(l) = 0.7662 ; Jq(2) = 0.2S89 ; /^(1.406) = 0.0000.
10. Prove, from the integrals, Jo(z) = - J^{x) and [x- V.]' a — s-V«^i.
11. Show that four terms in the asymptotic ezpaiurion of P 4* 4Q whta • ■ •
give the best result when x = 2 and ttiat tlie error nuty be about O.OOS.
12. From the asymptotic expansions compute Jq{Z) as accurately aa umf ba.
13. Show that for large values of x the solutions of J^{z) s 0 are iie*rly of tb*
form A:ir — i IT + i nir and the solutions of K^{x) = 0 of the form Inr -f i v -f Imt.
14. Sketch the graphs of y = Jq(x) and y = J^{x) by using tbe series of
ing powers for small values and the asymptotic expressions for large values of x
15. From Jo(x) = - | cos (x cos ^)d^ show I e-^JJbx)dz=: — =^=*
TT Jo Jo -^^ ^ Ift
16. Show r e-<«e/o(x)dx converges uniformly when a S 0.
Jo
17. Evaluate the following integrals ; (^) J •^o(**)^ = *^*»
(/9) r sin axJo(te) — = ^ or sin- i-asa>6>0or6>a>0,
Jo X 2 0
sin axJJJxt)dx = -^=:^= or 0 as a* > 6* or fc« > a*,
0 Va« - 6«
(5) r*cosaxJo(to)dx = — :^=: or 0 as 6« > a« or a« > ft».
18. If u = Vij,(ax), show ^ + /a» - ^^^^)«« = 0. If e = \^.(bi),
L^ _ u ^^1'= (6» - a«) r*xJ.(ax)J.(to)d*.
L dx dxjo Jo
19. With the aid of Ex. 18 establish the relations:
(cr) feJ,(a) J, + ,(fe) - aJ.(6)J. + i(a) = (6« - a«)jr x/.(ax)J,(te)d«,
(/9) aJi(a) = a* J /Jo(ax)dx = ^ xJo(«)dJfi
(7) J^.(a) J,+i(a) + a [J.(a) j:^i(o) - J:(a) J..,i(a)] = a^jT «(J.(«»)piL
2 r* sinxCdt ., 2 /** ootakK
20. Show /,(«)-?( -^==. AVx) = -j; ;;^==.
CHAPTER XV
THE CALCULUS OF VARLATIONS
155. The treatment of the simplest case. The integral
F(x,y,y')dx= I ^ (x, y, dx, dy\ (1)
A cJa
where ^ is homogeneous of the first degree in dx and dyj may be evalu-
ated along any curve C between the limits A and B by reduction to an
ordinary integral. For if C is given by y = f(x),
/ = f F(x,y,y')dx=f ^F{x, f(x), f(x)) d^ ;
cJa Jx^
and if C is given by a; = <^(^), y = ^(i),
1= f ^(x,y, dx, dy) = r '^ (<^, ^, c^', i/r') dt.
cJ A Jt^
The ordinary line integral (§ 122) is merely the special case in which
^ — Pdx + Qdy and F = P ■\- Qy'. In general the value of / will depend
on the path C of integration ; the problem of the calculus of variations
is to find that path which will make I a maximum or minimum, relative
to neighboring paths.
If a second path C^ be y =f(x) + -qix), where t^{x) is a small quan-
tity which vanishes at x^ and ic^, a whole family of paths is given by
y =/W + ocv{^\ - 1 ^ a ^ 1, ri{x^) = '^{x;) = 0,
and the value of the integral
Ar^^\
taken along the different paths of the family, be- -^i ^Jo Xx X
comes a function of a; in particular /(O) and /(I)
are the values along C and Cj. Under appropriate assumptions as to
the continuity of F and its partial derivatives F^, FJ, Fy,, the function
/(er) will be continuous and have a continuous derivative which may
be found by differentiating under the sign (§ 119) ; then
^'(«) = r \vK(?^^f+ o^-n,f -f an') -f rj'F'^(x,f+ an^f -h ar,')-]dx,
400
CALCULUS OF VARIATIONS 401
If the curve C is to give 1(a) a maximum or minimum value for all
the curves of this family, it is necessary that
^'(0) = f \riK(', y, y') + V^;(ar, y, y^]dx .0; (1)
and if C is to make / a maximum or minimum relative to all neigbboriof
curves, it is necessary that (2) shall hold for any function ^(t) whidi k
small. It is more usual and more suggestive to write ^(x)^!^, and to
say that Sy is the variation of y in passing from the curve C or y ■■/(<)
to the neighboring curve C or y =f(x) -f i|(:r). From the roUilioiH
y<=f{x), y'=/'(x) + ,'(x), Jy'-,'(*)-^.
connecting the slope of C with the slope of Cp it is seen that tlu vartatwm
of the derivative is the derivative of the variation. In di£ferential DOll^
tion this is dhj = &///, where it should be noted that the sign % applies
to changes which occur on passing from one curve C to another ounre C,,
and the sign d applies to changes taking place along a particii]ar ounre.
With these notations the condition (2) becomes
r V;«y -f F'M)^^ = P^Fdx = 0, (8)
where hF is computed from F, 8y, hf by the same rule as the differential
dF is computed from F and the differentials of the variables which it
contains. The condition (3) is not sufficient to distinguish between a
maximum, and a minimum or to insure the existence of either ; neither
is the condition g\x) = 0 in elementary calculus sufficient to answer
these questions relative to a function g{jr)\ in both cases additional coop
ditions are required (§ 9). It should be remembered, however, that
these additional conditions were seldom actually applied in discosstng
maxima and minima of g{x) in practical problems, because in SQeh eMSi
the distinction between the two was usually obvious; so in this oaae
the discussion of sufficient conditions will be omitted altogether, as in
§§68 and 61, and (3) alone will be applied.
An integration by parts will convert (3) into a differential Miuatioo
of the second order. In fact
Hence (''(FM + F:.i,/),l:t = j'' (^F;-j-^r^irU~fi, (SO
402 INTEGRAL CALCULUS
since the asstimption that hy = tj (x) vanishes at x^ and x^ causes the
integrated term [Fy,Sy] to drop out. Then
^^"di^'^'dy dxdy^ oycy^y dy^^ " "* ^*^
For it must be remembered that the function Sy = rj (x) is any function
that is small, and if F^ — — F^, in (3') did not vanish at every point
ctx
of the interval x^^ x ^ x^^ the arbitrary function By could be chosen
to agree with it in sign, so that the integral of the product would neces-
sarily be positive instead of zero as the condition demands.
156. The method of rendering an integral (1) a minimum or maximum
is therefore to set up the differential equation (4) of the second order
and solve it. The solution will contain two arbitrary constants of inte-
gration which may be so determined that one particular solution shall
pass through the points A and B, which are the initial and final points
of the path C of integration. In this way a path C which connects A
and B and which satisfies (4) is found ; under ordinary conditions the in-
tegral will then be either a maximum or minimum. An example follows.
Let it be required to render I = j - V 1 + y'^dx
a maximum or mmimum.
y ^y y^ W v VTTy^
Hence -IVl + y^'^ + J^ ^^ y' i ^^" = 0 or y2^'+y'2 + l = 0
y^ y^VTTV^ y{i-\-y^)^
is the desired equation (4). It is exact and tiie integration is immediate.
(yy^y +1 = 0, yy'-{'X = c^, y^ + {x^ c^)
^i'
The curves are circles with their centers on the x-axis. From this fact it is easy
by a geometrical construction to determine the curve which passes through two
given points A{Xq, y^) and 5(Zj, y^); the analytical determination is not difficult.
The two points A and B must lie on the same side of the x-axis or the integral I
will not converge and the problem will have no meaning. The question of whether
a maximum or a minimum has been determined may be settled by taking a curve
C^ which lies under the circular arc from A to B and yet has the same length.
The integrand is of the form ds/y and the integral along C. is greater than along
the circle C if y is positive, but less if y is negative. It therefore appears that the
integral is rendered a minimum if A and B are above the axis, but a maximum if
they are below.
For muny problems it is more convenient not to make the choice of x
or y as independent variable in the first place j but to operate symmetri-
cally with both variables upon the second form of (1). Suppose that the
integral of the variation of <I» be set equal to zero, as in (3).
CALCULUS OF VAKIATI0N8 40S
Let the rules hdx = dhr and Idtj a d%}/ be applied and lei the tenM
which cuntain dhr^ and dhy be intct^rated 1)v parU a« lH»f(»ra*
As ^ and B are fixed points, the integrated term diaappean. At the
variations &r and ly may be arbitrary, reasoning as above givei
*; - d^'u. = 0, ♦; - d*'^ a 0. (4*)
If these two equations can be shown to be essentially M^mti^Hil and to
reduce to the condition (4) previously obtained, the justification of the
second method will be complete and either of (4*) may be used to deter-
mine the solution of the problem.
Now the identity ♦(x, y, dc, dy) = F(x, y, dy/dx)dx givet, on dUfwwittetkw,
*: = F;,dx, ♦; = F;d/, ♦;, = f;, ♦;, = -ir;,^ + jp
by the ordinary rules for partial derivatives. Substitution In eech of (4') glT«
*;-d*:. = F;dx-dF; = (F;-lF;)dx = o.
*x - rf*i/x = K^ - d(F- F^vl = ^x*** - <IF+ >^,iy T r •'^
= F^dx - F;^dx - F;dy - F'^dy' + F;,d|r' + r'di^
= - F;dy + y'dF; = - {K-§,K)^^ = ^'
Hence each of (4') reduces to the original condition (4), as was lo be
/d» c V<te* + <^
— =1 . Then
y J f
where the transformation has been integration by parts, including ibe
of tlie integrated term which vanishes at the limits. The two
ydi ydM ^ yw <!
is the obvious first integral of the first. The integration nay tht
find the circles as before. The integration of the saoond eiiuatloo would aot be w
simple. In some instances iht advatOaife qf the ek§tet 9f 9m ^ t^M tm tqmUmt
nff'ercil by this method qf direct operation U
404 INTEGRAL CALCULUS
EXERCISES
1. The shortest distance. Treat ^(1 + y'^)^dx for a minimum.
2. Treat fVdr^ + rH<p^ for a minimum in polar coordinates.
3. The brachistochrone. If a particle falls along any curve from A to B, the
velocity acquired at a distance h below A is v = V2gh regardless of the path fol-
lowed.. Hence the time spent in passing from A to B is T = j ds/v. ^ The path of
quickest descent from ^ to B is called the brachistochrone. Show that the curve
is a cycloid. Take the origin at A.
4. The minimum surface of revolution is found by revolving a catenary.
5. The curve of constant density which joins two points of the plane and has a
minimum moment of inertia with respect to the origin is c^r^ = sec (3 0 + c^). Note
that the two points must subtend an angle of less than 60° at the origin.
6. Upon the sphere the minimum line is the great circle (polar coordinates).
7. Upon the circular cylinder the minimum line is the helix.
8. Find the minimum line on the cone of revolution.
I. Minimize the integral f - wi (— ) + - n^x^ \dt.
r
\
Cj^
>^.
k
ri
0
X
157. Variable limits and constrained minima. This second method
of operation has also the advantage that it suggests the solution of the
problem of making an integral between variable end-points a maximuTn,
or minimum. Thus suppose that the curve C which
shall join some point A of one curve V^ to some
point B of another curve r^, and which shall make
a given integral a minimimi or maximum, is desired.
In the first place C must satisfy the condition (4)
or (4') for fixed end-points because C will not give
a maximum or minimum value as compared with
all other curves unless it does as compared merely with all other curves
which join its end-points. There must, however, be additional condi-
tions which shall serve to determine the points A and B which C con-
nects. These conditions are precisely that the integrated temUf
r*;,Sx -I- *;,y8yl ^ = 0, for A and for B, (6)
which vanish identically when the end-points are fixed, shall vanish at
each point A or B provided &c and Bg are interpreted as differentials
along the curves r^, and r^.
CALCULUS OF VABIATI0X8 405
For example, in the case of / — = / IlJl tra*tad Above, thm t
J y J y
t-nn8, which were discarded, and the resulting condition* arv
Tdxix dyiyy dzto + dy«y"|« . dBte + tfrtvl *
Here (Ix and dy are differentials along the circle C and Ax and ly ai« to ur inter-
preted a8 (lifTerentialM along the curves r^ and T^ which reipeeCivclj paM Ikm^fc
.1 and B. The conditions therefore show that the HiigwiH to O umI T^9l A •!•
P(>r])cndicular, and KJmilarly for C and Tj at B. In oChar wovdi tht anr?* wMdl
renders the integral a niininmm and has its extremities on two flxad amrst b tiM
circle which ha8 its center on the x-axis and cuts both the cnnrss ortbofiNMllj.
Tu prove the rule for finding the conditions at the eod points it will bt mA-
cient to prove it for one variable point. Let the equations
determine C and C\ with the common initial point A and different termiani potoH
B and B" upon Tj. As parametric equations of Tj, take
a; = a;^ + a/ («), y = y^ + Inn («) ; ~ = at{B), -1 = 6imi).
where » represents the arc along Tj measured from B, and the functions I (s) and
m(«) vary from 0 at /J to 1 at B'. Next form the family
x = 0(O + i(«)f(Oi i/ = ^(0 + »n(«)i,(0, x' = ^' + «r, r'af + ai^.
wliich all pass through A for « = <„ and which for t = (j describe the ciure Pg.
Consider
p(«) = C\(x + /(a)f, y + m(«),,, X' + ff. !< + ai^^ »
which is the integral taken from ^ to Tj along the curres of the familj, wiMia
«» Vi a:', y' are on the curve C corresponding to « = 0. Differentiate. Tben
g'ii) = J'^(;r(a)f*; + m'{s)rfi^\ + r(«)r'*;. + m'(«) ,'♦;,](«,
where the accents mean differentiation with regard to « when upon 9, (, or m,
with regard to t when on x or y, and partial differentiation when on ♦, and wl
i the argument of * is as in (0). Now if 17(a) has a maximttm or minimum wWn
a = 0, then
j,'(0) = r '* [^'(0) r*;(x, y, X', lo + m'(0) IT*; + r(0) t'K -♦• "•'(<>) v*;.]* « • •
riu- chanije is made as usual by integration by parts. Now as
* (X, y, X', yO d< = ♦ (x, y, dJf, dy), so *'^dt = ♦; , ♦;. « ♦i,. •««.
406 INTEGRAL CALCULUS
Hence the parentheses under the integral sign, when multiplied by dt, reduce tc
(4') and vanish ; they could be seen to vanish also for the reason that f and 17 are
arbitrary functions of t except at t = t© and t = ij, and the integrated term is a
congtant. There remains the integrated term which must vanish,
no)nti)K + m'(0)n(«i)*;, = [^*;, + ^*;J^ = [*^,«x + Ky^yJ" = o.
The condition therefore reduces to its appropriate half of (6), provided that, in
interpreting it, the quantities 8x and Sy be regarded not as a = f (ij) and 6 = 17 (t^)
but as the differentials along r^ at B.
158. In many cases one integral is to be made a maximum or minimum
subject to the condition that another integral shall have a fixed value,
' ^(^> 2/> y') dx ^^y J= \ G{x,y, y')dx = const.
(7)
For instance a curve of given length might run from A to 5, and the
form of the curve which would make the area under the curve a maxi-
mum or minimum might be desired ; to make the area a maximum or
minimum without the restriction of constant length of arc would be
useless, because by taking a curve which dropped sharply from .4, in-
closed a large area below the a;-axis, and rose sharply to B the area
could be made as small as desired. Again the curve in which a chain
would hang might be required. The length of the chain being given,
the form of the curve is that which will make the potential energy a
minimum, that is, will bring the center of gravity lowest. The prob-
lems in constrained maxima and minima are called isoperivietric prob-
lems because it is so frequently the perimeter or length of the curve
which is given as constant.
If the method of determining constrained maxima and minima
by means of undetermined multipliers be recalled (§§58, 61), it will
appear that the solution of the isoperimetric problem might reasonably
be sought by rendering the integral
/ + A/ = r \f{x, y, y') + \G(x, y, y')^dx (8)
a maximum or minimum. The solution of this problem would contain
three constants, namely, \ and two constants c^, c^ of integration. The
constants c^, c^ could be determined so that the curve should pass through
A and B and the value of X would still remain to be determined in such
a manner that the integral J should have the desired value. This is
the method of solution.
CALCULUS OF VARIATIONS 401
To juHtify the inethud in the CMe of fixed end-poiiito, whleh Is Um <mUj cm|
that will be coiiHidered, the procedure U like that uf 1 166. Lei C; be g|f«i hf
y =/(x) ; consider
y =/(») + a, (X) + /Jr (X), Ho = f I = ft » ti « 0,
a two-parametered family of curves near to C. TImd
h{a, p) = jr''(?(x, y + an + ^f, 1^+011' + /Jn<*« = •^
would be two functions of the two variables a and fi. The ooodftlooe for Ike
iniuu ur niaximuu) of 9 (a, /3) at (0, 0) subject to the condition that A (a, /i)
an' n'ljuired. Hence
fir;(o, 0) + xa;(o, 0) = 0, fir;(o, o) + xa;(o. o) = 0.
or f'^iK + ^^,) + ^(Fi;. + XO;)dx = 0,
/'V(f; + xG?;) + r'C^;- + xG;)dx = o.
By integration by parts either of these equations gives
(F+XG);-£(F+XC);, = 0; (9)
the rule is justified, and will be applied to an example.
Uequired the curve which, when revolved about an axiH, wiii genermte a given
volume of revolution bounded by the least surface. The integrals are
I = 2 TT r ^yds, min., J = ^ f ^V^ const.
Make f'iyds + Xy^dx) min. or (*% {yds + Xff'dz) = 0.
Hence Xd(i/«) + d^=0 or d«-d^ + SXrlx = 0.
^ ' da dM
The second method of computation has been used and the
terms have been discarded. The first equation is simplest to Intsgrale.
X,. + ,_L= = c,X, ± /(^|-«^^ ^ds,
ViTTi Vy«-x«(c,-rv
The variables are separated, but the integration cannot be executed In terns d
domentary functions. If, however, one of the end-polnta is on ths «-axlB, tke
408 INTEGRAL CALCULUS
values Xq, 0, Pq or Xj, 0, y[ must satisfy the equation and, as no term of the equa-
tion can become infinite, c^ must vanish. The integration may then be performed.
^ , dx, l-XV = X2(x-C2)2 or (x-C2)2 + ya=-.
Vi - x2y=» ^
In this special case the curve is a circle. The constants Cj and X may be deter-
mined from the other point (Xj, y^ through which the curve passes and from the
value of J = u ; the equations yfi\\ also determine the abscissa x^ of the point on
the axis. It is simpler to suppose x^ = 0 and leave x^ to be determined. With this
procedure the equations are
(H-<',f + yi = l,- ^ = ^-5W-3vf + Sc|»j),
TT ZXj
and Xi = IT- i [(s » + V9 r2 + tt^j/ «)i + (s » - V9«2 + ^2j,6)ij ^
EXERCISES
1. Shovy that (a) the minimum line from one curve to another in the plane is
their common normal ; (/3) if the ends of the catenary which generates the mini-
mum surface of revolution are constrained to lie on two curves, the catenary shall
be perpendicular to the curves ; (7) the brachistochrone from a fixed point to a
curve is the cycloid which cuts the curve orthogonally.
2. Generalize to show that if the end-points of the curve which makes any inte-
gral of the form / ^(x, y)d8 a maximum or a minimum are variable upon two
curves, the solution shall cut the curves orthogonally.
3. Show that if the integrand *(x, y, c^, dy^ x^) depends on the limit Xj, the
condition for the limit B becomes *^^5x -|- ^'^y^y -V bx C W^\ = 0.
4. Show that the cycloid which is the brachistochrone from a point -4, con-
strained to lie on one curve r^, to another curve T^ must leave Tq at the point A
where the tangent to r^ is parallel to the tangent to Tj at the point of arrival.
5. Prove that the curve of given length which generates the minimum surface
of revolution is still the catenary.
6. If the area under a curve of given length is to be a maximum or minimum,
the curve must be a circular arc connecting the two points.
7. In polar coordinates the sectorial area bounded by a curve of given length is
a maximum or minimum when the curve is a circle.
8. A curve of given length generates a maximum or minimum volume oi
revolution. The elastic curve
B = <L+_Oi=_A or ta^-M^^m^.
CALCULUS OF VABUTI0N8 0^
9. A chain U<« in » oentrml field of force of which the potenUel p«r unii sMik
V(r). If tlie constant deiudtj of the chain la p, ahow that the form ot Om otrve to
0+c,
r — -
[cf(pr+x)M-i]^
10. I )i8cu88 the reciprocity of / and J, that ia, the queetiona of Btaklaf / a i
iiiuiii or minimum when J is fixed, and of making J a minimum or maiiao
/ is fixed.
1 1. A solid of revolution of given ma« and onifonn denirftj exerta a aaxiaHui
attraction on a point at ita axis. Awi. 2 X(je* -f y*)t -f x = 0, if the point ia at the
origin.
159. Some generalizations. Suppose that an integral
F(x, y, y\ z,z','-.)dx=j ♦(x, dx, y, rfy, a, i£a, • • ) (10)
(of which the integi-and contains two or more dependent rariablet
y, «, • • • and their derivatives y', «', • • with respect to tlie independtfot
variable x, or in the symmetrical form contains three or more variables
and their differentials) were to he made a maximum or minimuuL In
case there is only one additional variable, the problem still has a geo-
metric interpretation, namely, to find
2/=/W, ^ = 5'W, or x = <l>(t), y = ^(0, *-x(0.
a curve in space, which will make the value of the integral greater or
less than all neighboring curves. A slight modification of the previous
reasoning will show that necessary conditions are
n-£F;,=o and f:-±k-o ^^,^
or <d: - c?*;;, = 0, ^i-rf*:/, = 0, ♦;-rf*;^ = o,
where of the last three conditions only two are independent Each of
(11) is a differential equation of the second order, and the soltttum of
the two simultaneous equations will be a family of corves in space
dependent on four arbitrary constants of integration which maj bo so
determined that one curve of the family shall pass through the end-
points A and B.
Instead of following the previous method to establish those liflli» ao
older and perhaps less accurate method will be used. Let the varisd
values of ?/, %, y\ «', be denoted by
■y + Sy, ;. + &r, y' -f V, z' ^ hz\ Sy' = («y)', ««'-"W.
410 INTEGRAL CALCULUS
The difference between the integral along the two curves is
A/ = f\F(x, y + hj, y' -f hj\ z + &t, z' + 8^') - F{x, y, y\ z, z')-]dx
= r^Fdx = r\F',hj + f;%' + Fihz -h i^;,8«') dx + "-,
where F has been expanded by Taylor's Formula* for the four variables
y, y\ «, «' which are varied, and " H " refers to the remainder or the
subsequent terms in the development which contain the higher powers
of Sy, Zy\ Bz, Bz'.
For sufficiently small values of the variations the terms of higher
order may be neglected. Then if A/ is to be either positive or nega-
tive for all small variations, the terms of the first order which change
in sign when the signs of the variations are reversed must vanish and
the condition becomes
PiF^By + F;^y' + F^Bz -f i^'^S;^') dx = PhFd^ = 0. (12)
Integrate by parts and discard the integrated terms. Then
• In the simpler case of § 155 this formal development would run as
A/= f'\F'8y-hFLdy')dx + ^ [""'(F'^^Stj^ +2 F';^.Sy8y'-{-F;;y.5y'^)dx-{- higher terms,
Jx, -i! »/Xo
and with the expansion AI= SI + —8^1 + —S^I-\ it would appear that
3/ = r ''(F;«y + F'j,,dy')dx, ^1= f ""'(F^'^Sy^ + 2 F;,'^,Sydy' + F;,:^,Sy'^)dx,
Jx^ *fx^
m=C '*(F;ray« + 3 F'^,'^,Syny' + 3 F'^,my'^ + F;:^Sy'^)dz, •".
The terms 5/, 3^7, 3*/, • • • are called the first, second, third, • • • variations of the integral
/ in the case of fixed limits. The condition for a maximum or minimum then becomes
81= 0, just as dg = 0 is the condition in the case of g (x). In the case of variable limits
there are some modifications appropriate to the limits. This method of procedure sug-
gests the reason that 8x, 8y are frequently to be treated exactly as differentials. It also
suggests that 3^7 > 0 and 3^7 < 0 would be criteria for distinguishing between maxima
and minima. The same results can be had by differentiating (1') repeatedly under the
sign and expanding I{a) into series; in fact, 37= 7'(0), 3«7= 7"(0), • • • . No emphasis
has been laid in the text on the suggestive relations 8/= fsFdx for fixed limits or
81 — J i* for variable limits (variable in x, y, but not in 0 because only the most ele-
mentary results were desired, and the treatment given has some advantages as to
modernity.
CALCULUS OF VARIATlONb 411
As By and Bz are arbitrary, either may in partiealAr be takao mml to
0 while the other is assigned the same sign as its ooefltoiani in fttt
parenthesis ; and hence the integral would not vanbh nnlcss thftt nnofll
cient vanished. Hence the conditions (11) ar« derived, aad it it aeett
that there would be precisely similar conditions, one for eaeb ▼mrkble
i/,z,-- -, no matter how many variables might occur in the integnuid.
Without going at all into the matter of proof it will be stated as a
fact that the condition for the maximum or minimum of
j *(ar, rfx,y,rfy, «,<&,...) is ji^^O,
which may be ti-ansformed into the set of differential equations
of which any one may be discarded as dependent on the rest ; and
^'^Bx + <P'^^Si/ + <d:,,&s -f • • • = 0, at i4 and at B,
where the variations are to be interpreted as differentials along the kMsi
upon which A and B are constrained to lie.
It frequently happens that the variables in the integrand of an inte-
gi-al which is to be made a maximum or minimum are oonnacted bj an
equation. For instance
/
^{xy dx, y, dy, «, dx) min., 5(x, y, x) aa 0. (14)
It is possible to eliminate one of the variables and its differential bjr
means of ^' = 0 and proceed as before ; but it is usually better to
introduce an undetermined multiplier (§§ 68, 61). From
S (x, y,' «) = 0 follows S'J^ -f s\hy + y.&t = 0
if the variations be treated as differentials. Hence if
/"[(*; - d^',:)U + (*; - rf*:,,)«y + (♦; - ^z*;,)^] - o,
/'
[(*; - rf*;, + X5;)&r + (♦; - rf*;, + XA-)^
+ (*;-rf*i. + x5;)S«] = o
no matter what the value of X. Let the value of X be so chosen as to
annul the coefficient of hz. Then as the two remaining variatioiis aie
indei>endent, the same reasoning as above will cause the coeilUtients of
&r and Sy to vanish and
*; - "'*:/r + X5; = 0, ♦; - rf«i»^ -H xs; = 0, ♦;-rf*^-rA.^. = o (15)
412 INTEGRAL CALCULUS
will hold. These equations, taken with 5 = 0, will determine y and k
as functions of x and also incidentally will fix X.
Consider the problem of determining the shortest lines upon a surface
S(Xf y, z) = 0. These lines are called the geodesies. Then
d^ ^^ ^^
d^ + X5; = ^f^ + X5;^^^ + A5; = 0, and J^^_ds_^_ds.
ds "" ds ^ ds s' S' S'
*^x '^y "z
In the last set of equations \ has been eliminated and the equations,
taken with S = 0, may be regarded as the differential equations of the
geodesies. The denominators are proportional to the direction cosines
of the normal to the surface, and the numerators are the components of
the differential of the unit tangent to the curve and are therefore pro-
portional to the direction cosines of the normal to the curve in its oscu-
lating plane. Hence it appears that the osculating plane of a geodesic
curve contains the normal to the surface.
The integrated terms dxdx + dySy + dzdz = 0 show that the least geodesic which
connects two curves on the surface will cut both curves orthogonally. These terms
will also suffice to prove a number of interesting theorems which establish an analogy
between geodesies on a surface and straight lines in a plane. For instance : The
loCus of points whose geodesic distance from a fixed point is constant (a geodesic
circle) cuts the geodesic lines orthogonally. To see this write
J-p pP pP f>p p
ds = const., A I ds = 0, SI ds = 0, | 5ds = 0 = dxdx + dySy + dzSz .
0 Jo Jo Jo
The integral in (16) drops out because taken along a geodesic. This final equality
establishes the perpendicularity of the lines. The fact also follows from the state-
ment that the geodesic circle and its center can be regarded as two curves between
which the shortest distance is the distance measured along any of the geodesic
radii, and that the radii must therefore be perpendicular to the curve.
160. The most fundamental and important single theorem of mathe-
matical physics is Hamilton's Principle, which is expressed by means
of the calculus of variations and affords a necessary and sufficient con-
dition for studying the elements of this subject. Let T be the kinetic
energy of any dynamical system. Let A",-, y,., Z,- be the forces which
act at any point a;,, y^, «^ of the system, and let &«<, Sy,-, S«< represent
displacements of that point. Then the work is
CALCULUS OF VARIATIONS 418
Hamilton's Principle states that the tims integral
J \hT 4- hW)dt = r \tT + 2^ {Xlr + nu + Zix\^Jl - 0 /IT)
vanishes for the actual viotion of the system, it m |jarticttUr Mmrit it
a potential function K, then tW zs — IV and
r '8(r - v)dt = s r'(r - v)dt -. 0, (m
and Me /ime integral of the difference between the kinetie and potemtiml
energies is a viaximum or minimum for the aehtal moti&n of the ewetem
as compared with any neighboring motion.
Suppose that the position of a system can be expre«ed by bmmw of n
eht variables or co5rdinate8 9p 9^, • • •, v«. Let the kinetie
T= 2J im,p? =f\fMm = T{q^, 7,, .. ., 7., ^„ ^„ ..., ^).
a function of the coordinates and their derivatiTes with reqwci to tbo iiam. Lit
the work done by displacing the single coordinate QrhetWs: (^J^r* to thai tbtloCal
work, in view of the independence of the coordinates, is Q|39|4> <J^,-|- • • • 4> 1
Then
Perform the usual integration by parts and discard the intcgnUed tan
vanish at the limits t = Iq and t = f,. Then
+ • +(r;+<j.-|n.)»fc]*.
In view of the independence of the variations 9q^^ 69,, • • •, iq^^
ddTdT^ dBTBT^ 1 ?I ^ ?I - n (lA
These are the Lagrangian equations for the motion of a dynamical njalaai.* If
there is a potential function V (Vp 9,, • • •, 9,), then by deflnltloo
Hence ^ ^ - ?i = 0, l!i-^ = 0, ..., ^^-^=0. L.T-F.
The equations of motion have been expressed in terms of a dngle f uncUon L, whkh
is the difference between the kinetic energy T and potential ftmctloB K. By
• Compare Ex. 19, p. 112, for a dedoedoa of (1>) by
414 INTEGRAL CALCULUS
comparing the equations with (XT') it is seen that the dynamics of a system which
may be specified by n coSrdinates, and which has a potential function, may be stated
as the problem of rendering the integral jLdt a maximum or a minimum ; both the
kinetic energy T and potential function V may contain the time t without chang-
ing the results.
For example, let it be required to derive the equations of motion of a lamina
lying in a plane and acted upon by any forces in the plane. Select as coordinates
the ordinary coordinates (x, y) of the center of gravity and the angle 0 through
which the lamina may turn about its center of gravity. The kinetic energy of the
lamina (p. 318) will then be the sum ^Mv^ + llo^. Now if the lamina be moved a
distance Sx to the right, the work done by the forces will be X5x, where X de-
notes the sum of all the components of force along the x-axis no matter at what
points they act. In like manner Ydy will be the work for a displacement Sy. Sup-
pose next that the lamina is rotated about its center of gravity through the angle
5<f> ; the actual displacement of any point is r8<f> where r is its distance from the
center of gravity. The work of any force will then be Rrd<p where R is the com-
ponent of the force perpendicular to the radius r ; but Z?r = * is the moment of
the force about the center of gravity. Hence
T=:^M{x^-\-if^) + ll4>\ dW=X5x-\-YSy + ^Si>
and 3f^ = X, M^=Y^ I^.<.,
dt^ ' dt^ ' dt^
by substitution in (18), are the desired equations, where X and Y are the total
components along the axis and 4» is the total moment about the center of gravity.
A particle glides without friction on the interior of an inverted cone of revo-
lution; determine the motion. Choose the distance r of the particle from the ver-
tex and the meridional angle <p as the two coordinates. If I be the sine of the
angle between the axis of the cone and the elements, then ds^ = dr^ + r^l^dtp^ and
r^ = r* + r^t^^^. The pressure of the cone against the particle does no work ; it is
normal to the motion. For a change 50 gravity does no work; for* a change 5r it
does work to the amount — mg Vl — J^br. Hence
r=im(f2 + r2i202)^ 8W = -7ng^\-fibr or V=mg^l-l^r.
Then ^^_rp(^y=-,Vn:p, ±U^Jt\ = 0 or r«^ = C.
dt* \di) '' ' dt\ dij dl
The remaining integrations cannot all be effected in terms of elementary functions.
161. Suppose the double integral
z,p,q)dxdy, V^Yx "^^fy ^^^
extended over a certain area of the icy-plane were to be made a maxi-
mum or minimum by a surface « = « (a:, y), which shall pass through a
given curve upon the cylinder which stands upon the bounding curve
of the area. This problem is analogous to the problem of § 166 with
-^^n^,v,
CALCULUS OF VARIATIONS 415
fixed limits ; the procedure for finding the partial differaotSal
which z shall satisfy is also analogoos. Set
JJsFdxfhj ^JjiF'M -h F;«/> + F'fy)dxdy . 0.
Write ?P " "^ ' 5? = y and integrate by parts.
The limits ^4 and B for which the first term is taken are poinU upoD
the lx)undiiig contour of the area, and &e = 0 for i4 and B by rirtne of the
iussuinptiun that the surface is to pass through a fixed eorve abore
that contour. The integration of the term in S^ is similar. Henoe the
condition becomes
//—//("■-il-^^)— " <».
dF d dF d dF ^
by the familiar reasoning. The total differentiations gire
K -F^-F-^- F-;,p -F^q-F;^r-2F;;^-F-;f = 0.
The stock illustration introduced at this point is the miniranin surfiier,
that is, the surface which spans a given contour with tbe leaat area and
which is physically represented by a soap film. The real use, however,
of the theory is in connection with Hamilton's Principle. To study the
motion of a chain hung up and allowed to vibnit«», or of a piano wire
stretched between two i)oints, compute the kinetic and potential energiet
and apply Hamilton's Principle. Is the motion of a vibrating elaatae
body to be investigated ? Apply Hamilton's Principle. And ao in
electrodynamics. In fact, with the very foundations of meehanioi aon^
times in doubt owing to modern ideas on electricity, tbe one refuge of
many theorists is Hamilton's Principle. Two problems will be worked
in detail to exhibit the method.
Let a uniform chain of density p and length I be sospended bj
and caused to execute small oscillations in a vertical pliuie. At any
of the curve is y = y{x), and y = y{x, t) will be Uken to repreient tbeafaape of Um
curve at all times. Let y' = ^y/bx and \f - ^/H. As the Qecfllelkim art mmH,
the chain v^ill rise only slightly and the main part of tbe UiMCie eiMlf7 ^vUl bt la
the whipping motion from side to side ; the aMumpUoii ds ss d« may be Bade aad
the kinetic energy may be taken as
416 INTEGRAL CALCULUS
The potential energy is a little harder to compute, for it is necessary to obtain the
slight rise in the center of gravity due to the bending of the chain. Let Ti be the
shortened length. The position of the center of gravity is
Here ds = Vl + y^dx has been expanded and terms higher than y^ have been
omitted.
Then f\T-y).U=fyJ[lp(fJa.-lMl-^)(^J]<i^, (21)
provided X be now replaced in F by i which differs but slightly from it.
Hamilton's Principle states that (21) must be a maximum or minimum and the
integrand is of precisely the form (19) except for a change of notation. Hence
dxl ^^ 'axj dt\ dt/ gdt^ ^ ' dx^ dx
The change of variable I — x = u^, which brings the origin to the end of the chain
and reverses the direction of the axis, gives the differential equation
d^y , Idy 4S2y cPP ,1 dP , 4n2„ ^ .. „. ,
— ^H - = ^ or — - + P — 0 if y = P(M)cosn<.
du^ udu gdt^ du^ udu g ^ '
As the equation is a partial differential equation the usual device of writing the
dependent variable as the product of two functions and trying for a special type
of solution has been used (§ 194). The equation in P is a Bessel equation (§ 107)
of which one solution P{u) = AjQ{2ng'~2u) is finite at the origin u = 0, while the
other is infinite and must be discarded as not representing possible motions. Thus
y (x, t) — AJq (2 ng~ iu) cos nt, with y {I, t) = AJ^ (2 ng~ iii) = 0
as the condition that the chain shall be tied at the original origin, is a possible
mode of motion for the chain and consists of whipping back and forth in the peri-
odic time 2ir/n. The condition Jf^{2ng~U^) = 0 limits n to one of an infinite set
of values obtained from the roots of Jq.
Let there be found the equations for the motion of a medium in which
-.--///[(ir-^eMi)]-^-.
V = l^fffin + ^' + h^] dxdydz
are the kinetic and potential energies, where A and B are constants and
ar air
4iro = f
4.A = ?!?-M
dy dz
^ dz ax
h^ 5V
CALCULUS OF VARIATIONS 417
are relations connecting/^ (f, A with the dJ^tlaoeiiienU |, f, f aloag UMans of js. v i.
Then
is the expression of Hamilton's Principle. TheM Intflgnls v
(10), for there are three dependent variables |, iy, ftatd four
X, {/, z, t of which they are f unctiuns. It is Uierefore nnnc—ij to apply tbt
of variations directly.
After taking the variations an integration by paru will be applied to t^ raHa-
tion of each derivative and the integrated tenna will be djeoarded.
ffffi\^i^ + ^ + hd^dydxdt = ffffA {kik-^m-^W) Irfjtoif
= "" ////^ (^ + f»f + iH)dM>i9iwiL
ffffi i B(/^ + P* + h^)dxdvdzdt = ffffBiAf-k- 9»9 + AM)d»Mi«
=-////f,[(i-Sh^(i-l)''^g-S'^]«^
After substitution in (22) the coefficients of S{, iii, 8f may be aeTerally eqttaied la
zero because 5^, Srjy i^ are each arbitrary. Hence the equatiooa
With the proper determination of A and B and the proper Interpmalioa of |, f, f,
/, (/, A, these are the equations of electromagnetism for the free elhar.
EXERCISn
1. Show that the straight line is the shortest line In space and that the
distance between two curves or surfaces will be normal to both.
2. If at each point of a curve on a surface a geoderic be erected
to the curve, the locus of its extremity is perpendicular to tba {
3. With any two point* of a surface as foci conatniot a geoderie elUpM by tak-
ing the distances FP + F'P = 2 a along the geodedca. Show that the taageat lo
the ellipse is equally inclined to the two geodeelo focal radii.
4. Extend Ex. 2, p. 408, to space. If J F{x, y, x)d» = oooil^ Aow tlMt tke
locus of P is a surface nonnal to the radii, provided the radU be
make the integral a maximum or minimum.
5. Obtain the polar equations for the motion of a particle In a
6. Find the polar equations for the motion of a particle in ipaea.
7. A particle glides down a helicold (c = Jc^ In cylindrical
the o<iuation8 of motion in (r, ^), (r, «), or (f, ^)« and carry the Integratioa ea fat
as iK)s8ible toward expressing the poeltlon aa a function of the tiaa.
418 INTEGRAL CALCULUS
8. If « = ax^ + 6i/2 + . • • , with a > 0, 6 > 0, is the Maclaurin expansion of a
surface tangent to the plane z = 0 at (0, 0), find and solve the equations for the
motion of a particle gliding about on the surface and remaining near the origin.
9. Show that r{l -\- q^) -{■ t{l ■\- p^) — 2pqs = 0 is the partial differential equa-
tion of a minimum surface ; test the helicoid.
10. If p and S are the density and tension in a uniform piano wire, show thai
the approximate expressions for the kinetic and potential energies are
Obtain the differential equation of the motion and try for solutions y = P{x) cos nt.
11. If f , »;, fare the displacements in a uniform elastic medium, and
OX dy dz \dy dzj \dz dxj \dx Zyj
are six combinations of the nine possible first partial derivatives, it is assumed that
V = j i j Fdxdydz, where 2^ is a homogeneous quadratic function of a, 6, c,/, gr, A,
with constant coefficients. Establish the equations of the motion of the medium.
82| _ ^2^ s^ d'^F Shi _ ^F^ ^F_ S^
^d^~dxda dydh dzdg* ^ dt^ " dxdh dydb dzdf'
S^_ d^F S^ d^
^di^~^xdg dy^ dzdc'
12. Establish the conditions (11) by the method of the text in § 165.
13. By the method of § 169 and footnote establish the conditions at the end
points for a minimum of fF{x, y, y') dx in terms of F instead of *.
14. Prove Stokes's Formula I = CF'dr = CfVxF'dS of p. 345 by the calculus
of variations along the following lines : First compute the variation of I on pass-
ing from one closed curve to a neighboring (larger) one.
SI = S f F.dr = f (3F.dr - dF.5r) + f d(F.5r) = f (VxF).(5rxdr),
where the integral of d (F.5r) vanishes. Second interpret the last expression as
the integral of VxF.tZS over the ring formed by one position of the closed curve
and a neighboring position. Finally sum up the variations SI which thus arise on
passing through a succession of closed curves expanding from a point to final coin-
cidence with the given closed curve.
15. In case the integrand contains y" show by successive integrations by
parts that
where r = ??. Y^ = ^, Y- = ^, « = «y.
dy dy' ay"
PART IV. THEORY OF FUNCTIONS
CHAPTER XVI
INFINITE SERIES
162. Convergence or divergence of series.* Let a Berim
X" = ^ + '^ +".+ ••+ «.-i+ «.+ (1)
0
the terms of which are constant but infinite in number, be given. Let Um
sum of the first n terms of the series be written
•-I
5. = w^ + ttj + w,+ -+tt.-i = 2I*^ W
Then -5^, •S,,^,,...,^'.,^.^,,... '
form a definite suite of numbers which may appnxuh a d^nite nmu
lim S^ = S when n becomes infinite. In this case the series ii said to
converge to the value 5, and Sj which is the limit of the sum of the first
n terms, is called the sum of the series. Or 5. may not approach a limit
when n becomes infinite, either because the values of 5, beoome infinite
or because, though remaining finite, they oscillate about and fail to
settle down and remain in the vicinity of a definite Tains. In these
eases the series is said to diverge.
The necessary and sufficient condition that a series eomverf^ is that e
ralue of n may be found so large that the numerical value o/S^^^ — 5,
s/ia/l be less than any assigned value for every value ef p, (See § 21,
Theorem 3, and comjmre p. 356.) A sufficient condition that a series
diverge is that the terms u^ do not approach the limit 0 when » beeoinei
infinite. For if there are always terms numerically as great as soma
number r no matter how far one goes out in the series, there MiiBi
;ilways be successive values of .S', which differ by as much as r no
matter how large n, and hence the values of 5, cannot possibly settle
down and remain in the vicinity of some definite limiting valne A
• It will be useful to fend over Chap. II. ff !*"»• •»* tOLMn^tm It It aks sii
to compare many of th> result* for infinite
iuflnite integrals (Chap. XIII).
410
420 THEORY OF FUNCTIONS
A series in which the terms are alternately positive and negative is
called an alternating series. An altematiny series in which the terms
approach 0 as a limit when n becomes Infinite, each term being less than
its predecessor y will converge and the difference between the sum, S of the
series and the sum S^ of the first n terms is less than the next term w„.
This follows (p. 39, Ex. 3) from the fact that| 5„^.^ - -S^nl < ^^ and ic^ = 0.
For example, consider the alternating series
1 - x2 + 2x* - 3x« + • . . + (- l)«ruc2n + . . . .
If |x| ^ 1, the individual terms in the series do not approach 0 as n becomes infinite
and the series diverges. If jx| < 1, the individual terms do approach 0 ; for
1
lim nx2»* = lim = lim = 0.
f, = oo n = floX-2« n = « — 2x-2»logX
And for sufficiently large* values of n the successive terms decrease in magnitude
«nce ^_j J
nx2'» <(n — l)x2»-2 gives > x^ or n>
n l — X"
Hence the series is seen to converge for any value of x numerically less than unity
and to diverge for all other values.
The Comparison Test. If the terms of a series are all positive (or all
negative) and each term is numerically less than the corresponding terrh
of a series of positive terms which is known to converge, the series con-
verges and the difference S — S^ is less than the corresponding difference
for the series known to converge, (Cf. p. 355.) Let
^o + ^^i + ^aH h^«„_iH-^„H
and wj + «^i + ^2 H f- i^;_i + < H
be respectively the given series and the series known to converge.
Since the terms of the first are less than those of the second,
Now as the second quantity 5^^^ — S'^ can be made as small as desired,
80 can the first quantity S^^p — 5„, which is less ; and the series must
converge. The remainders
00
R.= S-S, = U, + i^n+l + •• • =X ^'
n
< = s-- 5; = < + <+i + --^ =i; ^^'
• It shoald be remarked that the behavior of a serjiBs n^ftr ^8 beginning is of no conr
Mquence In regard to its convergence or divergence ; the first iV terms may be added
and ooiMidered as a finite sum Sy and the series may be written as Sy + uy -f- uy+\ -f • • • ;
It U the properties otuy'^uy^i-\ which are impoHant, that la, the ultimate behavior
of the series. .'i-v <«'.<.. -in:,.^- ) ...• H;,f:..)a;
INFINITE SERIES
4S1
clearly satisfy, the stated relation iS. < 7^. Tb« aeriM whieb It
frequently used for comparisou with a given seriat it tbt
a -4- fl'* + «'** 4- «/** -f
a#*
which is known to converge for all valoet of r
For example, consider the seriea
1
0<r<l,
L
W
^+^+^r8+24r4+--^f.+
and
»+i+ri+r.ir,+
Fi+'+tM
Here, after the first two terms of the first and the flnt t«nn o< the aaeood, mgk
term of the second is greater than the corresponding term of the flnt.
first series converges and the remainder after the term I/r 1 is less than
2»
^<^ + o;Ti + -- = ^
1 1
1
A better estimate of the remainder after the term 1/m I may be had bjr
1.1. ... 1 . 1
K« =
(n + 1) ! ■*■ (n + 2) ! "^
with
(n + l)l (n + l)l(» + l)
i
Bl«'
163. As the convergence and divergence of a leriet are of vital im-
portance, it is advisable to have a number of tests for the
or divergence of a given series. The test
by comparison with a series known to con-
verge requires that at least a few types of
convergent series be known. For the estab-
iishment of such tyi)es and for the test
of many series, the terms of which are
positive, Cauchi/s integral test is usefuL
Suppose that the terms of the series are
decreasing and that a function /(n) which deoreatet otn be found tveli
that n^ =f(n). Now if the terms w, be plotted at unit intamlt aloof
the n-axis, the value of the terms may be interpreted at the arM of
certain rectangles. The curve y =/(n) lies above the raotangltt and
the area under the curve is
\
^.
\
■-^
■t~
■0
"-L
-^
/■
/(n)rfn>u,-hK,4---+v,. (4)
Hence if the integral converges /which in praetioe meant that if
Cf{n)dn^F(n), then ^/(ii) = F(oo) - F(l) It anita),
422 THEORY OF FUNCTIONS
it follows that the series must converge. For instance, if
be given, then m„ =f(n) = 1/n^t and from the integral test
provided p > 1. Hence the series converges if j9 > 1. This series is
also very useful for comparison with others ; it diverges if ^ ^ 1
(see Ex. 8).
The Ratio Test. If the ratio of two siiecessive terms in a series of posi-
tive terms approaches a limit which is less than 1, the series converges;
if the ratio approaches a limit which is greater than one or if the ratio
becomes infinite, the series diverges. That is
if lim -^^^ = y < 1, the series converges,
n = oo u^
u
if lim ^^^ = y' > Ij the series diverges.
For in the first case, as the ratio approaches a limit less than 1, it must be pos-
sible to go so far in the series that the ratio shall be as near to 7 < 1 as desired,
and hence shall be less than r if r is an assigned number between 7 and 1. Then
Un+l<rUn, Un + 2<rUn+l<r^Un,---
and w« + u„+i + Wn + 2 + --- < m„(1 + r + r2 + ...) = if„-
1 — r
Ihe proof of the divergence when m„ +i/w„ becomes infinite or approaches a limit
greater than 1 consists in noting that the individual terms cannot approach 0. Note
that if the limit of the ratio is 1, no information relative to the convergence or
divergence is furnished by this test.
If the series of numerical or absolute values
of the terms of a series which contains positive and negative terms
converges, the series converges and is said to converge absolutely. For
consider the two sums
^n^p - -5. = w» + • • • + w,+p-i and |w„| + • • • + |^»+p-i|.
The first is surely not numerically greater than the second; as the
second can be made as small as desired, so can the first. It follows
therefore that the given series must converge. The converse proposition
INFINITE SERIES 4t|
that if a series of positive and negative terms oooTerget, then Um wuim
of absolute values converges, is not true.
As an example on convergeDce consider the binomial mrim
12 12 8 l-f...m
where ljf^l = l^!iJlAl,x,, ||„ Ijlllll , ,„.
It is therefore seen that the limit of the quotient of two ■neeemlv* terme la Um
series of absolute values is |2|. This is leM than 1 for valuee of u BiUMtieaUj Um
than 1, and hence for such values the series converges and cooveifas
(That the series converges for potUive values of z less than 1 foUows f n
that fur values of n greater than m + 1 the series alternates and tJ
0 ; the proof above holds equally for negative values.) For values of s
greater than 1 the series does not converge absolutely. As a matter of fact
|x| > 1, the series does not converge at all ; for as tbe ratio of am
proachis a limit greater than unity, the individual terms cauiot approaeb 0. For
the values x = ±\ the test fails to give information. The condasions are tlMra-
fore that for values of |z|<l the binomial series converges absolutely, for valoas
of |x|> 1 it diverges, and for |2| = 1 the question remains doubtful.
A word about series with complex terms. Let
^'o + «i + ^a H + «.-! -f «. + • •
= w; H- i/'i -h w; H h <_i + «H
+ i{< + y'l + < -h • • • + «:'-i + t'l' + • • •)
be a series of complex terms. The sum to n terms is .S, = 5^ -♦- i.*C.
The series is said to converge if 5, approaches a limit when n becomes
infinite. If the complex number «S', is to approach a limit, both its real
part 5;, and the coefficient S'^ of its imaginary part must approaeh limits,
and hence the series of real parts and the series of imaginary parts
must converge. It will then be possible to take i» so large that for any
value of ^ the simultaneous inequalities
|s'.+,-s'.i<i. and |s:v,-s:i<j.,
where < is any assigned number, hold Therefore
i5.,,-5.|^i5;,,-s:i+|.-5:,,-.-5:i<c
Hence if the series converges, the same condition holda as for a
of real terms. Now conversely the condition
|5.^^-5,|<« implies |5;^,-5;|<., \s:^p- S:\Kt,
Hence if the condition holds, the two real series oonverge and tbe
plex series will then converge.
424 THEORY OF FUNCTIONS
164. As Cauchy's integral test is not easy to apply except in simple cases and
the ratio test fails when the limit of the ratio is 1, other sharper tests for conver-
gence or divergence are sometimes needed, as in the case of the binomial series
when a; = ± 1. Let there be given two series of positive terms
uo + Ml + ••• + M„ + ••• and vo + t)i + ••• + Un + •••
of which the first is to be tested and the second is known to converge (or diverge).
// the ratio of two successive terms u„ + i/u« ultimately becomes and remains less {or
greater) than the ratio r» + i/c», the first series is also convergent {or divergent). For if
M, B« M„+l Un + 1 Vn U„ + 1 »« + 2
Hence if Un = pVn, then u„ + i</ow„ + i, u„ + .2< pvn + i, •••,
and Un + Un + l + Un + 2 + • • • < P ("n + Un + 1 + V» + 2 + •••)•
As the c-series is known to converge, the pu-series serves as a comparison series
for the u-series which must then converge. If u„ + i/a„ > Vn + i/Vn and the r-series
diverges, similar reasoning would show that the w-series diverges.
This theorem serves to establish the useful test due to Raa^e, which is
if lim n I— 1) > 1, /S„ converges ; if lim n I— 1 ) < 1^ ^n diverges.
« = » \Un+l I n = » \M„+1 /
Again, if the limit is 1, no information is given. This test need never be tried
except when the ratio test gives a limit 1 and fails. The proof is simple. For
I = IS finite
J n(logn)i+<' a(logn)«J
/* dn ~l "
= log log n is infinite,
n log n J
hence -- — — — + • • • + — : — + . • • and
2(log2)i+« n (log 71)1+" 2 (log 2) n(logn)
are respectively convergent and divergent by Cauchy's integral test. Let these b«
taken as the tJ-series with which to compare the u-series. Then
Pn _ n + 1 /log(n + l)V+^
/log(n +J_)y+-^ / ^ 1\ /ioga±n)y+«
\ logn / \ n/\ logn /
and J!^ = fl + lV-28£±2)
t>»+i \ nj logn
in the two respective cases. Next consider Raabe's expression. If first
Ii,nn(-^-l)>l, then ultimately n{-^^-\\>^>\ and -^>1 + -
INFINITE SERIES 415
where c l» arbitrarily Hinall. Hence ulUmfttelj if > > t,
or V- + 1 <"•/«. + 1 or M.4i/M.<B.^i/gb,
and thu u-series converges. In like manner, ■econdly, If
lim n(— - - 1 1< 1, then nlUnuUely J!2- < 1 + ?, >< 1 ;
..., i+><(i + l\!^^(L+iL) or -^<_5i. or !!i±l>5L±i.
n \ n/ logn 11,41 b,*i «, iW
ll(u«(> as the o-series now diverges, the u-eeries must diverge.
Suppose this test applied to the binomial series for x =: — 1. Tbto
Xn — m f ■-», m
u, n + 1 ,.
— — = . lim
Wn + i n — m «-» , , I — .::
n
It follows that the series will converge if m > 0, but diverge lfm<0. Ifxw^l,
the binomial series becomes alternating for n > m + 1. If the series of ihsolla
values be considered, the ratio of successive terms lUa/Ua 41 1 Is still (r ^f !)/(» -> a)
and the binomial series converges absolutely if m > 0 ; but when SK 0 tiM series
of absolute values diverges and it remains an open question whether the allemM-
in^ series diverges or converges. Consider therefore the alternating series
This will converge if the limit of u. Is 0, but otherwise it will diverge. Now If
m ^ — 1, the successive terms are multiplied by a factor |m — « •(- \\/% K 1 f'
they cannot approach 0. When - 1< m < 0, let 1 + m = *, a frmctkm. Tkn the
jitli term in the series is
KI = U-*,(i-()-(.-3
-log|«.| = -log(l-»)-log(l-j) '"«('- O*
and
Each successive factor diminishes the term but diminishes It by so little that It aftj
not approach 0. The logarithm of the term Is a series. Now apply Caucliy*s te«.
J-_log(l_?)dn = [-nlog(l-?) + ^log(«- #)]•»•.
The series of logarithms therefore diverges and llm|ii,|«c-« ■ ©. Utmm Iht
terms approach 0 as a limit. The final results are therefore that wbsn • s — 1 Ifct
binomial series converges if m > 0 but diTeigst If m< 0 ; aod when x a ♦llt<g»
verges (absolutely) if m > 0, diverges If m < - 1, and eouftigea (not abntaUlf) W
- 1 < m < 0.
426 THEORY OF FUl^CTIONS
EXERCISES
1. State the number of terms which must be taken in these alternating series to
obtain the sum accurate to three decimals. If the number is not greater than 8,
compute the value of the series to three decimals, carrying four figures in the work :
^^^ 3 ~2T3«''"3T3«~4T3* "'"*'' ^^ 2 ~ 2^ "^ 3T2« ~ 4^2^ "^ " *'
(7) l-2^3-i + ---' <*) i^-i^ + i^-'"'
2. Find the values of x for which these alternating series converge or diverge;
a'' 3'
11 x^ x^ s!^
a; x + 1 x + 2 x + 3 'as x + 1 « + 2 a; + 3
3. Show that these series converge and estimate the error after n terms :
(«'i + ^. + l + P + -' <^>§ + lr5 + |i|i^+-'
11 11 /1\2 /1 . 2\2 /1 . 2 . 3\2
<^>2+2-:25 + 3-:i;+riJ + -' <»> (3) + (3-5) + (s-itt) + • -
From the estimate of error state how many terms are required to compute the
series accurate to two decimals and make the computation, carrying three figures.
Test for convergence or divergence :
(«) 8inl + 8ini + sini + --, (f) sin^ 1 + sin2- + sin^- + . . .,
2 3 2 o
(17) tan-^H- tan-»- + tan-i - + ..., {$) tanl + — ^tan- + — ^tan- + • ••,
2 3 y/2, 2 ^3 ^
^'^ in"*'2W2"^8T^"^'**' ^'^ ^rzT2'^3^3T2^^irr8i + '"'
X z'' X» X* ' X X* X« X*
4. Apply Cauchy's integral to determine the convergence or divergence :
^''^^+ 2P + 8P +^r+*'' ^^^^ + ^0^^ 8(l^g3)^ 4(log4)P+ '
INFINITE SERIES 4S7
JS. 1 • I
Aniognloglog.' ''■AalogaOaglota)*'
(.)cot-.l + cot-.» + .... (ni + _l_ + _i_ + _l_ + ....
6. Apply the ratio test to detennine oooTargwioe or dJTMftoM 1
, , 2! 3! 4! , 6! . ,^ «• 8* ¥
(.) Ex.8(a),(«,W,(i); Ex.4(»).(0, (f) ^+ ^ + |^ + ....
6. Where the ratio test fails, discuss the aboye exercises by any meClMMl.
7. Prove that if a series of decreasing positive terms conveffes, llm aaia s t.
8. Formulate the Cauchy integral test for diyergenoe and check the suMmmoI
on page 422. The test has been used in the text and in Sx« 4. Pram the IMI.
9. Show that if the ratio test indicates the diveifmoe of the mHm of
values, the series diverges no matter what the distribution of itgno bmj be.
10. Show that if Vu^ approaches a limit less than 1, the seiiei (of
terms) converges ; but if -y/u,, approaches a limit greater than 1, it
11. If the terms of a convergent series u^ 4* U| 4- S "^ " * ^' positive
multiplied respectively by a set of positive numbers a,, a,, a,, • • • all of ^
less than some number (?, the resulting series a^u^ + 0,111-1- <ij«4 + • • • c
State the corresponding theorem for divergent series. What if the given
terms of opposite signs, but converges absolutely f
12. Show that the series — — — + — — -r— -f • • • conveiijes
1* 2» 8" 4*
lutely for any value of x, and that the series l + «sln^ + «*slnX# + **iint# + -'
converges absolutely for any z numerically less than 1, no matter what # aiay be.
13. If Oq, a,, a,, • • are any suite of numbers sueh that \4<iw|
limit less than or equal to 1, show that the series a, + a^z + «,«• +
absolutely for any value of x numerically less than 1. Apply this to show tbal the
following series converge absolutely when jx| < 1 ;
(a) l + lx> + l;i«« + l^*« + .-, (0) l-J» + 8*«-4x«+- .
(7)l + i + 2P*« + 8»«» + 4i>i« + --, (»)l-*logl + «»k)t«-««loi»*
428 THEORY OF FUNCTIONS
14. Show that in Ex. 10 it will be sufficient for convergence if v^ becomes
and remains less than 7 < 1 without approaching a limit, and sufficient for diver-
gence if there are an infinity of values for n such that -\/iin > 1. Note a similar
generalization in Ex. 13 and state it.
15. If a power series Qq + a^x + a^x^ + a^x* + • • • converges for jc = X> 0, it
converges absolutely for any x such that |a;| < -X", and the series
ttfft + ^ OjX* + ^ ajjX^ + • • • and a^ -\- 2 a^x -\- 3 a^x"^ + • • • ,
obtained by integrating and differentiating term by term, also converge absolutely
for any value of x such that |x| < X. The same result, by the same proof, holds if
the terms Oq, ol^X^ a2-^^ * * * remain less than a fixed value G.
16. If the ratio of the successive terms in a series of positive terms be regarded
as a function of 1/n and may be expanded by Maclaurin's Formula to give
M„ 1 u/l\2 1
= a + /5--|--(-), \t. remaining finite as - = 0,
Wn+i n 2\n/ ° n
the series converges if a > 1 or « = 1, /3 > 1, but diverges if a < 1 or a: = 1, /3 ^ 1.
This test covers most of the series of positive terms which arise in practice. Apply
it to various instances in the text and previous exercises. Why are there series to
which this test is inapplicable ?
17. If /9q, /Oj, Pz,--- is a decreasing suite of positive numbers approaching a
limit X and Sq, -Sj, S^,- •• is any limited suite of numbers, that is, numbers such
that I -Sn I ^ Cr, show that the series
(Po - Pi) ^0 + (Pi - P2) -^1 + (P2 - P3) -^2 + • • ■ converges absolutely,
^Gip,-\).
and ^{pn-pn + l)Sn
0
18. Apply Ex. 17 to show that, p^, p^, />2i • • • heing a decreasing suite, if
Wo + Uj + Mg + • • . converges, PqUq + p^u^ + p^u^ + • • • will converge also.
N.B. po^o + Pl^l + -'-+PnUn = PoSi + Pi (Sg " ^l) + ' " ' + P« (S«+l - Sn)
= -Si (/»o - Pi) + • • • + Sn {pn-l - pn) + pnSn + l-
19. Apply Ex. 18 to prove Ex. 15 after showing that p„Uo + p^u^ + • ■ • must
converge absolutely if p^, + p^ + . . . converges.
20. If ttp Cj, ttg, • . . , an are n positive numbers less than 1, show that
(1 + aj) (1 + Oj). . . (1 + a„) > 1 + ai + aa + • • • + On
and (1 - tti) (1 - a,) . . . (1 - a„) > 1 _ ai - a^ a^
by Induction or any other method. Then since 1 + a^ < 1/(1 — a,) show that
1 - (at -f a, 4- ... -t- a,) > ^^ + ''i) (^ + ^2) • • ' (1 + ««) > 1 + («, + «, + • • • + ««)»
INFINITE SERIES 4t9
if a, + 0, + .+ a, < 1. Or If TT be the ^mbol for a
(l-2^) >lT(l + a)>l + 5;a, A4-2)«T'>'frO--)>l-]g«.
21. Lttfr(l + Ui)(l + u,)...(l+tt,)(l+M,^i)...be an loAiUto pr
let I\ he the product of the flret n factors. Show that IP. -fy — P.l << It
Hary and suflicient condition tliat P, approach a liDiit whan a beeoi
Show that Um must approach 0 aM a limit if P« approaches a UniL
22. In case P. approaches a limit different from 0, show that if t
a value of n can be found so large tliat for any value of p
Pn
-i|=rTy(i+u,)-i|<« or "ff (i+m)=i+f, if:<
I ln+l I •♦I
Conversely show that if this relation holds, P. must approach a limit ociit-r itwn o
The ir\finite product js said to converge when P. approaches a limit other than 0 : in
all other cases it is said to diverge, including the case where lim P. = 0
23. By combining Exs. 20 and 22 show that the neowry and m
dition that
P, = (l + aO(l + a,)...(l + a,) and gu = 0 -«,)(!- «,)---(t -O
converge as n becomes infinite is that the series Oj + a, + • • • +a, + •
verge. Note that Pn is increasing and Q. decreasing. Show that in ease 2m
P, diverges to oo and Q« to 0 (provided ultimately Oi < 1).
24. Define absolute convergence for infinite prodacts and aliow that If a prodort
converges absolutely it converges in its original form.
25. Test these products for convergence, divergence, orabaoluta
(7) tr[i-(^)']. (») (1 +«)(« + «^(i+ «•)(! + «^-
26. Given -i^ or -u*<u- log (1 + u) <- m« or -l^ according as « w a pn*.
1 + u 2 2 1 + M
tive or negative fraction (see Ex. 29, p. 11). Prove that If XaJ
w» + 1 + M„ + 2 +•• + w«+p - Jog (1 + M, + 1) (1 + lU + i)- •• 0 +"•♦#)
can be made as small as desired by taking n laige enoofb TCfafdl«0 at p, Hanee
prove that if 2u* convergoH, 17(1 + «») conveiijes If 2ii« doea, hot diver«ea to •
if 2m„ diverges to + oo , and diverge to 0 if Sn, diveruea to — • ; whersaa II X<
diverges while 2u« converges, the product divexgea to <l»
480 THEORY OF FUNCTIONS
27. Apply Ex. 26 to: (a) (l + l)(l " ^)(l + i)(l - ^) * ' ' ^
»(-^J(-^J(-^>- <"(-0(-f)(-f)('-?);-'
28. Suppose the integrand /(x) of an infinite integral oscillates as z becomesln-
finite. What test might be applicable from the construction of an alternating series?
165. Series of functions. If the terms of a series
Six) = u^(x) -h u^ (:x)+ . . . + u„(x) + • • • (6)
are functions of x, the series defines a function S(x) of x for every
value of X for which it converges. If the individual terms of the series
are continuous functions of x over some interval a ^ x ^ bj the sum
5, (x) of n terms will of course be a continuous function over that interval.
Suppose that the series converges for all points of the -interval. Will it
then be true that S(x), the limit of S^(x), is also a continuous function
over the interval ? Will it be true that the integral term by term, -
J/^b r*h pb
u^(x)dx + I u^(x)dx H , converges to , j S(x)dx?
a %J a %J a
Will it be true that the derivative term by term,
u'^{x) •\- u{ix) ■\- ' " ^ converges to S'(x)?
There is no a priori reason why any of these things should be triie ; for
the proofs which were given in the case of finite sums will not apply
to the case of a limit of a siun of an infinite number of terms (cf. § 1^).
These questions may readily be thrown into the form of (questions concerning
the possibility of inverting the order of two limits (see § 44).
For integration : Is f Mm Sn{x)dx= lim f Sn{x)dx? ■ ■'}
For differentiation : Is — lim S^ (x) = lim — S„ (x) ?
dXn = » n^oodx
For continuity : Is lim lim S„ (x) = lim lim Sn {x) ?
As derivatives and definite integrals are themselves defined afe limits, the existence
of a double limit is clear. That all three of the questions must be answered in the
negative unless some restriction is placed on the way in which S«(x) converjges to
8{x) is clear from some examples. Let 0 ^ x ^ 1 and
8n{z) = xn^e-'^, then limiS,(x) = 0, or S{x) = 0..
No matter what the value of x, the limit of 8n(x) is 0. The limiting funetipii:is
therefore continuous in thia case ; but from the manner in which\/6^M(x) converge?,
INFINITE SEBDSS
4S1
to 8 (x) it is apparent that under mUuble oondiUooi U
tinuou8. The area under the limit S(z) = 0 from 0 to 1 la of
limit of the area under S^ (z) ia
Urn f zn*e-**dx= lim re-"»(- wx— 1) 1 as 1.
The derivative of the limit at the point x = 0 is
of course 0 ; but the limit,
Ml bs
0; battte
lim f— (xn^c-**) 1
•— « Ldx Jjt.
lim \n^
Mia OC |_
e-'«(l- nx)
] = lim
T
A
.<•»
4
Jr» V
\T
^
r M
1 J
of the derivative is infinite. Hence in this oaae two of the questions have nifSllTw
answers and one of them a positive answer.
If a suite of functions such as ^^(ar), 5j(ar), • • • , 5, (ac), • • • oooTerge to a
limit S(x) over an interval a ^ x ^ b, the conception of a limit reqoiret
that when c is assigned and x^ is assumed it must be possible to talni m
so large that |^«(^o)l = I'^W ~ '^-Wl "^ * ^^^ *^** '^^ *"y Iwger a.
The suite is said to converge uniformly toward its limit, if this condition
can V)e satisfied simultaneously for all values of x in the ii. ' * ' tiat is,
if when e is assigned it is possible to take n so large tl ,>| < •
for every value of x in the interval and for this and any larger n. In
the alx)ve example the convergence was not uniform ; the 6gure sbowa
that no matter how great n, there are always values of x between 0 and
1 for which S^(x) departs by a large amount from its limit 0.
The uniform convergence of a conttnuoujf function S^(x) to its limu m
sufficient to insure the continuity of the limit S{x), To show that S{x) tk
continuous it is merely necessary to show that when c is assiglied it
is ]X)ssible to find a Ax so small that |5(x -f Ar) — 5(x)| < c But
\S(x -f Ao-) - 5(^)1 = |5,(x H- Aa:) - S,{x) -f 7?,(x + Aor)- i?.(x)|; and
as by hypothesis R^ converges uniformly to 0, it is possible to take n
so large that \K(x + A.r) | and | /?, (x) \ are less than \ c irrespectire of jr.
Moreover, as .s\ {x) is continuous it b possible to take Ax so small that
|5,(a;4- Aa;) - Sjx)\ < J c irrespective of ar. HemMj|5(x + Ajt) -5(x)|< c,
and the theorem is proved. Although the uniform convergence of ^ to 5
is a sufficient condition for the continuity of S. it is not a ueoMtirr con-
dition, as the above example show-
The uniform convergence of S^{x) tn its nmu tn^urf that
lim r \ (x) fte = r S (x) dx.
432 THEORY OF FUNCTIONS
For in the first place S(x) must be continuous and therefore integrable.
And in the second place when c is assigned, n may be taken so large
that I R^ (x) \< €/(b - a). Hence
I rs(x)dx- f S,(x)dx\ = \ f R,(x)dx\< f J^dx = €,
\%J a %Ja I \U a \ \J a
and the result is proved. Similarly if S\ (x) is continuous and converges
uniformly to a limit T(x), then T(x) = S'(x). For by the above result
on integrals,
r T(x)dx = lim r S'^(x)dx = lim S^(x) - 5„(a) == S(x) - S(a).
Hence T(x) = S'(x). It should be noted that this proves incidentally
that if 5j,(x) is continuous and converges uniformly to a limit, then
S(x) actually has a derivative, namely T(x).
In order to apply these results to a series, it is necessary to have a
test for the uniformity of the convergence of the series ; that is, for the
uniform convergence of S^(x) to S{x). One such test is Weierstrass's
M-test : The series
u^(x) + u^{x) + ---^u,{x)-\-.'' (7)
wtU converge uniformly provided a convergent series
i»fo + ^A+--- + ^n4---- (8)
of positive terms may he found such that ultimately \ui(x)\ ^ 3/^. The
proof is immediate. For
and as the M-series converges, its remainder can be made as small as
desired by taking n sufficiently large. Hence any series of continuous
functions defines a continuous function and may be integrated term by
term to find the integral of that function provided an il/-test series may
be found ; and the derivative of that function is the derivative of the
series term by term if this derivative series admits an 3/-test.
To apply the work to an exa^nple consider whether the series
-. cosa; co82x , cosSa; . cosru; .
s(^) = -^ + -^r- + -8r- + --- + -^r- + --- (^o
defines a continuous function and may be integrated and differentiated term by
term as
dx ^ ' 1 2 8 n "" ^ '
INFINITE SERIES 4$$
As |coez|^ 1, the convergent series l4.-.4.i.^....^JL4,... Bay bt
rui 3f-«erie8 for S{x). Hence 8{z)ia% continuous function ol s f or «ll iml
nf X, and the integral of S{x) may be uken u Um limit of Um UM«fral «| B^i^
that i8, as the integral of the series term by tenn as writt«ii. On tb* oUmt ^H.
:iii If-series for (7'") cannot be found, for tbe series 1 + | -f | 4- . . . Is Micosfw^
^'(•nt. It therefore appears that S'{i) may not be identical witb tb« tsrai bi f
• Itrivative of S (x) ; it does not follow that It will not be, — nnely that it may not bai
166. Of series with variable terms, the
/W = «o + «i(« - «^) + a,(« -«)* + ••• + «.(»-«)■-«.... (9)
is })erhaps the most important Here «, or, and tbe ooeffieieato «| bmij
be either real or complex numbers. This series maj be wriUeo mote
simply by setting a: = « — or ; then
f(x + a) = it>{x) = a^ + a,x 4- a,** + ••• + o.ar' -I- (y)
is a series which surely converges for x = 0. It may or may noi ooi>»
verge for other values of Xj but from Ex. 15 or 19 abore it ia aeea
that if the series converges for Xj it converges absolutely for any «
of smaller absolute value ; that is, if a circle of radius X be drawn
around the origin in the complex plane for x or about
the point a in the complex plane for «, the series (9)
and (\)') respectively will converge absolutely for all
complex numbers which lie within these circles.
Three cases should be distinguished. First the
series may converge for any value x no matter how
g:eat its absolute value. The circle may then liave
an indefinitely large radius ; the series converge for all values of 2 or «
and the function defined by them is finite (whether real or complex)
for all values of the argument. Such a fiuiction is called an imttgrml
fmu'tinn of the complex variable z or x. Secondly, tbe series may ooo-
verge for no other value than a- = 0 or « = a and therefore eannoi define
any function. Thirdly, there may be a definite largest rains for the
radius, say 7?, such that for any point within the rsspeotiTe eireles of
radius R the series converge and define a function, whereas for any point
outside the circles the series diverge. The circle of radios R is sailed
the circle of convergence of the series.
As the matter of the radius and circle of oonTeigenoe Is inpoitaat, ll will bt
well to go over the whole matter in detail. Consider tbe ■alts of numben
Let them be imagined to be located as pointa with coordinates betwesaOaad 4 •
:)n a line. Three possibilities as to the distribution of tbe poInU arte. FIflA Ifeif
4a4 THEORY OF FUNCTIONS
may be unlimited above, that is, it may be possible to pick out from the suite a set
of numbers which increase without Jimit. Secondly, the numbers may converge to
the limit 0. Thirdly, neither of these suppositions is true and the numbers from 0
to 4- 00 may be divided into two classes such that every number in the first class is
less than an infinity of numbers of the suite, whereas any number of the second
class is surpassed by only a finite number of the numbers in the suite. The two
classes will then have a frontier number which will be represented by 1/R
(see§§19ff.).
In the first case no matter what x may be it is possible to pick out members
from the suite such that the set v^|a,-|, -(/\aj\, "VlOfcl, • • • , with i<j<k---, increases
without limit. Hence the set \/|at||x|, ■(/[%] |xj, • • • will increase without limit ; the
terms a,-a;» ajxJ, • • • of the series (O') do not approach 0 as their limit, and the series
diverges for all values of x other than 0. In the second case the series converges
for any value of x. For let e be any number less than l/|x|. It is possible to go so
far in the suite that all subsequent numbers of it shall be less than this assigned e.
Then
|an+pa?»+i'|<€"+P|a;|«+P and e'»lxl"+e» + i|a;|«+i + ---, €\x\<l,
serves as a comparison series to insure the absolute convergence of (Q'). In the
third case the series converges for any x such that |x| < /? but diverges for any
xsuchthat|x|>B. For if \x\<R, take€<l? — |a;|so that|a;| < R — t. Now proceed
in the suite so far that all the subsequent numbers shall be less than 1/{R — e),
which is greater than 1/R. Then
.-"- |a„+pX«+i'|<-^-L-— -<1, and ^ ' '
, «. (R-e)« + P ^{R-e)^+P
will do as a comparison series. If |x] > /?, it is easy to show the terms of (G') do not
approach the limit 0.
Let a circle of radius r less than R be drawn concentric with the
circle of convergence. Then within the circle of radius r < R the power
series (9') converges uniformly and defines a continuous function; the
integral of the function may he had by integrating the series teiin by
term,
^(x)
XI 1 1
and the series of derivatives converges uniformly and represents the
derivative of the function,
4i\x) = ttj + 2 a^jsc -f- 3 aga;2 H f- na^x*"-^ H .
To prove these theorems it is merely necessary to set up an 3f-series
for the series itself and for the series of derivatives. Let .Y be any
number between r and R. Then
|aJ-MaJ.Y4-|s|.Y^ + -..-f|«H|-Y»-f- (10)
INFINITE 8ERIE8 4S6
converges because X<R; and furthermore \aj^\ < |a.| JP holds for aaj
./ such that \x\< X, that is, for all pointa within and on the eirale of
radius r. Moreover as \x\ < JT, '
\^^-'\ = \a.\ji^""x'<\a.\X'
holds for sufficiently large values of n and for any a mieh that |x| S r.
Hence (10) serves as an A/-series for the given seriea and the aeriee of
derivatives ; and the theorems are proved. It should be noUeed that it
is incorrect to say that the convergence is uniform over the eirele of
radius Ry although the statement is true of any circle within thai eirole
no matter how small R ^ r. For an apparently slight but none the
less important extension to include, in some oases, some points upon
the circle of convergence see Ex. 6.
An immediate corollary of the above theorems is that ony
series (9) in the coviplex variable which eanveiyei for other waUu
z = a^ and hence has a Jinite circle of converyenee or eomwytM all
the complex planCy defines an anal i/tie function /{m) of z in tko oenoo of
§§ 73, 126; for the series is dififerentiable within any circle within tibm
circle of convergence and thus the function has a definite finite and
continuous derivative.
167. It is now possible to extend Taylor^sand Maclaurni's Formulas,
which developed a function of a real variable x into a polynomial plus
a remainder, to infinite series known as Taylor's and Maclaurin's Seriea,
which express the function as a power series, provided the rematoder
after n terms converges uniformly toward 0 aa n beoomea infinite. U
will be sufficient to treat one case. Let
f(x) =/(0) +/'(0)x + ^/"(0)x« + . . . + J^^ZTp/*'''^^^^'' "•" ^
lim R^(x) = 0 uniformly in some interval — A S x S A,
where the first line is Maclaurin^s Formula, the second gives diffemet
forms of the remainder, and the third expresses the condition that the
remainder converges to 0. Then the series
/(0)+/'(0)a + |;/"(0)*'
1 /<.-i)(0)i— + l/«->(0)i' + (11)
(«-l)!
436 THEORY OF FUNCTIONS
converges to the value f{x) for any x in the interval. The proof con-
sists merely in noting that f{x) — R^ (x) = S^(x) is the sum of the first
n terms of the series and that |/?»(a;)| < c.
In the case of the exponential function e^ the nth derivative is e*, and the re-
mainder, taken in the first form, becomes
As n becomes infinite, Kn clearly approaches zero no matter what the value of h ,
and 2 8 a;»
is the infinite series for the exponential function. The series converges for all
values of x real or complex and may be taken as the definition of e^ for complex
values. This definition may be shown to coincide with that obtained otherwise (§ 74) .
For the expansion of (1 + a;)"* the remainder may be taken in the second form.
Rnix) = — ^^ ' ^^ -X^i —I (l-f^X)'"-!
1^(^)1 <
(n-1)
m (m — 1) • • • (m — n + 1) 1
/i»(H-/i)"'-S ^<i.
1 . 2 ... (n - 1)
Hence when h<l the limit of B„ (x) is zero and the infinite expansion
(1 -f- X)"» = 1 + TMX -I ^ x^ + — ^^ — x* + • . •
^ I o !
is valid for (1 -|- x)"* for all values of x numerically less than unity.
If in the binomial expansion x be replaced by — x^ and m by — |,
1 1 . 1 2 . 1-3 4 . 1-3-5 - , 1.3.5.7 - ,
VriT^ 2 2.4 2.4.6 2.4.6.8
This series converges for all values of x numerically less than 1, and hence con-
verges uniformly whenever (xj ^ ^ < 1. It may therefore be integrated term by
**""• . , .lx« 1.3x6 1.3.5x7 1.3.5. 7x»
sin-ix = X + h 1- + • • • .
2 3 2.4 62.4.6 72. 4. 6. 89
This series is valid for all values of x numerically less than unity. The series also
converges for x = ± 1, and hence by Ex. 5 is uniformly convergent when — 1 ^ x ^ 1 .
But Taylor's and Maclaurin's series may also be extended directly to
functions f(z) of a complex variable. If f(z) is single valued and has
a definite continuous derivative /'(«) at every point of a region and on
the boundary, the expansion
f(z) =f(a) +f'(a)(z - a) + .. . +/"■-> («)(^^~_">^"' + A>,
has been established (§ 126) with the remainder in the form
1 r" 3/£
l«.(*)l =
(z - «)■ f f(t)dt
2t Xit-ayit-z)
2ir p" p
(12)
INFINITE 8ERIE8 4t7
for all points % within the circle of radius r (Ex. 7, p. 906). At n
infinite, 11^ approaches zero unifortuly, and henoe the infinite
/(«) =/(«) +/'(«)(«-«) + ••• +/'-'(<.)i5^+...
is valid at all points within the circle of radius r and upon ite
ference. The expansion is therefore oonvergent and ralid fbr any •
actually within the circle of radius />.
Even for real expansions (11) the signifieanoe of this remit is grwa
l)e(;ause, except in the simplest cases, it is impossible toeQnpote/*^(#)
and establish the convergence of Taylor's series for real variablas. Tbe
result just found shows that if the values of the function be
for com])lex values z in addition to real values z, the circle of
genee will extend out to the nearest point where the oonditioot impotsd
on f{z) break down, that is, to the nearest point at which /(«) beeoaics
infinite or otherwise ceases to have a definite continuous deriratiYe /*(«).
For example, there is nothing in the behavior of tlie funciioo
(l-hx«)-» = l-x> + ar*-;r« + *»-...,
as far as real values are concerned, which should indicate why the expan-
sion holds only when |x| < 1 ; but in the complex domain the foneCkNi
(1 + ^'0~' becomes infinite ?it z = ± t, and hence the greatest eirek
about 2! = 0 in which the series could be expected to converge has a unit
radius. Hence by considering (1 -f «*)"* for complex Yalms, it can be
predicted without the examination of the nth derivatiTe that the lla»
laurin development of (1 -f x*)- * will converge when and only when m
is a proper fraction.
EXERCISES
1. (a) Doesx + x.(l - x) + x(l -«)«+•• con vergeiinlfonnlywheoOa«a It
03) Does the series (1 + fc)* = 1 + 1 + ?-^ + ^^ '"*^^!""**^ + ...eonfeV"*
formly for small values of it ? Can the derivation of the limit eof 1 4 thosbt asde
rigorous and the value be found by setting I; = 0 in the Mrfetf
2. Test these series for uniform convergence ; alao the Mites of derivmtlvw ;
(a) l + xsin^ + i«8in2^ + z««in3^-}- . Ix|SX<l,
^<7tfS^'r<«»*
o<tai«siX<«
(f) Consider complex m well m real values of the variable.
438 THEORY OF FUNCTIONS
3. Determine the radiiis of convergence and draw the circle. Note that in prac-
tice the test ratio is more convenient than the theoretical method of the text:
^^^ ^4727"^ +4.4!'^ 4.6!'^ + '
(iy) 1 - X + X* - x« + a;8 - a;9 + xi2 _ a;i8 ^ . . . ^
(19) (x-l)i-i(^-l)'' + H^-l)«-H^-l)* + ---,
(m-l)(m + 2) (m-l)(m-3)(m + 2)(m + 4)
(t)X — ^^ X + - X -...,
, , , x2 . X* x6
(f) 1-
22(m-|-l) 2*.2!(m + l)(wi + 2) 2^ • 3 ! (m + l)(m + 2) (m + 3) '
^ ^ 22 2*(2 !)2 \1 2/ 2«(3 !)2 \1 2 3/ 28(4 !)2 \1 "^ 2 "^ 3 "^ 4/ "^ ' " '
(u) 1 + J^x + ^(^ + 1)^0 + 1)^, ar(a + l)(ar + 2)/3(/3 + l)(/3 + 2)
1-7 1-2.7(7 + 1) l-2.3.7(7 + l)(7 + 2) "*" " *
4. Establish the Maclaurin expansions for the elementary functions:
{a) log(l-x), (/3) sinx, (7) cosx, (8) coshx,
(O a% (f) tan-^x, (v) sinh-ix, , (0) tanh-ix.
5. AbeVs Theorem. If the infinite series a^, + a^x + a^x^ + agX^ -|- . . . converges
for the value X, it converges uniformly in the interval O^x^X. Prove this by
showing that (see Exs. 17-19, p. 428)
\Rn{x)\ = \anX^ + On+ixn+i + • • • | < /-|V|a„X« + • • • + a„ + pX«+p),
when p is rightly chosen. Apply this to extending the interval over which the
series is uniformly convergent to extreme values of the interval of convergence
wherever possible in Exs. 4 (or), (f), (0).
6. Examine sundry of the series of Ex. 3 in regard to their convergence at ex-
treme points of the interval of convergence or at various other points of the circum-
ference of their circle of convergence. Note the significance in view of Ex. 6.
7. Show that/(x) = e x«, /(O) = 0, cannot be expanded into an infinite Mac-
laurin series by showing that R^ = e~^, and hence that iJ„ does not converge
uniformly toward 0 (see Ex.9, p. 66). Show this also from the consideration of
complex values of x.
8. From the consideration of complex values determine the interval of con-
vergence of the Maclaurin series for
r<I
INFINITE SERIES
9. Show that if two timllAr inflnlie power aeriM y..^,„ -tt
in any interval the coefBdenU \n the aeriee muat be equel (ef. flf).
10. From 1 + 2rco8X + r« = (1 + i^(l + re-*0 = '•(l + ^(l + '"^
prove log(H-2rcoflx + r*) = 2/rcoex-.^ooe2jt + -ooel« \
^'log(l + 2rcoe« + r«)dx = 2/rdn»-^«In«x + ^alnS« );
ami log(l + 2rco8z + »^ = 2loer + 2/??!^-22^ + 22!!*f «. \
j|j'loK(l + 2rco8z + r«)dx==2ilogr + 2(— -^l?f ^.!!!L?f \. ^
f log(l + 8inaco8Z^da; = 2xlogcog- -l-g/tAn-dng^fmnigfJgl* « \
•'o 2\2 IJ"/
11 P^v. r^_^-i_ 1 , 1-8 1.8.5 _r^_j»_
•^0 VT+^ 2. 6^2. 4. 9 2.4.«.1«^ Ji vm«'
12. p:valuate these integrals by expansion into aeriee (aee Sx. SS, p. 4M)
/•-e-«'8innc , r 1 /r\« 1 /r\» ,r
(a) / di; = __ (_)+(_) = t»n->-.
Jo X q 3\q/ b\q/ q
^0 COS 2 •/• 1 + ooa*s 4
/ e-«'^co8 2/Szdx = — -c V.;^ (,) f !og(l + Jrooa« + f^4i.
13. By formal multiplication (§ 168) show that
I -a*
= l + 2n:co«x + 2rt«ooa2x + . •,
1 — 2 a cos X + a*
asinx . . • , «
= a sin X + a' sin 2 / •
1 — 2aco8x + a*
14. Evaluate, by use of Ex. 18, these definite iniQgraU, m aa InUger:
/»» cosmxdx _ trct^ f*' ataJnadii *b»n a. v
*"^ Jo l-2aco8x + a«~n^«' ^' Jo l-2aooa« + «•" 5^' *•
, ^ /•» sin X sin fnx(ix w ,
^" Jo l-2crcoex + a« 2
/,^ f »^»'^
^ ' Jo (l-2aooa« + a^(l-2/Jooax-|.^
15. Ill Ex. 14 (7) let a = 1 - h/m and x = t/m. Obuin by a Umitlar
id by a similar method exercised upon Ex. 14 (a):
Jr«zsingdg_ w j^ /'•ooa£^_w ^
0 A« + «« "i*" ' Jf ik« + f«"«
('an the use of these liinitinir proceoaea be readily joatifladf
440 THEORY OF FUNCTIONS
16. Let h and x be less than 1. Assume the expansion
/(x, h) = ^ = 1 + APi(x) + h^PoSx) + • . . + hnP^x) + . . . ,
Vl-2xA + A2
Obtain therefrom the following expansions by differentiation :
i/^ = ? = P; + hP", + ;i2p; + . . . + /,n -ip; + . . . ,
'^ (l-2x/i + A2)5
/; = "^^ = Pj + 2/1P2 + 3A2P3 + . . . + nhn-'^Pn + '".
{l-2xh-\-h^)i
Hence establish the given identities and consequent relations :
^^= xF[-^h(xP',-P[)-^-.- + hn-l(xK-K-l) +••• =
(l±^y;;_/=-i + p; + A(p;-p,) +... + a«(p;+i + p;_i-p„) + ...=
2x/i/= /i(2x) +... + ;i«(2xP„_i).
Or nPn = xP'„-F'„_T, and P;+i + P;_i - Pn = 2xP;.
Hence xP; = P;+i - (n + 1) P„ and (x^ - l)r„ = n(xP„- P„_i).
Compare the results with Exs. 13 and 17, p. 252, to identify the functions with the
Legendre polynomials. Write
1 1 ^ 1
(1 - 2 x^ + /i^)* (1 - 2 A cos ^ + /i2)i (1 _ hei6) h{l-he- »«)^
= (1 + -h&& + —h^e^ie + . . A /i + i/ie-»« + ^h^e-^<9 + •••).
and show Pn(cos^) = 2^ ' ^" ' ^^^~ "^^ Jcosn^ + ^'^ cos(n- 2)tf + • • .}.
168. Manipulation of series. If an infinite series
5 = z., + t^i + t.2 + • • • + ^« -.1 + ^« + • • • (13)
converges J the series obtained by grouping the terms in parentheses with^
out altering their order will also converge. Let
5' = f/, 4- f^, + • • • + C/„, _ 1 + ^n^ + • • • (ISO
and S{, -Sa, --.j^;,, ••
be the new series and the sums of its first n' terms. These sums are
merely particular ones of the set 5^, S^,---, 5,,, ••■, and as n' < n it
follows that n becomes infinite when n' does if » be so chosen that
S^ = 5^,. As 5, approaches a limit, S'^, must approach the same limit
As a corollary it appears that if the series obtained by removing paren-
theses in a given series converges, the value of the series is not affected
by removing the parentheses.
INFINITE BEBIES 441
// two convergent infinite eeriet be given as
5 = tt^ + t«, + - •, and r-», + »,^...,
f^' ^n (Xu, 4- Mt^,) + (Xu, + M«'|) + • • •
will converge to the limit kS -^ ftT, and '"if! nrftnrryiT n/Wnfaf n/y jinn tfirf
both the given series converge absolutely. The proof if left to the
If a given series converges ahsolutely, the series formed by
the terms in any order without omitting any terms wiil eemmrf is tk§
same value. Let the two arrangements be
5 = tt^ + ttj + ti, + ... + «,., -H t«, + .. .
and 5 = w^ -f II,, -f M^ H (- «,,_, + u^^....
As S converges abeolutely, n may be taken so large that
|w,|H-|tt,+,| + --- < c;
and as the terms in S' are identical with those in S except for their
order, n' may be taken so large that S'^ shall contain all the temt in
S^. The other terms in S'^, will be found among the terms m., v. « , .
"^"^^ |5;,-5j<|u.n.|i/„,n- .-.<€.
As \S - 5J < e, it follows that|5 - S'^\<2€. Henoe S'^ appraeehet 8
as a limit when n' becomes infinite. It may easily be shown that S* also
converges absolutely.
The theorem in still true if the rearrangement of Sis into a eonoe rnms
of ivhose terms are themselves infinite series of terms seieeted firem S,
Thus let s'^U^-i-U^-hU,-^"--^ r,,_, + U^ + •••,
where Ui may be, any aggregate of terms selected from S. If C7| be an
infinite series of terms selected from A', as
^i = «« + «rt -I- tt« H h ^to H »
the absolute convergence of f/,- follows from that of S (cf. Ex. 22 below).
It is possible to take n' so large that every term in .S, shall ooeor in one
of the terms U^, U^, • •, ?/.,_,. Then if from
there be canceled all the terms of 5„ the terms which remain will be
found among i/„ i/, + „ • • • , and (14) will be lees than c Henoe M •'
becomes infinite, the difference (14) approaches lero as a limit and the
theorem is proved that
•5>' = ^0 + ^i •+• • -^■ ^^ - » ■^' ^'-^ -♦■ • • • ■ ^-
442 THEORY OF FUNCTIONS
If a series of real terms is convergent, but not absolutely, the number of posi-
tive and the number of negative terms is infinite, the series of positive terms and
the series of negative terms diverge, and the given series may be so rearranged as
to comport itself in any desired manner. That the number of terms of each sign
cannot be finite follows from the fact that if it were, it would be possible to go so
far in the series that all subsequent terms would have the same sign and the series
would therefore converge absolutely if at all. Consider next the sum Sn = Pi— ^m,
Z + m = n, of n terms of the series, where Pi is the sum of the positive terms and
NrH that of the negative terms. If both Pi and Nm converged, then Pi + N^ would
also converge and the series would converge absolutely ; if only one of the sums
Pi or Nm diverged, then S would diverge. Hence both sums must diverge. The
series may now be rearranged to approach any desired limit, to become positively
or negatively infinite, or to oscillate as desired. For suppose an arrangement to
approach I» as a limit were desired. First take enough positive terms to make the
sum exceed L, then enough negative terms to make it less than i, then enough
positive terms to bring it again in excess of X, and so on. But as the given series
converges, its terms approach 0 as a limit ; and as the new arrangement gives a
sum which never differs from L by more than the last term in it, the difference
between the sum and L is approaching 0 and L is the limit of the sum. In a similar
way it could be shown that an arrangement which would comport itself in any of
the other ways mentioned would be possible.
If two absolutely convergent series be multiplied, as
5 = i/o + i^i 4- ?^2 + • • • + ^n H y
T=Vq + Vi-{-V2-\ ]rV„-\ ,
and W = UqVq + UiVq -f u.2Vq -\ f- u^Vq -\
+ UqVi + Uj^v^ H- u^Vi H h u„Vi H
+
-f ?/o?;„ -f n^v^ -f u.,v„ H f- u„v^ H
+ •
and if the terms in W be arranged in a simple series as
or in any other manner whatsoever, the series is absolutely convergent
and converges to the value of the product ST.
In the particular arrangement above, S^T^, S^T^; S^T^ is the sum of
the first, the first two, the first n terms of the series of parentheses. As
lira 5„r, = ST, the series of parentheses converges to ST. As S and T
are absolutely convergent the same reasoning could be applied to the
series of absolute values and
KII''.l + l«.ll''.l+KIKI + l«olKI + l«.lKI + --
would be seen to converge. Hence the convergence of the series
INFINITE
is absolute and to the value ST when the pftraiithetM are onitlMl
Moreover, any other arrangement, such in particular an
would give a series converging absolutely to AT.
The equivalence of a function and its Taylor or Macknho
series (wherever the series converges) lends importanoe to the
of multiplication, division, and so on, which may be parfonnad oa Um
series. Thus if
A^) = «o + «i* -»- «i^ + «g»^ + • •» I'l < ^,.
^ W = *o + &,x + ft^ + A^ 4- • • •, \x\ < i?„
the multiplication may be performed and the seriet arranged as
f(x)ff(x) = 0,6, + (a,*, + afi;)z -h (a^, + a,4,'+ a/J«* +
according to ascending powers of x whenever x is numerically h
the smaller of the two radii of convergence /?,, 7?,, becaoie both eeriee
will then converge absolutely. Moreover, Ex. 5 abore ahowa that
form of the product may still be applied at the extremitSea of iti
val of convergence for real values of x provided the aeriea cooTeifea
for those values.
As an example in the multiplication of series let the pnM!urt»ii)x i**/ u- (. s;,!
sin^ = x-l«. + lx.-..., co.x = l-l,. + l..-l«. + ....
The product will contain only odd powers of x. The first few ienns ars
' '- (h + ty + (5I + 8-iTl + f:)'* - (fi + 6^. + 8lT. + •->' •
The law of formation of the coefficienU gives as the coefficient of i^* '^ *
(-1)* r. ■ (2ifc-n)2fc . (2ic4-i)(afc)(2fc-i)(>*-») . .<ii±L>l
(2ik + l)!L "^ 21 ■*■ 4l * "^ II J
But 2" + i ==(1 + !)«*+> = 1 + (2* + l)+2*.tM*+...+(l» + l) + l.
Hence it is seen that the coefficient of «•*♦> takes every other t«T» In thlsi
rical sum of an even number of terms and must therefore bt equal to kalf the mm.
The product mav then be written as the series
444 THEORY OF FUNCTIONS
169. If a function f(x) be expanded into a power series
f(x) = a^-^a^x + a^ + a^*^-'", \x\<R, (15)
and if a; = a is any point within the circle of convergence, it may he
desired to transform the series into one which proceeds according to powers
of (z — a) and converges in a circle about the point x = a. Let t = x — a.
Then x = a -\-t and hence
a^ = a^-\-2at + t^, x''= a^ -^ S aH -{■ S at^ -j- f, ••.,
f(x) = a^ -t- «i(a + 0 + ^\^' + 2 a:^ + ^ + . . . . (15')
Since |a| < R, the relation |a| -f- 1^| < R will hold for small values of t,
and the series (15') will converge for ic = |a:| + |^|. Since
«. + «i(i«i + I'D + «.(i«r + 2i«ii<i + I'D + • • •
is absolutely convergent for small values of t, the parentheses in (15')
may be removed and the terms collected as
f(x) = if>(t) = (a^-{'a^a-\-a^a^-{-ay-^---')-{-(a^ + 2a^a-\-Say-[-''')t
+ K + 3 V + • • O'^' + K + • • 0^' + • • •>
ot f(x) = <t>(x - a) = A^ + A^(x - a) + A^(x - af
+ ^3(^ -«)« + •••, (16)
where A^^ A^, ^2'"* ^^® infinite series ; in fact
^, =/("), ^, =/'(«), ^,=|j /"(«). ^, =!]/'"(«)>••••
The series (16) in a; — a will surely converge within a circle of radius
72 — I a] about x = a\ but it may converge in a larger circle. As a matter
of fact it will converge within the largest circle whose center is at a and
within which the function has a definite continuous derivative. Thus
Maclaurin's expansion for (1 + x'^~'^ has a unit radius of convergence;
but the expansion about x — \ into powers of ic — ^ will have a radius
of convergence equal to ^ Vs, which is the distance from a: = ^ to either
of the points x= ±i. If the function had originally been defined by
its development about a; = 0, the definition would have been valid only
over the unit circle. The new development about x = \ will therefore
extend the definition to a considerable region outside the original
domain, and by repeating the process the region of definition may be
extended further. As the function is at each step defined by a power
series, it remains analytic. This process of extending the definition of
a function is called analytic continuation.
INFINITE SERIES 445
Consider the expansion of a funetum of a fkmeHmL L0I
/W = «p + a,ar + a^4.a^+... |*| < JT,,
^ = ^(y) = *o-f-A^ + 6j^ + Ay+ .., |y|< if,,
and let \b^\ < R^ so that, for sufficiently small taIom of y, tbe poiot m
will still lie within the circle R^. By the tbeoram on nmltapUcatkiii, the
series for x may be squared, cubed, ■ • •, and the Miiet for «■,«•,.. . nay
be arranged according to powers of y. These leeulto may then be eob-
stituted in the series for/(x) and the result may be oideied aeeofdlaf
to powers of y. Hence the expansion for /[4(y)] it oblainad. l\m\
the expansion is valid at least for small values of y may be eeen hj
considering
f = l*.l+l».IW+l*.l|yr+-. |y| 8111.11.
which are series of positive terms. The radios of eooTefgeoee of the
series for/[<^(y)] may be found by discussing that function.
For example consider the problem of expandhig €***• to five teriM.
e>' = l + y+ Jy« + iy« + Ay* + ..., y = co«x = 1- 1*^+ ^««^....,
ya = l_xa+}x* , y» = l~|x«+lar« , y« = 1- S<« ^. l|s« .
ei' = l + (l-ixa + ,^x< )+l(l-«« + ix« ) + |(l-|/t+|^ )
+ A(l-2af« + I|r« ) + ...
= (1 + 1 + i + i + 1^ + . . •) - (i + i + i + A + • • •)«•
ev = CCO.X = 2JI - 1 Jx« + }f X*
It should be noted that the coefficient in tiiis series for e**' are raelly laflBlie
series and the finat values here given are only the approzlmale valoM foead ky
takin;; the first few terms of each series. This will alwaji be tbe eeet wtMa
y z=\^h^x-^ '" begins with \^(i\ it is also true In the ezpaaiioa aboot a aev
origin, as in a previous paragraph. In the latter gmb the diflfeolly eeaaot be
avoided, but in the case of the expansion of a function of a fenction It b «■»>
times possible to make a preliminary change which materially rimplUles the
result in that the coefficients become finite series. Thus here
eco«x = ei + « =ee*y « = cosz— 1 =— J«*+ A**~ Tli** + "*»
fico-' = ee« = e(l - 4«« + ix« - AV** + •)•
The coefficients are now exact and tbe computation to C* tanut
than to x^ by the previous method ; the advantage introdneed by <
be even greater if the expansion were to be carried aeveral terns lartiM-r.,
446 THEORY OF FUNCTIONS
The quotient of two power series f(x) by ff(x), if ff(0) ^ 0, may be
obtained by the ordinary alfforism of division as
fix) a,^a,x + a^^+'-' . ex . c^2 . .. . j, ^q
For in the first place as ^ (0) =?^ 0, the quotient is analytic in the neigh-
borhood of X = 0 and may be developed into a power series. It there-
fore merely remains to show that the coefficients c^^ c^, c^, • • • are those
that would be obtained by division. Multiply
(a, 4- a^x -h a^2 -f . . .) = (c, + c^x + c^^ _^ . • •) (b^ ^ h^x + b^' -{- ■ • ■)
= Vo + (Vo + Vi)^ + (Vo + Vi + Va)^' + • • • '
and then equate coefficients of equal powers of x. Then
% = Vo> «i = h% + .Vi» S = ^2^0 + ^1^1 + V2» • • •
is a set of equations to be solved for c^, c^, c^, - ■ ■ . The terms in f(x) and
^(x) beyond x" have no effect upon the values of c^^, c^, • • -, c„, and hence
these would be the same if ^„+i, ^„ + 2> • • • were replaced by 0, 0, • • •, and
a»+i,a« + 2j "'y(^2nj<^2n+ir" by such valucs < + i, < + 2, ..-, a'^^, 0, •••
as would make the division come out; even ; the coefficients c^^, c^, • • • , c„
are therefore precisely those obtained in dividing the series.
If y is developed into a power series in x as
y=f(x) = a^ + a^x + a^^ -{-•••, a^^O, (17)
then X may be developed into a power series in y — a^j as
X =/-'(y - «„) = *i(y - «o) + Kif -%y + - ■■■ (18)
For since a^ ^ 0, the function /(x) has a nonvanishing derivative for
05 = 0 and hence the inverse function/-^(y— - aj is analytic near x = 0
or y = a^ and can be developed (p. 477). The method of undetermined
coefficients may be used to find J^, b^,---. This process of finding
(18) from (17) is called the reversion of (17). For the actual work it is
simpler to replace (y — a^/a^ by t so that
t = x + a'^ -h a^x' -f or^x* -\ , aj = a.-/ai>
and x = t-{-b'^e-\- b\^ -}- b\t^ -\ , b\ = 5,aj.
Let the assumed value of x be substituted in the series for t ; rearrange
the terms according to powers of t and equate the corresponding coef-
ficients. Thus ,., „
t-^t^-{b\^- a'^t^ -h il>^ ^-2b'^a^ + a;)^»
-f {b\ -k-^b'^a'^ -f b'^a'^ 4- 3/>aa; + <)<* + • • •
or *; B= — oj, 6; = 2 a:} — «;, b\ = — h a^ -f- 5 a'^a^ — a\^ • ..
INFINITE SERIES 447
170. For some few purposes, which ar« tolermUj impottmni, m /krmmi
operational method of treating series is so useful ts to be alaoel Mfa-
pensable. If the series be taken in the form
with the factorials which occur in Maclaurin's inrnlnpiml anl with
unity as the initial term, the series may be written as
])rovided that a' be interpreted as the formal equiraleot of a,. TW
product of two series would then formally suggest
e-6^ = e<- + *)'=l4-(a4-*)'x + |j(a + A)V+.-., (19)
and if the coefficients be transformed by setting aV « «A, Umb
This as a matter of fact is the formula for the product of two
and hence justifies the suggestion contained in (19).
For example suppose that the development of
were desired. As the development begins with 1, the formal
may be applied- and the result is found to be
-^ = e", X =. «<»«>• _ e", (W)
er — 1
(/? + l)»-5« = 0, (B + 1)«-B* = 0, ..., (B + l)*-^-a,--,
or 25^4-1 = 0, 3^,+ 3B, + l = 0, 4fi,+ 6B, + 411, + 1 - 0,
or 5i = -i, ^,= i, ^, = 0, B,--A»- •
The formal method leads to a set of equations from which tb« siie-
cessive -B's may quickly be determined. Note that
?^-i-57^-!"-'|-|-'(-|) ™
448 THEORY OF FUNCTIONS
is an even function of £c, and that consequently all the S's with odd
indices except B^ are zero. This will facilitate the calculation. The
first eight even -B's are respectively
h -^V iV. -^> A^ -/^^, h -¥t^t^- (23)
The numbers By or their absolute values, are called the BernouUian
numbers. An independent justification for the method of formal cal-
culation may readily be given. For observe that e^e^'^ = e^*+^>^ of (20)
is true when B is regarded as an independent variable. Hence if this
identity be arranged according to powers of B, the coefficient of each
power must vanish. It will therefore not disturb the identity if any
numbers whatsoever are substituted for B^, B^, B^, • • • ; the particular
set B^f B^, ^s) " ^^y therefore be substituted ; the series may be rear-
ranged according to powers of cc, and the coefficients of like powers of
X may be equated to 0, — as in (21) to get the desired equations.
If an infinite series be written without the factorials as
a possible symbolic expression for the series is
= 1 + a^« + aV + aV H , a' = a..
1 — ax
If the substitution y = x/(l-\-x) ov x = y/(l — y) be made,
1 _ 1 _ 1-y
l-«^ l-a-i^ l-(l-f-a)y* (24)
Now if the left-hand and right-hand expressions be expanded and a be
regarded as an independent variable restricted to values which make
|aic| < 1, the series obtained will both converge absolutely and may be
arranged according to powers of a. Corresponding coefficients will then
be equal and the identity will therefore not be disturbed if «»• replaces
a*. Hence
1 -h «,x -f a^'' -f ... = (1 - y)[H. (1 + a)y + (1 + a)y + •••],
provided that both series converge absolutely for a, = a*. Then
1 4- SX + a-r^ -f V« + . . . = 1 + ay 4- a(l 4- a)y^ + a(l 4- «)'/ 4- • • •
= 1 -h a,y 4- (ai4- aa)^' + K + 2a,4- a3)/4- •••,
or a^x 4- aj^» + a^» 4- • •• = a^y 4- {a^ 4- ^3)^'
-f(aj 4- 2a, 4- «.)/ + ••-. (26)
INFINITE 8ERIJS8 449
This transformation is known as Eultt^a tratufonnation. lu grau
iul vantage for computation lies in tlie fact thai loaiettnMS Um iMoad
series converges much more rapidly than the fiftt This is cspadttllj
true when the coefficients of the first seriet are inch aa to make tlie
coctKcients in the new series smalL Thus from (25)
log(H-x) = x-.4a:«+ix"-Jx*+i«»-J«« + ...
= y + iy*+iy* + Jy* + »/ + */ +
To compute log 2 to three decimals from the first seriee would reqoiie
sevenil liundred terms ; eight terms are enough with the seoond s^ieSw
An additional advantage of the new series is that it may eoottnoe to
(!on verge after the original series has ceased to couTerge. In this ease
the two series can hardly be said to be equal ; bat the seoond series of
course remains equal to the (continuation of the) function defined bj
the first. Thus log 3 may be computed to three decimals with abooi a
dozen terms of the second series, but cannot be oomputed firom the first
EXERCISES
1. By the multiplication of series prove the following reiauona:
(a) (1 + X + x« + x« + . . .)* = (1 + 2x + 8jc« + 4«« + ...)«(! -^-t,
(/9) cos^x + sin^x = 1, (7) ^H* = «« + », («) 2riD«* = 1 - ootts.
2. Find the Maclaurin development to terms in *• for the fmtcHonit
(a) e' cos X, (/3) e' sin x, (7) (1 + «) log (1 + x), (I) oos« 1
3. Group the terms of the expansion of coax in two different wajs
cos 1 > 0 and cos 2 < 0. Why does it then follow that cot ( = 0 when 1< |< t f
4. Establish the developments (Peirce's Not. 786-780) of tl
(a) e-inx, 03) c««*, (7) «*"*•» (*) «*•"*••
5. Show that if g(x) = «wc" + ^.+iX- + > + • • • »nd/(0) ?« 0, Umb
/(x)_ a»-|-a,x4-a,x^-t--- , c— ^ g^iJELL? .f. . . . + gzl a ^ 4. ly 4. . .
g{x) 6«x'« + 6.^+iX- + » + ... X- «-- » «
and the development of the quotient has negatire powen of z,
6. Develop to terms in x« the following funetioos:
(a) sin (Jb sin x), (^) log coax, (7) >/««&. (*)(!-*■ i**^)" *•
7. Carry the reversion of these series to tenns In the fifth power:
(a) i/ = sinx = x-ix» + .-., {ft) y = tan-«* = «-*«« + - •
(7) y = e- = l + x + ix« + ..., (J)if = a* + ««« + 4** + »«*-^
460 THEORY OF FUNCTIONS
8. Find the smallest root of these series by the method of reversion:
(a) 1 = JT V^dx = X - ix. + ^x. - ^x' + . . ,
9. By the formal method obtain the general equations for the coefficients in the
developments of these functions and compute the first five that do not vanish :
. . sinx ... 2e^ x*
^ — 1 ^' e?= + l 1— 2xea;4-e2*
10. Obtain the general expressions for the following developments:
, , ,, 1 , » X« 2X6 JB2„(2x)2n
(a^ coth x = -H +
X 3 46 946 (2n)!x
vp; wt* X 3 46 946 ^ ' (2n)!x
(7) logsmx = logx }- (— 1)" ^ — ,
^^' ^ ^6 180 2836 ^ ' 2n.(2n)I
x^ X* X* B2«(2x)2»
(a) logsinhx = logx + ^- — + — + ''**^ ^ .
^' ^ ^ ^6 180 2836 2n.(2n)!
11. The Eulerian numbers E^n are the coefficients in the expansion of sechx.
Establish the defining equations and compute the first four as — 1, 6, — 61, 1386.
12. Write the expansions for sec x and log tan (i t + ^ x).
12 1
13. From the identity = derive the expansions:
ef^-l e^^-1 ea^ + 1
(a) -^ = i + BJ2^ - 1)— + BA2* - 1) ^ + . . . + 52n(22» - 1)?^ + ....
^'e^ + 1 2 ^^ '2! *^ '4! ^ '2n! '
(/3) -^^ = i - B2(2^ - 1) — - 5.(2* - 1) — J52n(22« - 1)'^^ + • . . ,
^'^^ e* + l 2 *^ '21 *^ '4! ^ ' 2n! '
(7) tanhx = (22-1)225 * + (2*^ 1)2*5,^ + . . . + (22»- l)22"J52n^^ + • • .
Z I 4 ! ^ 71 I
x' 2x6 173.7 x^"-!
(a) tanx = x + - + ^4-^ + ....+ (-l)-H2^'»-l)22«l?2»^ + ...,
(e) 10gC08X=-^-^-— (-1)'»-1(22«-1)22«»B2« ^^"
2 12 46 ^ / V ' '"2n.2n!
(r) log ton X = logx + ^ + ^ + . . . + (- l)«-l(22n-l _ l)22»52n -^ +
(i,)c8cx = l(cot| + ton|)=l + ^ + ... + (-l).-i2(2«-i-l)B2«|^.
(0) log cosh X, (i) logtanhx, (k) cschx, (X) sec^x.
INFINITE SERIES 45|
obHerve that the Bernoullian numben aflord a
triguiioinetric and hyperbolic funcUoiw and their lagariUunt wtek Um «aaptfaa il
the sine and cosine (which have known deTelopmenu) and Um ■tfini (i
{uircs the Eulerian numbers). The Importance of then wmbHi ll
il>parent.
14. The coefficienta P^{y), P,(|/), ... P.(y) in the i1oTiln|MMl
^fj = y + Pi(y) X + P,(y) x« + . . . + P.(y) «. + ...
are called Bernoulli's polynomials. Show thu (a -f 1) t P»(y) zs (B 4- yV» ♦* ~ ••♦I
and thus compute the first six polynomiala in y.
15. If 1/ = i^ is a positive integer, the quotient in Ex. U la i
n 1 P^{N) = l + 2- + 8- + ... + (J<r-iy
is easily shown. With the aid of the pol3rnomlaU found abof« eonpotat
(a) l + 2*+3«+ • + 10«, (^ l + J« + 3» + ... + 9*,
(7) l + 22 + 3« + ... + (^-l)«, (t) l + J« + «t + ... + (y-|>i.
16. Interpret = — I = V ^.
1-axl — te x(a — 6)Ll — ojc a — brj ^ a-^
17. From r*e-o-«)'(tt = eatabliah formally
Jo 1 — ax
1 + a^x + ajX« + a^x* + • • • = fe-'Fixl^dl = ^ j'%'»F(u\Ai,
where F{u) = 1 + a^u + — a,u« + — o,u« + • • ..
Show that the integral will converge when 0 < x < 1 provided |a<| S 1.
18. If in a series the coefficients ai=f t*f{t)dL,
1 + ttjX + a^« + a,x* + • • • = J^ i^^**
i-j
19. Note that Exs. 17 and 18 convert a seriee into an integral. Show
■ ^''' '^2p^8p 4P^ r(p)Jo l-«t ^ M Jt
^^' 1 + 1» l + 2« l + 8« Jo l-x« ! + •• J*
: ^^' ^^6"^+ 6(6 + 1) Mft + l)(6 + 2)
J
r<a)
r(6) /»«!— «o-<>»— »^
t{h^a)J% l-«l
452 THEORY OF FUNCTIONS
20. In case the coefficients in a series are alternately positive and negative show
that Euler's transformed series may be written
ajX — OiX^ + osz* — 042;* + ••• = «!?/ + A ai2/2 + A^aiy^ ^ A^aii/* + • • •
where Aai = 01 — 02, A^oi = Aai — Aoa = ai — 202 + ag,- • • are the successive
first, second, • • • differences of the numerical coefficients.
21. Compute the values of these series by the method of Ex. 20 with x = 1, y = J.
Add the first few terms and apply the method of differences to the next few as
indicated :
add 8 terms and take 7 more,
0.6049, add 5 terms and take 7 more,
13, • add 10 and take 11 more,
and compute forp = 1.01 with the aid of five-place tables.
22. If an infinite series converges absolutely, show that any infinite series the
terms of which are selected from the terms of the given series must also converge.
What if the given series converged, but not absolutely ?
23. Note that the proof concerning term-by-term integration (p. 432) would not
hold if the interval were infinite. Discuss this case with especial references to
justifying if possible the formal evaluations of Exs. 12 (a), (5), p. 439.
24. Check the formula of Ex. 17 by termwise integration. Evaluate
1 /»• -- Mr* 1
- I e ='jQ{bu)du = 1 - ^62x2 + ^ . f = (1 + b^x^)-i ■
X vQ A !
by the inverse transformation. See Exs. 8 and 15, p. 399.
(a) 1-
-V
\-v
0.69316,
(/3) 1-
1
Vi
-^-
'+... = 0
Vi
(»r
= 1-
vv
V-
• . = 0.78
CHAPTER XVII
SPECIAL INFINITE DEVEL0PMEHT8
171. The trigonometric functions. If m is an odd intcfer, my
,n = 2 7t -f 1, De Moivre's Tlieorem (§ 72) gives
sin 7WA ... (m — l)(m — 2) . . . .
7/t sin ^ 3! ▼" ^ ^ » I*/
where by virtue of the relation cos*^ = 1 — sin*^ the right-hand meia-
ber is a polynomial of degree n in sin' ^. From the left-hand side it is
seen that the value of the polynomial is 1 when sin ^ as 0 and that the
n roots of the polynomials are
sin* 7r/m, sin* 2 w/m, • , sin* mr/in.
Hence the polynomial may be factored in the form
sin vi4^ ^ A sin* ^ \L si"*^ \ {\ "°'^ V (2)
msin<^ \ sin*7r/m/\ Biv^^tr/m) \ sin'inr/a/ ^'
If the substitutions <j> = x/m and ^ = %x/m be made,
sinx _ /-I _ sin* y/wA / _ sin*g/m \ A _ sin'x/i \
7/1 sin a-/7/i ~ \ sin*7r/m/ \ &in* 2 ir/m) \ sin^nw/m/ ^ *
sinha;
7/isinha;/7»
/ sinh*a;/m\/ 8inh*jrAn\ / sinh'x/my
-(,^+>i^V^j(,^"^sin*2x/mj--A -i"^
Now if m be allowed to become infinite, passing through sneeessiTo
odd integers, these equations remain true and it would appear thai Um
limiting relations would hold :
=(-5)('-i4) -V(-s>> <•>
X
sinh
since lim . — = lim
/£_l£!+...Y
454 THEORY OF FUNCTIONS
In this way the expansions into infinite products
8ina= = a:Tr(l--^). sinh a, = a. IT (l + ^) (5)
would be found. As the theorem that the limit of a product is the prod-
uct of the limits holds in general only for finite products, the process
here followed must be justified in detail.
For the justification the consideration of sinha;, which involves only positive
quantities, is simpler. Take the logarithm and split the sum into two parts
sinhx
sinh2 — \ / sinh^.
,og-^;:i;i^=2:.odi+-£uy.odi+
msinh— ^ \ sm^ — / p + i \ sin*
m \ m / \ m,
As log (1 + a) < or, the second sum may be further transformed to
/ sinh2-\ „ sinh2£ „
p+i \ ^inan/. i>+i sin^^— ; % + isin2 —
\ m/ m m
Now as n < J m, the angle kir/m is less than \ ir, and sin ^ > 2 f /tt f or f < ^ tt, by
Ex. 28, p. 11. Hence
E < sinh2 - > — = — sinh2- > — < — sinh*— | —
m A' 4A:2 4 m 4^, A:* 4 mJp A:«
p / sinh2-^\
1 sinhx xr^ I , . wij m* . , „ x
Hence log X 1 +— ^ < — smh^ — .
msmh— 1 \ . sin* — / ^
m \ m I
Now let m become infinite. As the sum on the left is a finite, the limit is simply
logE^ _y (i + ^\ < ^ ; and log^i^ = y fl + -^\
'^ X -WV k^irV 4p' '^ X ^\ k^irV
then follows easily by letting p become infinite. Hence the justification of (4').
By the differentiation of the series of logarithms of (5),
, sin a; ^ , / . x" \ , sinh a; ^ , /, a^^ \ ,^,
the expressions of cot x and coth x in series of fractions
2x ., 1 . ^ 2a;
-*^ = S-?ifcVW' eoth. = i-fy-^, (7)
SPECIAL INFINITE DEVELOPMENTS 4^
are found. And the differentiation is legitimate if
unifoiinly. For the liyi)erbolic function the uniformity of tlie
gence follows from the ^/-test
The accuracy of the series for cot x may then be inferred by the suUiti-
tution of ix for x instead of by direct ezaminatioii. Aa
-2ar 11 ♦•
H --;— » ooix
;tV_ar^ x--kir^ x->tkir' "^ ^i^S
In this expansion, however, it is neoessary still to aMooiate the
for k = -\-n and A; = — n ; for each of the series for I; > 0 and (or
/.• < 0 diverges.
172. hi the series for cotha; replace ar by \x. Then, by (22), p. 44 T
If the iirst series can be arranged according to powers of x, an eipres-
sion for B.^^ will be found. Consider the identity
which is derived by division and in which ^ is a proper fiacUuu u i is
positive. Substitute t = x^jX kV ; then
I
Let |;^ = l + ^ + ^+- = V
|coth|-l = -2y.sJ=^'-2»...(^-.
* The 0 is still a proper fraction »ino* emch $1, b. Tbe latorehMg* •'•^
sunimatiou is legitimate becaiue the mtIm would sliU eoaTWg* II all tifM wii»
siuee ^k~ '^*' is convergent.
456 THEORY OF FUNCTIOi^S
As S^^ approaches 1 when n becomes infinite, the last term approaches
0 if « < 2 TT, and the identical expansions are
2|*V-l)'-'(^ = |^a,g^, = |coth|-l. (10)
Hence -B,, = (-l/-' g^S,,
(2t)»
"-1 ^ip
(11)
and _coth- = l+^B,, — + eA.27r!- (^2)
The desired expression for ^g* is thus found, and it is further seen
that the expansion for ^ x coth J a; can be broken off at any term with
an error less than the first term omitted. This did not appear from the
formal work of § 170. Further it may be noted that for large values of
n the numbers B^n are very large.
It was seen in treating the F-f unction that (Ex. 17, p. 385)
log T(n) = (W — J) log 71 — 71 + log V2 TT + O) (7l),
where a)(n)= I f-coth- — l}e^-^»
J- CO Jo ^'^"'' n'P^-'
the substitution of (12), and the integration gives the result
"W- 1.2 + 3.4 +--- + (2^-3)(2^-2) + (2^-l)2^- <^^>
For large values of 7i this development starts to converge very rapidly,
and by taking a few terms a very good value of w {n) can be obtained ;
but too many terms must not be taken. Compare §§ 151, 154.
EXERCISES
1. Prove coso; = = 7T 1 1 i.
28inx 0 \ (2A; + 1)2W
2. On the assumption that the product for sinhx may be multiplied out and
collected according to powers of x, show that
SPECIAL INFINITE DEVELOPKENTS 46T
a nvHi.iofEx.21(»),p.4M,ibow: (a) I + i + i ^. i ^ ... . •[!.
r W 4^ •
^^^ ^ + ^"^i5 + 4 + ••• = ?• (7) I-i + i-i + ...-??.
«how cHcx = 1 (cot? + un?U y inill = ? + V i:ili^.
6. From ^^ = J) (-*)* + (-. 1)-^ =2) (- ,y» + (- |y.|^
J/»i^a-i ^ (_ n* 1
--— dx = > ^— -^ , and compute f or a = - by Ki. 11, p, 4«t.
7. If a is a proper fraction so that 1 — a Is a proper f rmcUon, §hom
^"' Jo 1 + x ^a-k Ji 1 + x ' ^'J. l+« daar
8. When n is large iJj, = (- 1)"-M V^/— V'tpproiinUcty (Ki. IS).
9. Expand the terms of - coth - = 1 + V -—r-- by divWoo wImb « < t »
2 2 '^4*V + x«
and rearrange according to powers of x. Is it easy to Juitlfy thla deriTatioa of (11) f
10. Find u{n) by differentiating under the sign and lubstitiitli^. Emm fiC
E:(!0 = iogn-±-A^A hizi iillL.
r(n) * 2n 2n« 4n* (2p-2)i»«*-« l^ii^*
11. From 1^+7= f * l^^^^— da of § 149 show that. If « l» lnt«ff»l,
r (n) ^/o 1 — rr
by taking n = 10 and using the neoBwiry number of lenai of Bz. lOl
12. Prove log r (n + i) = n(logw - 1) + log Viir + «, («>. ^hmrm
458 THEORY OF FUNCTIONS
0 10
13. Shown! = V2^0ye^ or V2^ (^^±1^ ""^"e"^"*"^^^. Notethatthe
results of § 149 are now obtained rigorously.
1 Vn e-*^ V^ g-(n-l)x
14. From = > e-*^ + = > e-^+ 0 , and the formulas
of § 149, prove the expansions
(.).ogr(„ + i) + .«=|g-,og!^). „)_L_ = ^fr(i + ge-l
173. Trigonometric or Fourier series. If the series
oo
f(x) = i «o + 2 (^* ^^^ ^^ "I" ^* ^^^ ^^)
= ^ a^ + Oj COS o; + 0^2 COS 2 ic 4- ftg COS 3 ic H ^ ^
+ Jj sin X -\- b^ sin 2 a; + i^ sin Sx -\
converges over an interval of length 2 tt in x, say 0 ^ x < 2 tt or
— TT <^ X ^ TTj the series will converge for all values of x and will de-
fine a periodic function f(x + 2 tt) = f(x) of period 2 tt. As
Jr^' 7 • 7 J /^ J r^^cosA^o^cosZaj , ^ ^. ^,
I cos AJic sin Ixdx = 0 and I . , . , aa; = 0 or tt (15)
„ Jq sin A;ic sin to ^ ^
according sls k ^ I ot k = I, the coefficients in (14) may be determined
formally by multiplying f(x) and the series by
1 = cos 0 Xj cos X, sin x, cos 2 a;, sin 2 a;, • • •
successively and integrating from 0 to 2 tt. By virtue of (15) each of
the integrals vanishes except one, and from that one
^k = ~ I f(x) cos kxdx, ^k = ~ I /(^) sin kxdx. (16)
Conversely if f(x) be a function which is defined in an interval of
length 2 tt, and which is continuous except at a finite number of points
in the interval, the numbers a^. and hj^ may be computed according to
(16) and the series (14) may then be constructed. If this series con-
verges to the value of /(a;), there has been found an expansion of /(a-)
over the interval from 0 to 2 tt in a trigonometric or Fourier series*
The question of whether the series thus found does really converge to
* By special devices some Fourier expansions were found in Ex. 10, p. 439.
SPECIAL IXFIXITE DEVELOPMENTS 45g
the value of the function, and whether that Mrias am be intMiHiiJ or
differentiated term by term to find the integiml or derivative of Um
function will be left for special inveetigation. At pieeeat it vill l»
assumed that the function may be repreeented by the ieri«, Qml tbe
series may be integrated, and that it may be differentiated ifthediffbfw
entiated series converges.
For example let tF be developed in the Interval from 0 to t v. Btf
or
and
Hence
This expansion is valid only in the interval from 0 t<> 2 r ; oataide tliat Inierral Uw
series automatically repeats that portion of the function which U« la tha latcrval.
It may be remarked that the expansion doea not hold for 0 or tv but gfvw ilw
point midway in the break. Note further that If the eerica wei« dilbiwitlaiad tW
coefficient of the cosine terms would be I -f \/k' and would not appivedl 0 «lMi
Ac became infinite, so that the series would apparently oadllate. Ini^rnukm ftnm
0 to X would give
+ CO0X4> ooaSx + ^-^— eoaSx 4
and
the term ^ x may be replaced by ita Fourier aeriea If deaired.
As the relations (15) hold not only when the integration ia frooi 0
to 2 7r but also when it is over any interval of 2ir from a to c -f 2 v,
the function may l)e ex {landed into series in the interval froin « to
a 4- 2 7r by using these values instead of 0 and 2 ir as limits in the
formulas (16) for the coefficients. It may be shown that a fnoetioB
may be expanded in only one way into a trigonometrio series (14) valid
for an interval of length 2 w ; but the proof is somewhat intricate and
will not be given here. If, however, the expansion of the ftanetioii is
desired for an interval a < x < fi leas than 2w, there are an infinili
number of developments (14) wliich will answer: for if ^{x^ W a
460 THEORY OF FUNCTIONS
function which coincides with f(x) during the interval a < x < p,
over which the expansion of f(x) is desired, and which has any value
whatsoever over the remainder of the interval p. < x < a -{- 2 tTj the
expansion of ^ (x) from a to a + 2 tt will converge to f(x) over the
interval a < x < p.
In practice it is frequently desirable to restrict the interval over
which f(x) is expanded to a length tt, say from 0 to tt, and to seek an
expansion in terms of sines or cosines alone. Thus suppose that in the
interval 0 < x < tt the function <^ (x) be identical with f(x), and that
in the interval — 7r<a:<0itbe equal to /(— ic) ; that is, the func-
tion <f> (x) is an even function, <f>(x) = <j) (— x), which is equal to f(x)
in the interval from 0 to tt. Then
X + ir /»«r •»»
<^ (x) cos kxdx = 21 tf>(x) cos kxdx = 2 I f(x) cos kxdxj
X + w pit r*v
<l> (x) sin kxdx =1 <t>(x) sin kxdx — I <t>(x) sin kxdx = 0.
^ Jo Jq
Hence for the expansion of <^ {x) from — tt to + tt the coefficients h^. all
vanish and the expansion is in terms of cosines alone. As f(x) coin-
cides with ^ (x) from 0 to tt, the expansion
f(x)—^aj^coskx, 0'k — ~ I f(x) cos kxdx (17)
0^ "^ Jo
of f(x) in terms of cosines alone, and valid over the interval from 0 to
TT, has been found. In like manner the expansion
f(x)=Vb,^smkx, h = - I f(x) sin kxdx (18)
r '^ Jo
in term of sines alone may be found by taking <f> (x) equal to f(x) from
0 to TT and equal to — /(— x) from 0 to — tt.
Let i X be developed into a series of sines and into a series of cosines valid over
the interval from 0 to ir. For the series of sines
bk = - I - X 8in kxdx = -^^--^, =y\±— —
IT Jo 2 k 2 ^ k
or ix = 8ina!— i8in2x + ^sinSx — J8in4x-f- .... (A)
o >.»i o -1 fO, fceven
AlBO 00 = -/ -aMix = -, at = - / - x cos Axcdx = -^ 2 , ,,
IT Jo 2 2 IT Jo 2 1 T'*^ odd.
u»»<. 1- "■ 2r . cosSx . cosSx . cosTx . "l .„.
Hence -« = -- -|^co8i. + -^ + -^ + ^j- + ... I . (B)
SPECIAL INFINITE DEVEL0PKENT8
4*ii
Although the two expaiulon* define the ■ame foneUoii | g over Um laiervai v w^
they will define different functlona in the Inienral 0 to ~ v, m la the figara.
The development for \ x* may be bad bj Intcgraiiog either aeite (A) er (IL
l2« = 1 - coex - i(l - ooe2«) + 1(1 - ooe8«) - ,1^(1 - «e4r> -f^
...-![,
These are not yet Fourier seriee becauae of the tenna | «x and the vailoM Tn Fer
I irx iu 8ine series may be mibetituted and the tenna 1 - | -f | mmj he col*
lected by Ex. 8, p. 467. Hence
(-T.ir)
-x' = - co8 2 + -coe2z— -oos8«-|-— ooa4<«->>
12
16
(AO
^'_1^8inx-^8ln2x + (^-l)»ln8x-^dn4x + ...].(B0
The differentiation of the series (A) of sines will gire a aerlea In which
terms do not approach 0 ; the differentiation of the aeriee (B) of codaee gfvea
ix = sinx + i8in8x + J8in6x+ |ain7x + .•.
alld that this is the series for ir/4 may be verified by direct caleolatloo.
ence of the two series (A) and (B) is a Fourier eeriea
fix)
X 2r , coe8x . "I r. , rintx ^ 1
(O
which defines a function that vaniahea when 0 < x < » but le equal 14) — «
0 > X > - IT.
174. For discussing the converigenoe of the trigononetrie mrim •» foraMllf
calculated, the sum of the first 2 n + 1 tenna maj be written aa
S«=l r"ri+coe(f-x) + ooe«(<-x) + ..+coe«(l-x)l/(0*
Un(2n + 1)
2 sin
(~x
•in(Sa4>l>ii
462 THEORY OF FUNCTIONS
where the first step was to combine a* cos kx and 6* sin kx after replacing x in the
definite integrals (16) by t to avoid confusion, then summing by the formula of
Ex.9, p. 30, and finally changing the variable to u = \{t — x). The sum Sn is
therefore represented as a definite integral whose limit must be evaluated as n
becomes infinite.
Let the restriction be imposed upon f{x) that it shall be of limited variation in
the interval 0 < x < 2 tt. As the function f{x) is of limited variation, it may be
regarded as the ditference P{x) — N{x) of two positive limited functions which
are constantly increasing and which will be continuous wherever /(x) is continu-
ous (§ 127). If f{x) is discontinuous at x = x,,, it is still true that/(x) approaches
a limit, which will be denoted by /(x^ — 0) when x approaches x^ from below ; for
each of the functions P(x) and N{x) is increasing and limited and hence each
must approach a limit, and /(x) will therefore approach the difference of the limits.
In like manner /(x) will approach a limit /(x^ + 0) as x approaches x^ from above.
Furthermore as /(x) is of limited variation the integrals required for -S„, a;t, &* will
all exist and there will be no difficulty from that source. It will now be shown that
limS,(Xo) = lim lr~Jf(x,+ 2uf^^^^±^du=\ [/(x^ + 0)-/(Xo- 0)].
n=oo n=ao7r«/— — sinu Ji
This will show that the series converges to the function wherever the function is con-
tinuous and to tJie mid-point of the break wherever the function is discontinuous.
T 4. *i , o \ ^^'^(^w + l)^ ., , „ . u sin(2n + l)M _, .sinAru
Let /(Xo + 2u) ^ '— =f{xQ-\-2u)- ^^ ^ = F(u) ,
smi* " sinw u u
then iS„(Xo) = - ( a, ^F{u) du - - I F{u) du, - 7r<a<0<6<7r
TV J--^ U IT J a U
As /(x) is of limited variation provided — 7r<a^M^6<7r, so must /(x^ + 2 u)
be of limited variation and also F{u) = w//sinM. Then F{u) may be regarded as
the difference of two constantly increasing positive functions, or, if preferable, of
two constantly decreasing positive functions ; and it will be sufficient to investigate
the integral of F{u)u-'^ sin ku under the hypothesis that F{u) is constantly de-
creasing. Let n be the number of times 2 ir/k is contained in b.
X: k k
J '2'^ /"*"■, , fSnn- /uXsinUj r^ ^, .sinfcu^
+ 1 +•••+/ F(-) du+L^^F{u) du.
0 J2w JiCn-Dn \k/ U J inn ^ ' ^
As F{u) is a decreasing function, so is u-^F{u/k), and hence each of the integrals
which extends over a complete period 2 7r will be positive because the negative ele-
ments are smaller than the corresponding positive elements. The integral from
2 nir/k to b approaches zero as k becomes infinite. Hence for large values of ifc,
r V(„)«i^d„ > f""F(lY^du, p fixed and less than n.
Jo u Jo W u
SPECIAL IXFIMTE DEVELOPMENTS 46$
Here all the teniM except the flnt and last un negativ* iia*»itt Um
menu of the intep^ls are larger than the po«iU?e atcoMBta. Bmm* fort
In the inequalities thus esUbllshed let k become Infinite. Then m/k^O tnm
above and F(u/k) = f (+ 0). It therefore follows thai
Althou<rh p \r fixed, there in no limit to the size of the number at which Itlsised.
Hence the inequality may be transformed into an eqoalltj
m„ /V(u)'-!ni:!!d« = F(+o) f "^l^d. = ?F(+«).
jt-ooJo u Jo II ^ \^ r
Likewise lim rV(«)!l^du = F(- 0) f'^d, = ^r(-«.
Henco lim rV(i/)?l^<fu = - [F(+ 0) + F(- 0)1
or Mm 1/;- V(x. + 2u)!i5ii^-*. = ![/(x.+ a)+/<^-0,).
N=ioe7«/— -J sinii X
Hence for every point Xq in the interval 0 < 2 < 2 r the series eo
function where continuous, and to the mid-point of the break where
As the function f(i) has the period 2 ir, it is natural to anppoei
vergence at x = 0 and x = 2ir will not differ materiallj from that at aaj
value, namely, that it will be to the value | [/(+ 0) +/(*«'— 0)]. This
shown by a transformation. If k is an odd int^^r, S a -f 1.
sin (2 n 4- 1) u = sin (2 n + 1) (v — m) = >in (2 N -t- 1 f M .
lim / F{u) — ^ -^—i-du = lim I F{u') — ^ ' Ai'»-i^(ii'» ■» f|.
Hence lim r'F(u) ''"<'" "^ '>''d.. = Hm fV /*' = ? [r(+ 0» + Jr(» - •>).
N>=ao«/0 U ■■•Jo Jft 2
Now for x = 0or« = 2xthe«umS, = i f '/(«•») ^^^Tr^^^*«^ •^ »*»•
»Jo ^ ' dan
will therefore be I [/(+ 0) +/(2«' - 0)) as predieted abore.
The convergence may be examined more cloeely. In fact
^' irJ-s ainn m wJacci •
464 THEORY OF FUNCTIONS
Suppose 0<a:^x^/3<27rso that the least possible upper limit 6 («) is ir — ^ /S
and the greatest possible lower limit a (x) is — ^ cr. Let n be the number of times
2 v/k is contained in tt — i /5. Then for all values of x in a ^ x ^ /3,
J^(Sp-i)w / uXsinu, . C^^'^-p, .sinfcu,
' Fix,-] du + e< / F{x,u) du
0 \ kj u •/o u
where e and ti are the integrals over partial periods neglected above and are uni-
formly small for all x's of a ^ x ^ /3 since F(x, u) is everywhere finite. This
shows that the number p may be chosen uniformly for all x's in the interval and
yet ultimately may be allowed to become infinite. If it be now assumed that /(x) is
continuous f or or ^ x ^ /3, then F(x, u) will be continuous and hence uniformly
continuous in (x, u) for the region defined by a ^ x ^ /3 and — ^x^M^Tr— Jx.
Hence F{x, u/k) will converge uniformly to jP(x, + 0) as A: becomes infinite. Hence
^/ rvv /**sinu, , /•^^*>Ti/ ,sinA^, _. ^, /"»sinu,
F(x, +0)/ dM + c'< / F{x,u) du<F{x,+0)i du + V
Jo u Jo u Jo u
where, if 8 > 0 is given, K may be taken so large that |e'| < 8 and [Vl < * for fc > jBT ;
with a similar relation for the integration from a (x) to 0. Hence in any interval
0<a^x^/3<27r over which /(x) is continuous Sn{x) converges uniformly
toward its limit /(x). Over such an interval the series may be integrated term by
term. If /(x) has a finite number of discontinuities, the series may still be inte-
grated term by term throughout the interval 0 ^ x ^ 2 tt because Sn (x) remains
always finite and limited and such discontinuities may be disregarded in integration.
EXERCISES
1. Obtain the expansions over the indicated intervals. Integrate the series.
Also discuss the differentiated series. Make graphs.
, s "^^ 11 .lo 1 o.l J
(a) — :— — = cos X -h - cos 2 X cos 3 x + — cos 4 x
^ ' 2 8inhir 2 2 6 10 17
— IT to +V,
12 3 4
4- - sin X sin 2 X 4- — sin 3 x sin 4 x H ,
2 6 10 17
(^) ^w, as sine series, 0 to ir, (y) ^ir, as cosine series, 0 to tt,
/>\ .<»» ^ ri cos2x cos4x cos6x 1 t^ .
(8) Binx = — , 0 to TT,
7rL2 13 35 6-7 J'
(«) cos X, as sine series, 0 to ir, (f) c*, as cosine series, 0 to 7r,
(n) X sin X, — IT to IT, (6) X cosx, — ir to tt, (t) tt 4- x, — t to w,
(«) sin^x, — vtotr, $ fractional, (X) cos^x, — tt to tt, ^ fractional,
(o) — log(28in-) = cosx + -co82x -I- -co83x + -C084x + •••, 0 to ir.
SPECIAL INFINITE DEVELOPMENTS 466
(r ) from (o) find ex|Mui8ioiM for log eoi | x, log ten c, log ua | #. Xoi* ikM te
these caAes, aa in (o), the function do« not namin flaita, tei li« Imifiil dMiu
2. What peculiarities occur in the trlgoaooMtito
for an odd function for which /(x) B/(r ~ s)f for ao eTM faaotfoa for
/(x)=/(ir-z)?
3. Show that/(x)= V6*8in— with 6» = ? f'/{x)dn — iK to Um UteB-
nometric sine series for/(z) over the interval 0<s<e and tbat tba tmmetkem IkM
defined is odd and of period 2 c. Write the correapondiag rMnlls for IIm m^m
series and for the general Fourier series.
4. Obtain Nos. 808-812 of Peirce's TaMes. Graph the 111111 oi Noi. 800 tad $!•.
5. Lete(z)=/(jc)- Ja<,-ajCoex a.oosiix-&iSlil«**«*-K<teM
be the error made by taking for /(x) the first 2 a •(- 1 terms of a trlfooooMCHfl ssrtaa.
1 r-^'
The mean value of the square of e(x) i« r— I [e{x)]*dx and Is a tvattkom
2» •/-»
F{aQ, ai, • ' , On, \, • ' , b^) of the coefficients. Show that If this msaa wqmn
error is to be as small as possible, the constants a,, a|, • • • , s^, k|, • • • , Iw MM ht
precisely those given by (16) ; that is, show that (1«) Is aqairalMit to
dF^_dF_ _^^^J_^ ~-0
6. By using the variable X in place of x in (10) deduce tl
/(aj) = J- J* /(X) cos 0(X - x)dX + l]|j Jll /(X)oos*(X - x)dX
= ±2^ j'7(X)c-*(A-')HfX == l-2e»*»<jr' /(x)e*Mdr ;
and hence infer /(x) =2^a*e»*^, ^*^2^J« •^<')'*^'^
— •
7. Without attempting rigorous analysis show formally that
f* <t>{a)da= lim [• . • + ♦(- n- Aa)Aa + ^(- »-»-l-Aa)Aa + •• • + #<-l-
^'^'* + ^(0 . Aa)Aa + ♦(! • Atf)Atf + ... + #(»• Aa)Aa 4 • • •)
= lim T0(fcAa)Aa= lim V #(*;);
is the expansion of /(x) by Fourier series from - e to e. Hence laier
fix)
= _L f r /(X)c*-<A-.)<d>da = lim r^ 2 r'-^^)** •
"-• 'aJ
466 THEORY OF FUNCTIONS
is an expression for/(a5) as a double integral, which may be expected to hold for
all values of x. Reduce this to the form of a Fourier Integral (Ex. 16, p. 377)
f{x) = - r * r " /(X) cos a{\-x) d\da.
IT Jo J— 00
8. Assume the possibility of expanding /(x) between — 1 and + 1 as a series of
Legendre polynomials (Exs. 13-20, p. 252, Ex. 16, p.440 ) in the form
fix) = aoPo(x) + a^P^ix) + a^P^{x) + • • • + a^Pn{x) + • • • .
By the aid of Ex. 19, p. 263, determine the coefficients as a* = f f{x) P*(x) dx.
For this expansion, form e (x) as in Ex. 6 and show that the determination of the
coefficients a,- so as to give a least mean square error agrees with the determi-
nation here found.
9. Note that the expansion of Ex. 8 represents a function /(x) between the
limits ± 1 as a polynomial of the nth degree in x, plus a remainder. It may be
shown that precisely this polynomial of degree n gives a smaller mean square error
over the interval than any other polynomial of degree w. For suppose
gr„(x) = Cq + CjX + . . • + CnX"" = 6o + \P^ + • • • + hnPn .
be any polynomial of degree n and its equivalent expansion in terms of Legendre
polynomials. Now if the c's are so determined that the mean value of [/(x) — sr„(x)]2
is a minimum, so are the 6's, which are linear homogeneous functions of the c's.
Hence the &'s must be identical with the a's above. Note that whereas the Maclaurin
expansion replaces /(x) by a polynomial in x which is a very good approximation
near x = 0, the Legendre expansion replaces /(x) by a polynomial which is the
best expansion when the whole interval from — 1 to + 1 is considered.
10. Compute (cf. Ex. 17, p. 252) the polynomials P^ = x, Pg = — J + f x^,
P3 = - |x + f x8, p^ = I _ Y x' + ¥«S ^6 = ¥« - ¥a;» + -¥««.
r^ 2 / 6 \ 2
Compute I X* sin irxdx = 0, - ( 1 ^ I, 0, - , 0 when i = 4, 3, 2, 1, 0. Hence show
that the polynomial of the fourth degree which best represents sin ttx from — 1
to + 1 reduces to degree three, and is
sinrx = ?x - - fi^ = lV-a;» - -x") = 2.69x - 2.89x8.
It v\7r2 /\2 2 /
Show that the mean square error is 0.004 and compare with that due to Maclaurin's
expansion if the term in x* is retained or if the term in x* is retained.
11. Expand sin i ttx = ^ P, - — /— - l\ P, = 1.663x - 0.562x«.
12. Expand from — 1 to + 1, as far as indicated, these functions :
(or) cos irx to P^, (p) e^ to P^, (7)log(l + x) to P^,
(«) Vl-x^ toP^, (e) cos-^x toP^, (f) tan-»x to P^,
(i»)-i= toP,, (<9)— l=toP3, (0-4= toP..
V 1 + X Vl-x2 Vl + x2
What simplifications occur if /(x) is odd or if it is even ?
SPECIAL INFINITE DEVELOPMENTS 467
175. The Theta f unctiont. It hai been seen Uuit a fuoecioii vitli Uw
period a IT may be expanded into a trigoooaielne aeriee; Uai if Um
function is odd, the series contains only sinea ; and if, fsitlMffaaML
the function is symmetric with respect to x » J v, the odd ailtiniai
of the angle will alone occur. In this case let
/(x) = 2 [a,8in ar - flj sin 3x + ...+(-!)• a. sin (2» + l), + ... J.
As 2 sin w* = - i{e^-e"^), the series may be written
/(') =22^ (- l)"a.8in (2n + l)x = - • 2^(-. l)'a.«<»-*«>-«.,«^,,
This exponential form is very convenient for many porpoasa. Lei ip
be added to x. The general term of the ' ' '
Hence if the coefficients of the series satisfy a,_i«~*^«tf,, the new
general term is identical with the succeeding term in the given
multiplied by — e^e~***. Hence
f(x-\-ip) = ^e^e'*'*f(x) if a.., = «.««r
The recurrent relation between the coefficients will determine
in terms of a^. For let q = e~^. Then
The new relation on the coefficients is thus compatible with the
relation a_^ = ««-i. If % = q^, the series thus beoomes
/(x) = 2^* sina; - 2y* sin 3a; + • +(-l)'27^'*''8in(2a + l;x i- ..
f(x-^2w)=f(x), /(x + ^) = -/(x), /(x + W--r'*-V('>.
The function thus defined formally has important propertiei.
In the first place it is important to discuss the eonvergenea of the
series. Apply the test ratio to the exponential form.
For any x this ratio will approach the limit 0 if y is numerically le«
than 1. Hence the series converges for all values of 9 provided |f | < 1.
Moreover if \x\ < ^0^ the absolute value of the ratio is lata than |ff*«*,
which approaches 0 as n becomes infinite. The terma of the aariM
therefore ultimately become lesa than thoae of any '"' "" *~^
468 THEORY OF FUNCTIONS
series. This establishes the uniform convergence and consequently the
continuity of f(x) for all real or complex values of x. As the series for
/' (x) may be treated similarly, the function has a continuous derivative
and is everywhere analytic.
By a change of variable and notation let
«(«) =/(!!). 2 = «"'^', (19)
i/(«) = 2,isin||-2,isin^ + 2,¥sin|^-.... (20)
The function H(u)j called eta of w, has therefore the properties
H{u + 2K)=- H(u), H{u H- 2 iW) = - y-^e'^V (w), (21)
init
H(u + 2 mK + 2 inK') = (- l)"» + "^-«e~^ V(i^), m, n integers.
The quantities 2 K and 2 iK^ are called the periods of the function. They
are not true periods in the sense that 2 tt is a period otf(x) ; for when
2 iJL is added to u, the function does not return to its original value, but
is changed in sign ; and when 2 iK' is added to u, the function takes
the multiplier written above.
Three new functions will be formed by adding to ti the quantity K
or iK' OT K -{- iK', that is, the half periods, and making slight changes
suggested by the results. First let Hj^(;u) z= H(u-\- K), By substitution
in the series (20),
H^(u) =2q^cos — -{-2q^ cos Y^ + 2 y '^ cos -j^ + . . . . (22)
By using the properties of H, corresponding properties of H^,
H^(u-{-2K)=:- H^(u), H^(u-h2 iK') = + r'e"'^X W. (23)
are found. Second let iK' be added to u in H(u). Then
J(2n + 1)» (2n+l)^(u + tJf) n' + n+J -«-(n+i)5 (2n + l>^i.
q e ^^ = q e ^e ^^
is the general term in the exponential development of H(u + iK')
apart from the coefficient ± i. Hence
H(u + iK') = i X(- l)V'"*e"'^V"^''
— 00
, iri 00 , ^ iri
= tq e ^^ 2^(—l) q e *^ .
SPECIAL INFINITE DEVELOPMENTS
Let e(«) = - U,K'^'h{u + .A-) - 5; (_ !///•«• .
— •
Tlie development of %{u) and further propeHitw ara evid«aUy
0(u + 2 /i:) = e(u), e(u + 2 ijc') - - r *«"?••(«). (15)
Finally instead of adding K + t/C' to m in ^(ii), add i^ in %(m^.
0,W = l + 2yco8 1^ + 29* COS l^ + 2/o«|^^
I'tii
0/t^ 4- 2 A-) = e^(M), e,(u 4- 2 i/C) = + y- «•- J- «,(«). (27)
For a tabulation of properties of the four functions tee Ex. 1 below.
176. As H (u) vanishes for u = 0 and is reproduced eseepi for a
Unite multiplier when 2 mK + 2 niK' is added to «, the table
H(u) = 0 for M = 2^1/: + 2 twit',
H^(n) = 0 for m = (2 w + 1) /: + 2 miC',
e(w) = 0 for u = 2mK + (2n'^l)iK',
0,(w) = O for tt = (2m + l)A' + (2iH-l)iiC',
contains the known vanishing points of the four functions. Now it is
possible to form infinite products which vanish for these Yaloee. Prom
such ])roducts it may be seen that the functions have no other ranisb-
ing points. Moreover the products themselves are usefuL
It will be most convenient to use the function B,(ii). Now
^*|(t-ur+-Ar+«-.ir + .-^o ^ _ ^„.,„^ - oe < » < ao .
Hence e?" -f y- <«-+») and «"?" + y-*"'*^ n fi 0,
are two expressions of which the second vanishes for all the roots of
®^(i() for which n ^ 0, and the first for all roots with n < 0. Heaoe
TT = C ^-(1 + y«-+»«Sr-)(l + ^•♦Vtt)
0
is an infinite product which vanishes for all the roots of H,«,«> Tbr
product is readily seen to converge absolately and uniformly. In par-
ticular it does not diverge to 0 and consequently has no olbar roots
than those of 0i(m) above given. It remains to show that the prodiKi
is identical with 9^(u) with a proper determination of C.
470 THEORY OF FUNCTIONS
Let 6i(u) be written in exponential form as follows, with z — e^ :
0 (2) = ei(u) = 1 + ^ (z + 1) + g* (z2 + ^^ ^. . . . + ^„^ ^2- + ^W . . . ,
^(z) = C-iTT(u) = (1 + qz){l + q^z){l + q^z)- • .(1 + q^^-iz).. .
x(-I)(-9(-f)-(-'-^)-
A direct substitution will show that <p {q^z) = q-^z-^^ (z) and ^ (q^z) = q-'^z-'^yp {z)
In fact this substitution is equivalent to replacing m by m + 2 iK' in Si. Next con-
sider the first 2 n terms of \}/{z) written above, and let this finite product be ^»(2)
Then by substitution
(g2« + qz)MQ^z) = (1 + 92« + i2;)f„(2).
Now>„(z) is reciprocal in z in such a way that, if multiplied out,
M^) = flto + «i(2' + ^) + «2/z2 + iW • • • + an(z» + ^V an = r''
Then {q^* + qz) ^ aiiq^^z^ + q-^^z-^) = (1 + q^n + iz) V «,- (z* + z-%
0 0
and the expansion and equation of coefficients of z* gives the relation
,2i-in _rt2M-2t + 2> y .11 v^ y i
^2i-l(l_^2«-2» + 2) ^^ ^_„ t=l
«»=«<-i ; ^TTTiTT ^ or af = ao ._^
■pT (l_^2« + 2t + 2)
"■JT (l_52n + 2* + 2) ^,-2"fp*(l_^2« + 2. + 2t)
From a» = g«', a^ = ^^^-^^ , a.- = — ^^^-^
Now if n be allowed to become infinite, each coefficient a,- approaches the limit
lim Oi = ^ , C = TT (1 - 9^") = (1 - q^) (1 - q') (1 - ?«) • • • •
o 1
Hence e^iu) = fT(l - 92«) • fr(l + 92« + ieF'')(i + q2n + ie~r'*)^
1 0
provided the limit of f „ (z) may be found by taking the series of the limits of the
terms. The justification of this process would be similar to that of § 171.
The products for ®y H^j H may be obtained from that for ©^ by sub-
tracting Kf iK\ K -f iK^ from u and making the needful slight altera-
tions to conform with the definitions. The products may be converted
into trigonometric form by multiplying. Then
H(u) = C 2 y» 8in ^ ft (l - 2 j=- cos 1= + <,'"), (28)
SPECIAL LNKLNXTE DEVELOPMENTS 47|
//.(«)=C2,.co.^fr(l + 2^.oo.|^+^.). (n^
©(«) = cfr(l_2j-.oo.|=!+,.-..j. ^M)
•,(«) = eft (l+ 2 y-*'oo.|^ +/.♦.). „,.
//,(0) = r 2 ,y» fr (1 + ^.)«, e (0) = c TT (1 _ /.♦•)•,
//'(O) = C 2 j» ^ ft (1 _ ,,-■»•. «,(0) » c ft (1 + ^« ♦ »)•
The value of //'(O) is found by dividing H(u) by u and letting ■ * 0
Then
"'W = 2^ ".(")«(«)«.(•) (W)
follows by direct substitution and cancellation or combinalkm.
177. Other functions may be built from the tbeta fonolaoiiA. Lei
&{K) 0,(0) ^^ e,(0) \jfc - 11,(0) • ^**>
V^0W NA- e(u) «(•») ^^
The functions sn ii^ en u, dn u are called elliptic functioiiB* of «. As iSf
is the only odd theta function, sn u is odd but en u and dn m are even.
All three funetloth^ have two actual periods in the same sense thai sinx
and cos x have the ])eriod 2 ir. Thus dn u has the periods 2 K and 4 UC*
by (25), (27); and sn u has the periods 4 A' and 2 iK* by (25), (21).
That en u has 4 /C and 2 A' + 2 I'A' as periods is also mMHj Tertfied.
The values of u which make the functions vanish are known ; tbej U9
those wliieh make the numerators vanish. In like manner the valnes
of u for which the three functions become infinite are the known
of 0(w).
If q is known, the values of Vib and VP may be found fktMi
definitions. Conversely the expression for V?,
«,(0) 1 + 2, + 2^ + 2^+...' ^ '
• Th« studjr of tba alUptic tunctloM Is eoaUaMd la CImHw XIX.
472 THEOKY OF FUNCTIONS
is readily solved for q by reversion. If powers of q higher than the
first are neglected, the approximate value of q is found by solution, as
is the series for q. For values of k' near 1 this series converges with
great rapidity; in fact if k^ ^ ^, A;' > 0.7, VP > 0.82, the second term
of the expansion amounts to less than 1/10* and may be disregarded
in work involving four or five figures. The first two terms here given
are sufficient for eleven figures.
Let d denote any one of the four theta series H^ H^^ 0, 0^. Then
n^) = <l>(z) = X^n^^^ ^ = e-^" (38)
— 00
may be taken as the form of development of »^^; this is merely the
Fourier series for a function with period 2 K. But all the theta func-
tions take the same multiplier, except for sign, when 2 iK' is added to u;
hence the squares of the functions take the same multiplier, and in par-
ticular <t>(q^z) = q~^z~^<t>(z). Apply this relation.
It then is seen that a recurrent relation between the coefficients is found
which will determine all the even coefficients in terms of h^ and all the
odd in terms of b^. Hence
^\u) = b^^{z) -F b^{z), b^, b^y constants, (38')
is the expansion of any i5»* or of any function which may be developed
as (38) and satisfies <t>(q^^) = q~^z-^<l>(z). Moreover ^ and ^ are iden-
tical for all such functions, and the only difference is in the values of
the constants b^, by
As any three theta functions satisfy (38') with different values of the
constants, the functions ^ and ^ may be eliminated and
a^l (u) + )Sd| (u) H- y&i (u) = 0,
where a, fi, y are constants. In words, the squares of any three theta
functions satisfy a linear homogeneous equation with constant coeffi-
cients. The constants may be determined by assigning particular values
to the argument u. For example, take i/, H^ 0. Then*
* For brevity the parenthesis about the arguments of a function will frequently be
omitted.
SPECIAL INHKITE DEVELOPMENT!* 47t
H'K ^{u) ^ Ufo 1^ ■ ^» "" •" • + «"« - 1. r3»)
By treating H, e,, 8 in a similar manner may be ptored
A:«8n«tt + dn«ti-il and k*-^k'^^\. (40^
The function S(u)»(u — a), where a is a constant, latltto the rtii
tion <f> 0/h) = f/-*z-* Ci» («) if log C - iwa/K, Reasoning like
for treating 6* then shows that between any Uuee sn^ mx]
there is a linear relation. Hence
aH(u)H(u - a) + pH^(u)H^(u - a) - y%(u)9(u - •),
1^ = 0, ^/^,(0)//,(a)«ye<0)e(«),
u = A', a//,(0)//,(a) = ye,{0)e,(a),
©OQiO0,<///(t/)A/(?f-a) eH) //|(w)//,(ii~a) eO iy,4i
//fO0«0(*/)0(tt ~ a) ^ /fjo e(u)e(ii - a) ■ H,0 •• '
or (in a sn w SB (« — a) + cnii cn(« — a) — en a. (41)
In this relation replace a by — r. Then there results
en ucn(u ^ v) + sn ii dn v sn (u + r) = en v,
or en V en (ii + v) -f- sn v dn u sn (u + r) as en «,
, / . X en* u — en* v =5 sn* r — sn* « ,,„
a 11(1 sn (u + v) = ' — - — , (42)
^ snt^cnudnu — sn«cnrdnr ^ '
by symmetry and by solution. The fraction may be reduced by multiply-
ing numerator and denominator by the denominator with the middle
sign changed, and by noting that
sn'* V cn^ u dn** u — sn* u en* v dn* v = (sn* r — sn* n) (1 — Jt*sn* 11 sn* r).
_, , ^ sm/cnrdnv + sn vcntf dnn ..^
Then sn (u -f r) = --^-5 = . (4S)
^ ' ^ 1 — ^•*sn*tlsn*v ^ '
sn u en r dn r — sn V en M dn M
and sn (u — v) = — ; 75 — 5 ; t
^ ^ 1 — Ar8n*fisn*if
A / . X / X 2 sn r en li dn ti ....
and sn (u -f r) — sn(u — r) = pi — = 7- • (4*)
^ ^ / ^ ^ 1~ A:*sn*iisn*r ^ '
The last result may be used to differentiate sum. For
sn (u -h Au) — sn u _ sn ^ Am en (m -f 4 AM)dn(ti 4- 1 Aw)
Am "■ (Am 1- A:*sn*t AMsn*(M + J A«)'
-t-snM« aonvdnw, a^lim-^ — • (45)
474 THEORY OF FUNCTIONS
Here g is called the multiplier. By definition of sn u and by (33)
^ /fj(0) ©(0) 2X^^ ^ ^^^'^
The periods 2 K, 2 iK' have been independent up to this point. It will,
however, be a convenience to have ff = 1 and thus simplify the formula
for differentiating sn w. Hence let
? = 1, ^^ = &,(0) = l + 2q + 2q* + .... (46)
Now of the five quantities K, K\ k, k\ q only one is independent.
If q is known, then k} and K may be computed by (36), (46) ; k is de-
termined by k^-^-k'^^ 1, and K\ by irK'/K = - log y of (19). If, on the
other hand, k^ is given, q may be computed by (37) and then the other
quantities may be determined as before.
EXERCISE^
1. With the notations X = g~*e ^^ ^ ii = q-'^e ■*" " establish:
JT(- u) =- ir(u), fl^(u + 2X) =- JT(m), H(u + 2iir') =- ^ff(M),
e(-M) = +e(u), e(u-i-2^) = +e(M), e (m + 2 i^O =-/*©("),
Gj (- u) = + 01 (u), e^ (M + 2 iT) = + Gi (m), Gi (m + 2 iX') = + mOj (u),
^■(u + -ST) = + Ifi (u), Kiyi + i-BT') = ixe (m), ir(M + iT + i^') = + XG^ (u),
ITj (u + ^) = - ir(u), ITj (u + iiT') = + XGi (w), H^^ (u + iT + i^^ = - iXG (u),
G (u + iT) = + Gi (u), G (u + iZ^') = iXir(M), G (m + X + i^') = + XH^ (u),
Gi(u + ^) = + G(m), Gi(M + i-K'')=+X^i(w), ei(u + ir+t-S:') = +iXJJ(M).
2. Show that if u is real and g = J, the first two trigonometric terms in the
series for If, H^^ G, Gj, give four-place accuracy. Show that with g ^ 0.1 these
terms give about six-place accuracy.
3. Use ; = Q' sin a: + g2 sin 2 a -f- o* sin 3 a + • • • to prove
l-2gcosa + g2'' ^^ ^^ ^ ^
liru „ . Sttw
liogew
4. Prove the double periodicity of en u and show that :
«n(u + in = ^^» 8n(u + iirO = T-^' sn(u-f--S: + iJr') ^""
dn u Jk sn u A; en u
cn(u + ^ = i^^l^iij', ^^(^ ^ .^,) ^ ::::ldnu^ ^^ . ^ -w_
dnw ifcsnu ikcnu
dn(u + iO = -^. dn(w + iiTO = -i^^. dn(u + ^ + iX') = fJfc'^^.
SPECIAL INFINITE DEVELOPMENTS 47i
5. Tabulate the values of sn m, en n, dn m ai 0, If , UC\ K 4 ilT'.
6. Compute kf and JfltoTq = \ and q s 0.1. Bane* Aow Uhm two
tcnns in the theta aeries give four-plaoe meeunej U l(^ S |*
7. Prove cn(. ■!■,) = '" '"""-■'"■"^"^*.
1 — li^ n* II n* t
:ui(l dn(u + «)s
dnwdnp — l!*w««fcniiaif
8. Prove — cnu=— snudnu, — dnM=— HunMCQiL fvl.
(/u du
9. Prove sn-^u = f ^" fruio (46) with f « 1.
•^» V(l-u«)(l-A«i««)
10. If $r = 1, compute k^ f, JT, JT', for 9 = 0.1 and « = 0.01.
11. If flr = 1, compute Jf, q, K, K\ for ic* = |, f , |.
12. In Exs. 10, 11 write the trigonometric exprwioni which gtvt « i^ en ■, da ■
with four-place accuracy.
13. Find an 2 u, en 2 u, dn 2 u, and hence sniic,en|tt,dn|«, and dMm
8ni£' = (l + ikO"i cnlJJ' = VP<l + IO"*. dnJX»VP.
14. Prove — A- fanudn = log(dntt + JEcnit); alio
e2(0)/f (u + a)^(u - a) = 0«(a)JEr«(ii)- i7a(«)ei(ii>,
e«(0)e(u + a)Q{u - a) = 0«(«)0«(a)- H«(ii)H«(«).
CHAPTER XVIII
FUNCTIONS OF A COMPLEX VARIABLE
178. General theorems. The complex function u (cc, y) + iv (x, y),
where u (a;, y) and v (x, y) are single valued real functions continuous
and differentiable partially with respect to x and y, has been defined
as a function of the complex variable z = x -^ iy when and only when
the relations w^ = v^ and u^ — — v'^ are satisfied (§73). In this case
the function has a derivative with respect to z which is independent
of the way in which A« approaches the limit zero. Let w = f(z) be a
function of a complex variable. Owing to the existence of the deriva-
tive the function is necessarily continuous, that is, if c is an arbitrarily
small positive number, a number 8 may be found so small that
l/W -/(*„) I < ' when |a - s„| < 8, (1)
and moreover this relation holds uniformly for all points z^ of the
region over which the function is defined, provided the region includes
its bounding, curve (see Ex. 3, p. .92). .
It is further assumed that the derivatives i4) %^ ^x? ^y ^^^ continuous
and that therefore the derivative /'(«) is continuous.* The function
is then said to be an analytic function (§ 126). All the functions of a
complex variable here to be dealt with are analytic in general, although
they may be allowed to fail of being analytic at certain specified points
called singular points. The adjective "analytic" may therefore usually
be omitted. The equations
^ = /W or u = u(x, y), v = v(xj y)
define a transformation of the ccy-plane into the wv-plane, or, briefer, of
the «-plane into the i/;-plane; to each point of the former corresponds
one and only one point of the latter (§ 63). If the Jacobian
=«)'+«)'= I /'(«)r (2)
• It may be proved that, in the case of functions of a complex variable, the
continuity of the derivative follows from its existence, but the proof will not be
given here.
476
COMPLEX VARIABLE 477
of the transformation does not ranUh at a point <^ tht tqi
be solved in the neighborhood of that pointy and Kfj^j^t io
of Mh* second pkue corresponds only one of tim fiivt:
X=x(tt,v), yr.y(||, r) Of M wm ^(w).
Therefore it is seen that if w ma /(«) m analytit m U# mifMtviktti
o/« = «o» "^ */ ^'*<' derivative /\x^ does not vanisA^ tMs,fimt§i§m «flf A#
ifolved as x=z <^(ir), where ^ is the inverse funetkm of/ and b »%r-
wise analytic in the neighborhood of the point w^w , It naj Nsdily
1x3 shown that, as in the case of real functions, the derirativca/'(s) tnit
^'(tr) are reciprocals. Moreover, it may be seen that ths I^WM^ra«>
tion is confonnaly that is, that the angle between any two fmnm is
unchanged by the transformation (| 63). For ^v^TJdw (
As Ax and Au; are the chords of the curres before and after
tion, the geometrical interpretation of the equation, apart frooi the influx
itesimal ^, is that the chords A« are magnified in the ratio \f{m^\ to 1
and turned through the angle of /*(«J to obtain the chords Air (| 72).
In the limit it follows that the tangents to the fixnrrm are htiTliirrl ai
an angle equal to the angle of the corresponding »«iirTes plus Um aagle
oif{z^. The angle between two curves is therefore unchanged.
The existence of an inverse function and of the geametrio inlafpr»>
tation of the transformation as conformal both become iUoaory at points
for which the derivative /*(«) vanishes. Points where /*(«) a 0 are
called critical points of the function (§ 183).
It has further been seen that the integral of a fonelkm wldeh isiB^
lytic over any simply connected region is independent of the path and
is zero around any closed path (§ 124); if the region be noi simply con-
nected but the function is analytic, the integral about any dosed palb
which may be shrunk to nothing is zero and the integrals about any
two closed paths which may be shrunk into each other are equal (1 125).
Furthermore Cauehy's result that the value
of a function, which is analytic upon and within a doted path* may \m
found by integration around the path has been deri?od (1 116). By a
transformation the Taylor development of the funetkm has been found
whether in the finite form with a remainder (| 13«) or as an
series (§ 167). It has also been seen that any infinite
478 THEORY OF FUNCTIONS
which converges is differentiable and hence defines an analytic function
within its circle of convergence (§ 166).
It has also been shown that the sum, difference, product, and quotient
of any two functions will be analytic for all points at which both func-
tions are analytic, except at the points at which the denominator, in the
case of a quotient, may vanish (Ex. 9, p. 163). The result is evidently
extensible to the case of any rational function of any number of analytic
functions.
From the possibility of development in series follows that if two
functions are analytic in the neighborhood of a point and have identical
values upon any curve drawn through that point, or even upon any set
of points which approach that point as a limit, then the functions are
identically equal within their common circle of convergence and over all
regions which can he reached by (% 169) continuing the functions analyti-
cally. The reason is that a set of points converging to a limiting point
is all that is needed to prove that two power series are identical pro-
vided they have identical values over the set of points (Ex. 9, p. 439).
This theorem is of great importance because it shows that if a function
is defined for a dense set of real values, any one extension of the defi-
nition, which yields a function that is analytic for those values and for
complex values in their vicinity, must be equivalent to any other such
extension. It is also useful in discussing the principle of permanence of
form; for if the two sides of an equation are identical for a set of
values which possess a point of condensation, say, for all real rational
values in a given interval, and if each side is an analytic function, then
the equation must be true for all values which may be reached by ana-
lytic continuation.
For example, the equation sin x = cos {^tt — x) is known to hold for the values
O^x ^ I IT. Moreover the functions sin z and cos z are analytic for all values of z
whether the definition be given as in § 74 or whether the functions be considered
as defined by their power series. Hence the equation must hold for all real or
complex values of x. In like manner from the equation e^e*' = e^ + f which holds
for real rational exponents, the equation e^e"' = e« + «" holding for all real and im-
aginary exponents may be deduced. For if y be given any rational value, the
functions of x on each side of the sign are analytic for all values of x real or com-
plex, as may be seen most easily by considering the exponential as defined by its
power series. Hence the equation holds when x has any complex value. Next
consider x as fixed at any desired complex value and let the two sides be con-
sidered as functions of y regarded as complex. It follows that the equation must
hold for any value of y. The equation is therefore true for any value of z and w.
179. Suppose that a function is analytic in all points of a region ex-
cept at some one point within the region, and let it be assumed that
COMPLEX VARIABIJ5 479
the function oeaftes to be analytic at that point li>i*imft It rmMtm to bt
continuous. The discontinuity may be either finite or iofiiilte. la mm
the discontinuity is finite let |/(«)|< a m the nelgfaborliood o# Ikt
point « =: a of discontinuity. Cut the {loint out
with a Hiiiall circle and apply Cauchy's Intef^ral to
a ring surrounding the point. Tlie integral is appli-
cable because at all points on and within the ring
the function is analytic. If the small circle be
replaced by a smaller circle into which it may he
shrunk, the value of the integral will not U» cluinged.
/<-)-.i.[XS-XS4
1. ?
Now the integral about y^ which is constant can be made ■• mmII
as desired by taking the circle small enough; for \/(t)\< O aad
\t — z\^\a — z\ — Viy where r, is the radius of the ein-le y^ an
the integral is less than 2 7rr,c;/[|x — a|~ rj. As the integtml m
st^\nt, it must therefore be 0 and may be omitted. The remmininf
gral al)out C\ however, defines a function which is analjrtie at e ■■ «.
Hence if f{a) be chosen as defined by this integral instead of the
original definition, the discontinuity disappears. Finite distoniimmitim
may therefore be cons'ulered as due to bad judytneni in dejimim^ «
function at some point; and may therefcre be disregarded.
In the case of infinite discontinu;' * ' ' ' ^<yomr
injinife for all methods of approar ^ . or il
may become infinite for some methods of approach and remain /I nit* fir
other methods. In the first case the function is said to have ajMl* at
the i)oint z =z a of discontinuity; in the second case it is Miid to have
an essential singularity. In the case of a pole consider the reciprocal
function
The function F(») is analytic at all points near s « e and remains
finite, in fact approaches 0, as « approaches «. As F(«) « 0, it is seen
tliat F{z) has no finite discontinuity at s s a and is analytic also at
z = a. Hence the Taylor expansion
F(z) = a,(* - a)- + a.^.(s -«)-« + .•
is proj>er. If E denotes a function neither lero nor infinite at a » «,
the following tmnsformations may be made.
480 THEORY OF FUNCTIONS
F{z) = (z- aYE^{z), f{z) = {z- a)--E^(z),
jy") {z-ay {z-ay-^^ ^z-a
^C^-^C^(z-a)+C^(z-ay+....
In other words, a function which has a pole at « = a may be written
as the product of some power (z — a)~"* by an ^-function; and as the
^-function may be expanded, the function may be expanded into a
power series which contains a certain number of negative powers of
(z — a). The order m of the highest negative power is called the order
of the pole. Compare Ex. 5, p. 449.
If the function f(z) be integrated around a closed curve lying within
the circle of convergence of the series C^ -f C^(z — a) -\ , then
+ r [Co + C^{z - a) + . . .]c?^ = 2 TTiC.i,
Jo
or ff(z)dz==2 7riC_i', (4)
Jo
for the first m — 1 terms may be integrated and vanish, the term
C_i/(z — a) leads to the logarithm C_ilog(^ — a) which is multiple
valued and takes on the increment 2 7riC_i, and the last term vanishes
because it is the integral of an analytic function. The total value of
the integral of f(z) about a small circuit surrounding a pole is there-
fore 2 7rtC_i. The value of the integral about any larger circuit within
which the function is analytic except at « = a and which may be shrunk
into the small circuit, will also be the same quantity. The coefficient
C_i of the term (z — a)~^ is called the residue of the pole ; it cannot
vanish if the pole is of the first order, but may if the pole is of higher
order.
The discussion of the behavior of a function f(z) when z becomes
infinite may be carried on by making a transformation. Let
z' = -f « = — ,
z z'
/(«)=/ (J) = ■?'(*')• (6)
To large values of z correspond small values of z' ; if f(z) is analytic
for all large values of «, then F(z') will be analytic for values of z' near
the origin. At ;?j' = 0 the function F(z') may not be defined by (5) ; but
if F(z') remains finite for small values of «', a definition may be given
80 that it is analytic also at «' = 0. In this case F(P) is said to be the
COMPLEX VARIABLE 4«t
value of /(«) when x is infinite and the noUlkm /(«) » 1^(0) mm
be used. If F{z') does not remain finite bat haa a pole at •' » 0^ Ums
f{x) is said to have a pole of the same oitlef at • » «; and if F{i^
has an essential singularity at «' » 0, then /(«) is said to bire an caM»>
tial singularity at « = oo. Clearly if f(x) has a pole at s ■• «o, the vmtes
of /(^) must become indefinitely great no matter how a hnmmia t^fl-
nite; but if /(«) has an essential singukrity at s at oo, Umm wOl be
some ways in which * may become infinite so that /(«)
while there are other ways so that /(«) beoomee infinite.
strictly speaking there is no point of the «-pUne which
to ;:;' = 0. Nevertheless it is convenient to speak as if thetv w«i« «it4t
a }K)int, to call it the point at infinity^ and to designate it as a m od. If
then F^x") is analytic for «' = 0 so that /(«) may be said to be analjtie
at infinity, the expansions
F(«') = C^ 4- C,«' + C^« + . . . ^. c^-- + . . . «
are valid ; the f imction f(x) has been expantM ahomi tks p^imi cf i^^
ity into a descending power series in x, and the series will oonTerge for
all points z outside a circle |«| = 7?. For a pole of order m at inilai^
/W = C_^z'^ + C_^^,zr-' 4. . . . + C.,a + C. + ^ + ^ .
Simply because it is convenient to introduce the eonoept of the point
at infinity for the reason that in many ways the totality of fatffs valttss
for z does not differ from the totality of valoes in the neighborhood of
a finite point, it should not be inferred that the point at infinity has
all the properties of finite points.
EXERCISS8
1. Discuss sin (x + v) = sin X 008 y + cos 2 sin y for pennanenos ol fona.
2. If f{z) has an essential singularity at r s a, show that l//(f) hss aa mhbIM
singularity at z = a. Hence Infer that there Is •oro« mMliod of apfmeek 10 • • a
sucli that /{z) ± 0.
3. By treating f{z) — e and [/(«) — c]-> show
function may be made to approach any avifrn^ value e bf %
approaching the singular point z = a.
4. Find the order of the poles of theae functlona ai Um orifta :
(a) cot «, ifi) cK^t log (1 - X), (y) «(dn t - laa S)-».
482 THEORY OF FUNCTIONS
5. Shew that if f{z) vanishes at z = a once or n times, the quotient f{z)/f{z) has
the residue 1 or n. Show that if f{z) has a pole of the mth order at z = a, the
quotient has the residue — m.
6. From Ex. 6 prove the important theorem that : If f{z) is analytic and does
not vanish upon a closed curve and has no singularities other than poles within
the curve, then
1 /» f'lz)
——, I -f-' dz = rii + rig + • ■ • + wt - mj - wig mi = N - M,
2 in Jo f(z)
where N is the total number of roots of f{z) = 0 within the curve and M is the
sum of the orders of the poles.
7. Apply Ex. 6 to 1/P(z) to show that a polynomial P{z) of the nth order has
just n roots within a sufficiently large curve.
8. Prove that e« cannot vanish for any finite value of z.
9. Consider the residue of zf'{z)/f{z) at a pole or vanishing point of /(z). In
particular prove that if /(z) is analytic and does not vanish upon a closed curve
and has no singularities but poles within the curve, then
1 r zf'(z)
-—, I -zrr^z = n^a^ + n^a^ + • • • + 71*^* - m^\ - m^\ m^u
^mJo f{z)
where a^, a^, • • • , ak and n^, rig, • • • , n*; are the positions and orders of the roots,
and &!, b^, ' • • tbi and wij, mg, • • • , m^ of the poles of /(z).
10. Prove that 6i(z), p. 469, has only one root within a rectangle 2 ^ by 2 iK\
11. State the behavior (analytic, pole, or essential singularity) at z = oo for :
{a) z2 + 2 z, (iS) es (7) z/(l + z), (5) z/(z8 + 1).
12. Show that if /(z) = (z - a)^E(z) with - 1< fc < 0, the integral of /(z) about
an infinitesimal contour surrounding z = a is infinitesimal. What analogous theo-
rem holds for an infinite contour ?
180. Characterization of some functions. The study of the limita-
tions which are put upon a function when certain of its properties are
known is important. For example, a function which is analytic for all
values of z including also z = co is a constant. To show this, note that
as the function nowhere becomes infinite, | f(z) | < G. Consider the dif-
ference /(«o) — /(O) between the value at any point z = z^ and at the
origin. Take a circle concentric with z = 0 and of radius R > |«ol-
Then by Cauchy's Integral
By taking R large enough the difference, which is constant, may be
made as small as desired and hence must be zero; hence f(z) =/(0).
COMPLEX VAKIAHLK 4St
Any rational function /(«) a P(«)/Q(«), wbera /»(«) and Q(t) m
polynomials in ;;; and may be ft^nm^ to be devoid of ifpgBKm bilan.
can have as singularities merely polat. Thero will be a pole a mA
point at which the denomiDator vanishes; and if tho degim of lb*
numerator exceeds that of the denominator, then will be a pole at in-
finity of order equal to the difference of those degrees. Convenslj il
may \)e shown that ani/ function which has no other »im^lariif tk^m m
pole of the mth order at infinity must he a pdymnmitU o/tks wUk •nlf r
that if the only simjularities are a finite nttmher 0/ pmlst, wktiktr m tm.
finityorat other points^ the function is a rational fynetion; and ftaallj
that the knowledge of the xeros and poles with the mtiltmiitfiiK er
of each is sufficient to determine the function ejeeepi /br •
multiplier.
For, in the first place, if /(z) Is analytic except for a pole of Um Mth orter ai
infinity, the function may be expanded aa
/(z) = o_«r" + • + a_i« + 0, + o^r-* + a^* + • .
or f(z) - [a_,»z* + • • • + a_i«] = a, + a,f-> + a^* + • • •
The function on the right is analytic at infinity, and to miMl Ha aqosl on the Ml
be. The function on the left is the difference of a fnncUon which la aoaljtk for
all finite values of z and a polynomial which ia alao analytic for flaito valuta.
Hence the function on the left or ita equal on the right la aoalyUe for aU
of z including z = ao, and is a constant, namely Og. Henoe
/(z) = ao + a-iz + ■ ' +a.MZ« ia a polynomial of order ai.
In the second place let Zp z,, • • • , zj^ » l>o Po^«« of /(«) of **>• reepoetlvo
»Wp Wj, • • , wjt, m. The function
0 (Z) = (Z - «,)">(« - «^-« . . . (« - «»)■*/(!)
will then have no .singularity but a pole of order ■»,-|'a4<f---'f«ff«
at infinity ; it will therefore be a polynomial, aod /(i) b ratkmsl. Aa Um
numerator <p{z) of the fraction cannot vanish at z,, c,, •• •, <*, hot mw* have
ryij -I- wij + • • + njt + m root*, the knowledge of theee rooU will
numerator 0 (z) antl hence /(z) except for a conaUot multiplier. It
noted that if f{z) has not a pole at infinity but haa a two of order as Che
reasoning holds on changing m to — m.
When f(z) has a pole at a = a of the stth order, the expansion of
f{z) about the \x)\e contains certain negative powers
and the difference /(«) - P(* - a) is analytic at « - «. Tko tani
P(z- a) art' lulled the principal part of the JhtnOian /(b) mi tkejmU «.
484 THEORY OF FUNCTIONS
If the function has only a finite number of finite poles and the prin-
cipal parts corresponding to each pole are known,
<t>(z) =f{z) - P^{z - z^) - Plz -z^ Pj,{z - z^
is a function which is everywhere analytic for finite values of z and
behaves at « = oo just as f{z) behaves there, since P^, P^j '" •> Pk ^^^
vanish at « = oo. If f(z) is analytic at « = oo, then <^(«) is a constant;
if f{z) has a pole at ^ = oo, then <^ (z) is a polynomial in z and all of
the polynomial except the constant term is the principal part of the
pole at infinity. Hence if a function has no singularities except a finite
number of poles, and the principal parts at these poles are known, the
function is determined except for an additive constant.
From the above considerations it appears that if a function has no
other singularities than a finite number of poles, the function is ra-
tional ; and that, moreover, the function is determined in factored form,
except for a constant multiplier, when the positions and orders of the
finite poles and zeros are known ; or is determined, except for an addi-
tive constant, in a development into partial fractions if the positions
and principal parts of the poles are known. All single valued functions
other than rational functions must therefore have either an infinite
number of poles or some essential singularities.
181. The exponential function e" = e^(cos y + ^sin y) has no finite
singularities and its singularity at infinity is necessarily essential. The
function is periodic (§ 74) with the period 2 iri, and hence will take on
all the different values which it can have, if z, instead of being allowed
all values, is restricted to have its pure imagi-
nary part y between two limits yQ^yKyQ-h^Tr;
that is, to consider the values of e* it is merely -"
necessary to consider the values in a strip of
the «-plane parallel to the axis of reals and of breadth 2 tt (but lacking
one edge). For convenience the strip may be taken immediately above
the axis of reals. The function e* becomes infinite as z moves out
toward the right, and zero as z moves out toward the left in the strip.
li c = a -\- bi is any number other than 0, there is one and only one
point in the strip at which e* = c. For
e* = Va'-* -f- b'^ and cos y + i sin ?/ = , -f- i ,
have only one solution for x and only, one for y ii y he restricted to an
interval 2 tt. All other points for which e* = c have the same value for
X and some value y ±2n7r for y.
2in
z+2Tn
COMPLEX VARIABLE 405
Any rational function of e", as
will also have the period Iwi, When « movet off to the Wft b tiM
strip, /2(e*) will approach Ca./6. if 6. ^ 0 and will beeone iniaite If
h^ = 0. When « moves off to the right, R{f) must beeooM ittftaila if
n > niy approaeli C if n s m, and approach 0 if ti < ». Tha ^Hrfm\-
nator may be factored into terms of the form («* — «)*, and if Uw fia^
tion is in its lowest terms each such factor will repreae&i a pola of tiM
Ath order in the strip because «" — or «■ 0 has jnrt one aimple lool In
the strip. Conversely it may be shown that: Any fimeiion /{m) whkA
has the period 2 TTt, which further has no timgnlarUim kmi m Jiniis
number of poles in each strip^ and whieh eiiker ktetmu i^/Jmiit or m^
proaches a finite limit as x moves off to ths right or to iko i^/t, wsmti hs
f{z) = R{'^)i « rational function of e^.
The proof of this theorem requires aeveral stops. L«t It lint bs tmamsA tlMt/u)
remains finite at the ends of the strip and his no poles. Then /(f) b flnlls owr sU
values of z, including z = ao, and must be merely consuni. Nest lei /(s> mMla
finite at the ends of the strip but let it have poles at some points la tiM saipu It wfll
be shown that a rational function R{(^) may be constmcted sueli tliat/(t)-> ili^
remains finite all over the strip, including the portions at Infinity, and thol Ibsw
fore /(z) = /{ (e«) + C. For let the principal part of /(s) at any pate s s e bt
is a rational function of e' which remains finite at both ends of the strip and Is
such that the difference between it and P{t-^e) or/(s) has a polo of not W0tm
than the {k — l)8t order at z = c. By gubtracting a number of sQeb IsraH fnaai
/(z) the pole at z = c may be eliminated without introducinf any Mw pole.
Thus all the poles may be eliminated, and the result Is proved.
Next consider the case where /(z) becomes Infinite at one or at both ends of Ihs
strip. If /(z) happens to approach 0 at one end, comrfder /(«) + C, whIeh caaBot
approach 0 at either end of the strip. Now if /<x) or /(f) •!• C, as tho CMS ■>! jK
had an infinite number of zeros in the strip, these leros would bs i
finite limits and would have a point of condensation and tl
identically. It must therefore be that the function has only a iaila anaihirol
zeros ; ita reciprocal will therefore have only a finite nonhsr of polSB In ths rtrip
and will remain finite at the ends of the strips. Hones the looiptneal
quently the function itself Is a rational function of #•. The theorem Is
demonstrated.
If the relation /(« + «) = /(«) » satiafiod by a fanetkm, tba fono-
tion is said to have the period u. The function f(2wiM/m) will t^aa
have the period 2^1 Hence it follows that •//(«) kao tU pmod m^
becomes infinite or remains finite at the ends of a strip ^f\
486 THEORY OF FUNCTIONS
01, and has no singularities but a finite number of poles in the strips the
function is a rational function of c*'^**/". In particular if the period
is 2 TT, the function is rational in e*"^, as is the
case with sin z and cos z ; and if the period is
TT, the function is rational in e*^/^, as is tan z.
It thus appears that the single valued elemen-
tary functions, namely, rational functions, and
I'ational functions of the exponential or trigonometric functions, have
simple general properties which are characteristic of these classes of
functions.
182. Suppose a function f(z) has two independent periods so that
/(« + «,) =/(«), /(« + 0,') = /(«).
The function then has the same value at z and at any point of the
form z -f- mm -f- ww', where m and n ape positive or negative ilitegers.
The function takes on all the values of which it is capable in a parallel-
ogram constructed on the vectors w and cu'. Such z^-u-^u'
a function is called doubly periodic. As the values g+w'
of the function are the same on opposite sides of
the parallelogram, only two sides and the one in-
cluded vertex are supposed to belong to the figure.
It has been seen that some doubly periodic func-
tions exist (§ 177) ; but without reference to these
special functions many important theorems concerning doubly periodic
functions may be proved, subject to a subsequent demonstration that
the functions do exist.
If a doubly periodic function has no singularities in the parallelogram,
it must be constant; for the function will then have no singularities at
all. If two periodic functions have the same periods and have the same
poles and zeros (each to the same order) in the parallelogram, the quo-
tient of the functions is a constant; if they have the same poles and the
same principal parts at the poles, their difference is a constant. In these
theorems (and all those following) it is assumed that the functions
have no essential singularity in the parallelogram. The proof of the
theorems is left to the reader. If f(z) is doubly periodic, f(z) is also
doubly periodic. The integral of a doubly periodic function taken
around any parallelogram equal and parallel to the parallelogram of
periods is zero; for the function repeats itself on opposite sides of the
figure while the differential dz changes sign. Hence in particular
X^<-""«' i^^"«' i^--'-
COMPLEX VARIABLE 48f
The first integral shows that the twm of tks rmUMm 0/ iks faim im Ot
parallelogram ia xero ; the second, that tks imwther ^ wtrm it 0fmmJ tm
the number of poles provided multiplicities are taken into aeeonal} tk»
third, that the number of zeros off(s) — C U the «mm •» lAe nMnkre/
zeros or poles off(x), because the poles of /(«) and/(«) - Cam tks
The common number m of poles of /(a) or of aeit» of /(t) or of
of f(x) = C in any one parallelogram is called tks or^er •/ fA#
periodic function. As the sum of the residues ranishss, H is
that there should be a single pole of the first order in the p*>f|j>tliy***
Hence there can be no functions of the first order and ths
possible functions would be of the seoond order with the
lH-«o + ^i(^-'^)+-- or — --4-«, + ... and — ~4-r'.
in the neighborhood of a single pole at s a a of the seoond ordsf or of
the two poles of the first order at « =: a, and s « e^ Let it be asMOMd
that when the pt^riods cd, J are given, a doubly periodie fnnetioo f (Si*)
with these periods and with a double pole at s es « exists, and sfanilarly
that h{Xy a^y a^ with simple poles at a^ and a, exists.
Any doubly periodic function fix) with the periods m, m* wutp he sat*
pressed as a polynamial in the functions g(x, a) and A (a, «,, «J of the
second order. For in the first place if the function /(a) hm a pole of
even order 2 A: at « = «, then /(«) — C[^(«, a)]*, where C is properly
chosen, will have a pole of order leas than 2 ib at a » a and will hav«
no other poles than /(«). Hence the order of /(«) — C[y(«, «)]* is leas
than that of /(«). And if /(«) has a pole of odd order 2A + lai«"«>
the function f{x) - C[i/(«, a)]*A(«, a, b), with the proper ehoiee of C,
will have a pole .of order 2 A or less 2X x^a and will gain % staipla
pole at « = /;. Thus although /- C/A will generally not be of \amm
order than /, it will have a complex pole of odd order split into a pole
of even order and a pole of the first order; the order of the former
may l)e reduced as before and pairs of the latter may be iWBOved. By
repeated applications of the process a fnnofcion may be obteined whiek
has no poles and must be constant The theorem is therefore proved.
With the aid of series it is possible to write down some donbly peri»
odic functions. In particular consider the series
w
and p
'(«)°-2X(,-J-iM.y
488 THEORY OF FUNCTIONS
where the second 2 denotes summation extended over all values of
m, n, whether positive or negative or zero, and 2' denotes summation
extended over all these values except the pair m = ti = 0. As the sum-
mations extend over all possible values for m, n, the series constructed
for z -\- <a and for z + w' must have the same terms as those for z, the
only difference being a different arrangement of the terms. If, there-
fore, the series are absolutely convergent so that the order of the terms
is immaterial, the functions must have the periods w, w'.
Consider first the convergence of the series p'(z). For z = mw + nu>% that is, at
the vertices of the net of parallelograms one term of the series becomes infinite
and the series cannot converge. But if z be restricted to a finite region R about
z = 0, there will be only a finite number of terms
which can become infinite. Let a parallelogram P
large enough to surround the region be drawn, and
consider only the vertices which lie outside this par-
allelogram. For convenience of computation let tHe
points z = mu) + nu' outside P be considered as ar-
ranged on successive parallelograms P^, P^, • • • ,
Pjfc, • . • . If the number of vertices on P be v, the
number on Pj is v + 8 and on Pj^ is v + 8A;. The
shortest vector z — mot — nw' from z to any vertex of Pj is longer than a, where
a is the least altitude of the parallelogram of periods. The total contribution of
Pj to p\z) is therefore less than {y + 8)a-3 and the value contributed by all the
vertices on successive parallelograms will be less than
^^y+8 i>-h8-2 i>,+ 8-3 V+8.A;
a« (2a)8 ((3a)8 {ka^
This series of positive terms converges. Hence the infinite series for i)'(z), when
the first terms corresponding to the vertices within P^ are disregarded, converges
absolutely and even uniformly so that it represents an analytic function. The
whole series for p'{z) therefore represents a doubly periodic function of the third
order analytic everywhere except at the vertices of the parallelograms where it
has a pole of the third order. As the part of the series p\z) contributed by ver-
tices outside P is uniformly convergent, it may be integrated from 0 to z to give
the corresponding terms in p (z) which will also be absolutely convergent because
the terms, grouped as for p'(z), "^ill be less than the terms of IS where I is the
length of the path of integration from 0 to z. The other terms of p'{z), thus far
disregarded, may be integrated at sight to obtain the corresponding terms of p{z).
Hence p'{z) is really the derivative of p (z) ; and as p (z) converges absolutely ex-
cept for the vertices of the parallelograms, it is clearly doubly periodic of the
second order with the periods w, w', for the same reason that p\z) is periodic.
It has therefore been shown that doubly periodic functions exist,
and hence the theorems deduced for such functions are valid. Some
further important theorems are indicated among the exercises. They
lead to the inference that any doubly periodic function which has tho
COMPLEX VARIABLK
periods oi, a>' and has no other singulariliM than polat mar ht
as a rational function of p(z) and p'(')* <M^ M an iiratioiial fiuMliaa af
jj {;:) alone, the only irrationalities being square roots. Thus bgr «i^
ploying only the general methods of the thaotj of fwMilioiis of a
(roniph'X variable an entirely new category of
actci'i/cd and its essLMitial propt^rtieM liave been
KXKKCISKS
1. Find the principal parU at x = U (ur Umj funcUoM of Kx. 4, p. Ml.
2. Prove by Ex. 6, p. 482, that e« - c = 0 has only one raoi la Um mHp,
3. How does e^*") behave as t becomes infinite In the strip?
4. If the values K (e*) approaches when s becomes infinite In ihm strip sfe
exceptional values, show that R (e*) takes on every valtM oUwr tbsa Um
tional values k times in the strip, k being the greater of Uie two biUBbm a, m.
5. Show by Ex. 0, p. 482, that in any parmUelogfam of periods th« sum uf Um
positions of the roots leas the sum of the positions of the poles of a doaMy psri«
odic function is nua + nu\ where m and n are Integert.
6. Show that the terms of p'{z) may be assoristed in sncii a way sa to pfufe
that p'{—z) = — p'{z)y and hence infer that the ezpansloos are
p'(z) = - 2z-« + 2cj« + 4c^ + . . . , only
an<l j)(«) = z-« + Ci«« + f^ + ..., only
7. Examine the series (6) f or p'(z) to show tbatp'(| m) = p'(| ••') = p'(J • + | ^O - ••
Why can p'(z) not vanish for any other poinU in the parallelogianf
8. Let p(i «) = e, p(i "O = <^i P(i « + i "O = «"• ^^^^ **»• »d«>«*«J o* »^
doubly periodic functions [p'(2)]* and 4 [p{z) - e][p(<) - «1[P(«) - O-
9. By examining the series defining p{z) show that any two points t • e
z = a' such that p(a)=p {a') are symmetrically situated In the paiaUelaflBa
respect to the center z = ^ (w + w^. How could this be Infened frOM Bs. ft t
10. With the notations g(z, a) and A(«, Oj, o,) of the teztjhow:
^'^' pW-pW pW-J>(<H)
11. Demonstrate Uie final theorem of the u:i.i
490
THEORY OF FUNCTIONS
12. By combining the power series for p{z) and p'{z) show
[P'(2)]* - 4 [P(2)]' + 20c^p{z) + 28C2 = Az^ + higher powers.
Hence infer that the right-hand side must be identically zero.
13. Combine Ex. 12 with Ex. 8 to prove e + e' + e" = 0.
14. With the notations g^ = 20 c^ and g^ = 28 Cj show
dp
!>'(«) = V4i>«(2)- 8^2^(2) -^3 or
d
V4p8_ g^p-g^
= dz.
15. If i*(«) be defined by U^)=Pi^) or f (z) = — fp{z)dz, show that
dz •^
f (z + w) — f (2) and ^-(2 + w') — f (z) must be merely constants ij and ij'.
183. Conformal representation. The transformation (§ 178)
w = /(z) or w -j- iv = ti (x, y) + *t^ (x, y)
is conformal between the planes of z and w at all points « at which
f*(z) T^ 0. The corresponde'nce between the planes may be represented
by ruling the «-plane and drawing the corresponding rulings in the
i<;-plane. If in particular the rulings in the ;5;-plane be the lines x = const.,
y = const., parallel to the axes, those in the i<;-plane must be two sets
of curves which are also orthogonal ; in like manner if the «-plane be
ruled by circles concentric with the origin and rays issuing from the
origin, the z^-plane must also be ruled orthogonally ; for in both cases
the angles between curves must be preserved. It is usually most
convenient to consider the w-plane as ruled with the lines u = const.,
V = const., and hence to have a set of rulings u (a:, y) — Cj, v (x, y) = c^
in the «-plane. The figures represent several different cases arising from
the functions •
w-plane (1) z^plane w-plane (2) z-plane
(1) w = az = (a^-{' a,^i) (x + iy), u = a^x - a^, v = a^c + a^y,
(2) w = \ogz=: log Va^ + j/'-f- i tan-* ^> ?< = log Vx* -h y% v = tan"* - •
X X
Consider w = «•, and apply polar co5rdinates so that
w = n (cos * -f i sin *) = r*(cos 2 <^ + i sin 2 <^), 72 = r^, * = 2 </»
COMPLEX VARIABLE 4t|
To any point (r, ^) in the •.plana oomtpoodi (/t » f', # . 2 4) faiiW
M'-plane ; circles about MwmQ beoome otralat abooi w « 0 and imvs b>
suing from « = 0 become rays iMuing fitm i# « 0 at tviee tU a^la.
(A figure to scale should be supplied by the raader.) The deritHite
w' = 2z vanishes at « b 0 only. The tranefbrmalioo k coali
all points except « = 0. At « = 0 it ie elear that the angle
two curves in the «-plane is doubled on paaeing to the
curves in the ir-plane ; henoe at s iv 0 the trantfoimatioo k _
formal. Similar results would be obtained from wmit^ eaeept that the
angle between rays issuing from w^O would be m tioMe the angle
between the rays at « = 0.
A point in the neighborhood of which a funetioQ w « /(«) fe aa»>
lytic but has a vanishing derivative /*(«) is called a erUieml mtiai of
/(«); if the derivative f{x) has a root of multiplicity k at any p^nf^
that point is called a critical point of order k. Let s a c he a cntimJ
point of order k. Expand /*(«) as
f(z) = a,(z - «^* + a,^,(, - ,^*i + a,^,(, « ^♦t + . . . j
then /(*)=/(*,) 4- j^(«-«J**> + -^(,-.J»*« + ....
or w^w^->r{z^x^^^^E{z) or ir - ir,=s (« _ «j--'£-(i), (J)
where £: is a function that does not vanish at «,. The point m * s^goes
into w = tv^. For a sufficiently small region about «, the
tion (7) is sufficiently represented as
w-w^=C(z^ «o)**S C* = JJ(s^.
On comparison with the case «;=:«", it appears that the angle
two curves meeting at z^ will be multiplied by i; -f 1 on passing to the
corresponding curves meeting at w^. Henoe at a eriiieml jwfal ^ CAe
kth order the transformation ia not eoi\formal but amglm art mmltipjitd
by k -\-l on passing from the z-plane to the w-plame.
Consider the transformation w = x* more in detail. To eaeh point e
corresi3onds one and only one point fr. To the points u in the tot
quadrant correspond the points of the first two quadrants in the ir-
plane, and to the upper half of the x-plane corresponds the whole ipplene
In like manner the lower half of the «r-phme will be mapped upon the
whole It-plane. Thus in finding the points in the w-ptutt whieh eor-
res})ond to all the ))oints of the c-phine, the Mvphuie ia eoforad twieti
This double counting of the tc^plane may he obviated hy a
vice. Instead of having one sheet of paper to reprseent the
492
THEORY OF FUNCTIONS
let two sheets be superposed, and let the points corresponding to the
upper half of the «-plane be considered as in the upper sheet, while
those corresponding to the lower half are considered as in the lower
sheet. Now consider the path traced upon the double t^;-plane when z
traces a path in the «-plane. Every time z crosses from the second to
Y
w— surface
z— plane
the third quadrant, w passes from the fourth quadrant of the upper
sheet into the first of the lower. When z passes from the fourth to
the first quadrants, w comes from the fourth quadrant of the lower
sheet into the first of the upper.
It is convenient to join the two sheets into a single surface so that
a continuous path on the «-plane is pictured as a continuous path on
the w;-surface. This may be done (as indicated at the right of the
middle figure) by regarding the lower half of the upper sheet as con-
nected to the upper half of the lower, and the lower half of the lower
as connected to the upper half of the upper. The surface therefore
cuts through itself along the positive axis of reals, as in the sketch on
the left* ; the line is called the junction line of the surface. The point
w = 0 which corresponds to the critical point « = 0 is called the branch
point of the surface. Now not only does one point of the ;5;-plane go
over into a single point of the ?/;-surface, but to each point of the sur-
face corresponds a single point z] although any two points of the vi-
surface which are superposed have the same value of w, they correspond
to different values of z except in the case of the branch point.
184. The 2^;-surface, which has been obtained as a mere convenience
in mapping the «-plane on the ?<?-plane, is of particular value in study-
ing the inverse function z = ^tv. For Vw^ is a multiple valued func-
tion and to each value of w correspond two values of «; but if w be
* Practically this may be accomplished for two sheets of paper by pasting gummed
Ktrips to the sheets which are to be connected across the cut.
COMPLEX VARIABLE
4M
regarded as on the u^«vir£Boe initeid of rnmHy in the w-phim, Umv fti
only one value of x oorresponding to a point w npoo the tnrlbML
the functwn Vt^ which ts daubis valusd oper tJks u^plmtu hnmm
valued over the wsurface. The irniurfaoe is oallad ilie i?iraMMi tmtjkm
of the function x = Vtr. The conttniotion of Biemaiin MifMat it i»-
{K)rtant in the study of multiple Tallied fnnntinni hfianMii the
keeps the different values apart, so that to each fiotnt of the
corresponds only one value of the ftmction. Consider
(The student should make a paper model bjr foUowing the st«|M as
indicated.)
Let 10 = z*^ Sz and plot the uvcarfsce. Ftrat solve /'(f) ss 0 to iad tiM etilkal
lM>int8 z and substitute to find the branch points w. Now If Um brsaek poAals W
considered as removed from the uvplsne, the plane Is no loogsr ibeplj
It must be made simply connected by drawing pn^wr Uoss la tlw l%af«
be accomplished by drawing a line from each bnuieb polat lo laiailj or hf eoa>
necting the successive branch points to each other as
the point at infinity. These lines are the junction Unas. In this
critical points are z = + li — 1 and the branch poInU are w s ~ 1, >f S, aa4 tlw
junction lines may be taken as the straight lines Joining w ss • S and v a 4> S to
i,n,m
I n HI
d^r\e tr-o
T^a
I'u'm'
a/p b
q
i',ii',in'
tc- surface
infinity and lying along the axis of reals as in the figure. Next
site number of sheeU over the u^plane and cut them along the
M> = z« — 3z is a cubic in z, and U) each value of w, except the braach vshMS» Ihccv
correspond three values of z, three sheeU are needed. Now find la tfcs s^pisae Iks
image of the junction lines. Tlie Junction lines are lepfUMOted by t«0; !■!
v = Sz^y-y*-Zy, and hence the line y = 0 and the hyperbola ««• - |^ ■ t wlB
be the images desired. The t-plane Is dlrided Into six pleess which wOl bs ssm Id
c(»rrespond to the six half sheeU over the le-plane.
Next z will be made to trace out the Inuges of the juaetlon lines sad lo tiif»
about the critical polnU so that ie will trace out the Joaetloa
the branch jwints in such a manner that the connectioM bscwssa lbs dU
sheet* may l>e made. It will b© convenient to regard f awl ^ as prmi
along their respecUve paths so that the terms "right" sad "left" hav*
494 THEORY OF FUNCTIONS
Let z start at z = 0 and move forward to z = 1 ; then, as/'(z) is negative, lo starts
at 10 = 0 and moves back to lo = — 2. Moreover if z turns to the right as at P, so
must 10 turn to the right through the same angle, owing to the conf ormal property.
Thus it appears that not only is OA mapped on oa, but the region 1' just above OA
is mapped on the region I' just below oa ; in like manner OB is mapped on 6b.
As ab is not a junction line and the sheets have not been cut through along it, the
regions 1, 1' should be assumed to be mapped on the same sheet, say, the upper-
most, I, I'. As any point Q in the whole infinite region V may be reached from 0
without crossing any image of oft, it is clear that the whole infinite region 1' should
be considered as mapped on Y ; and similarly 1 on I. The converse is also evident,
for the same reason.
If, on reaching A, the point z turns to the left through 90° and moves along ^C,
then w will make a turn to the left of 180°, that is, will keep straight along aCf
a turn as at B into 1' will correspond to a turn as at r into I'. This checks with
the statement that all 1' is mapped on all I'. Suppose that z described a smali
circuit about + 1. When z reaches D, w reaches d ; when z reaches jE", w reaches e.
But when w crossed ac, it could not have crossed into I, and when it reaches e it
cannot be in I ; for the points of I are already accounted for as corresponding to
points in 1. Hence in crossing ac, w must drop into one of the lower sheets, say
the middle, II ; and on reaching e it is still in II. It is thus seen that II corre-
sponds to 2. Let z continue around its circuit ; then II' and 2' correspond. When
z crosses AC from 2' and moves into 1, the point w crosses ac' and moves from 11'
up into I. In fact the upper two sheets are connected along ac just as the two
sheets of the surface for w = z*-^ were connected along their junction.
In like manner suppose that z moves from 0 to — 1 and takes a turn about B so
that w moves from 0 to 2 and takes a turn about h. When z crosses BF from 1' to 3,
w crosses bf from Y into the upper half of some sheet, and this must be III for the
reason that I and II are already mapped on 1 and 2. Hence Y and III are con-
nected, and so are I and III'. This leaves II which has been cut along 6/, and III
cut along ac^ which may be reconnected as if they had never been cut. The reason
for this appears forcibly if all the points z which correspond to the branch points
are added to the diagram. When to = 2, the values of z are the critical value — 1
(double) and the ordinary value z = 2 ; similarly, lo = — 2 corresponds to z = — 2.
Hence if z describe the half circuit AE so that w gets around to e in II, then if z
moves out to z = 2, lo will move out to lo = 2, passing by lo = 0 in the sheet II as
z passes through z = Vs ; but as z = 2 is not a critical point, lo = 2 in II cannot
be a branch point, and the cut in II may be reconnected.
The lo-surf ace thus constructed for w == /(z) = z* — 3 z is the Riemann surface
for the inverse function z =/-i(io), of which the explicit form cannot be given
without solving a cubic. To each point of the surface corresponds one value of z,
and to the three superposed values of w correspond three different values of z ex-
cept at the branch points where two of the sheets come together and give only
one value of z while the third sheet gives one other. The Riemann surface could
equally well have been constructed by joining the two branch points and then
connecting one of them to oo. The image of r = 0 would not have been changed.
The connections of the sheets could be established as before, but would be dif-
ferent. If the junction line be — 2, 2, + oo, the point lo = 2 has two junctions
running into it, and the connections of the sheets on opposite sides of the point are
not independent. It is advisable to arrange the work so that the first branch point
HH
COMPLEX VARIABLE
which is encircled «hAll have only one joneUoa nmnliif tnm iu Tlik wmw h» 4aa»
by taking a very large circoit In t ao that w wlU dcacribe a laift rtnaji mA iMMi
cut only one junction line, namely, from S to ao, or by takli^l a mmM diwll ^btm
2 = 1 so that to will take a small turn about w s - t. Lai iha iMIar Biitei %•
chosen. Let z start from f = 0 at O and more to t b 1 - ^ : Itif wstartin wf
and moves to tr = — 2. The oonmpoodmim btlWMB 1' and r la iImi MiaMtakai.
Let z turn about A ; then 10 tuma about i0B-.fat«« Aslba Um — t to — a^or «
is not now a junction line, 10 movea from V
into the upper half I, and the region acroaa
AC from V should be labeled 1 to corre-
spond. Then 2', 2 and 11', II may be filled
in. The connections of I-II' and II-I' are
indicated and 1 1 I-II T is reconnected, aa the w-turfaee
branch point Ih of tlie first order and only two
sheets are involved. Now let t move from fsOtoss — 1 and lain m
B ; then w nioveK from to = 0 to 10 = 2 and takes a turn about b, Tba rifloa
r i8 marked 3 and V is connected to III. Paaaing from 8 to f for « b aqali
to passing from III to III' for to between 0 and 6 where thaaa aheaUara eoaai
From 3' into 2 for z indicates III' to II acroaa the Junction frooi v a t to ao^
leaves I and 11' to be connected acroas thia junction. The fffnnnctiw bid ceo^
pletc. They may be checked by allowing a to deacribe a laifa dreoit ao IktlL tka
regions 1, 1', 3, 3', 2, 2', 1 are succeealTely traTwaed. That I, l\ III, III', II, IT, I
is the corresponding succession of aheeta la clear from the oonnactlooa batwa
w = 2 and oo and the fact that from to = — 2 to — oo there b no JuncUfci.
Consider the function to = z* — 8z^-)-8z*. The critical polnta ara a a <^ I, I,
— 1,-1 and the corresponding branch points are 10 = 0, 1, 1, 1, 1. Draw the JOBO*
tion lines from to = 0 to — 00 and from 10 = 1 to <|- oo along tiia axia i»f rrmlM. To
find the image of 0 = 0 on the z-plane, polar ooOrdlnataa may be uarol
« = r(co80 + i8in0), to = u + to = r<e«*'-8r«««*' + f rM**.
V = 0 = rS[r« sin 6^ - Sf'ain 4^ + 8ain2#]
= r* sin 2 0[f*(3 - 4 ain 2 #) - « f* ooa # + 8).
The equation t) = 6 therefore breaka up Into the equation ain S ^ = 0 awl
3co820i \/88in2^ VS ain (60 i2#)__ _ ^
3-48in32^ 2 8in(60 + 2^)ain(00-2#) tda(60±t#>
r2 =
Hence the axes 0 = 0° and ^ = W and the two rectangular hyptrbohM IncllMd at
angles of ± 15° are the iniagea of c = 0. The i-plaiie ia thoa divMad iMo ilz par*
tions. The function to is of the alxth order and alz ahead moift ba ipraad otar iha
t«-plane and cut along the junction linea.
To connect up the sheets It is merely neoeaaary to get a atari. T%a Um « ■ t
to 10 = 1 is not a junction line and the aheeta have not been col throofh alo«f IL
But when z is small, real, and Increaaing, w la alao anall, '••'•■■f *"•*"*■••
Hence to OA corresponds oa In any sheet deaired. If oieoiver the leglea above OA
will correspond to the upper half of the sheet and the ragloa below (U Ip the
lower half. Let the sheet be ehoaen aa III and place the ntmben 8 aad r aa^ le
correspond with III and III'. FUl In the numbers 4 and 4' aioond « . 0.
496
THEORY OF FUNCTIONS
It VI
z turns about the critical point z = 0, to turns about lo = 0, but as angliis are doubled
it must go around twice and the connections III-IV, IV-III' must be made. Fill
in more numbers about the critical point z = 1 of the second order where angles are
tripled. On the lo-sur-
face there will be a
triple connection III'-
II, II'-I, r-III. In
like manner the criti- W ' ///
cal point z = — 1 may
be treated. The sur-
face is complete except
for reconnecting sheets _/_^__/
I, II, V, VI along MJ = 0 -^ » Vi
to 10 = — CO as if they
had never been cut. w—surfa/ce z— plane
\\i///
EXERCISES
1. Plot the corresponding lines for : (or) lo = (1 + 2 i)z, (j3) lo = (1 — ^ i)z.
2. Solve for x and y in (1) and (2) of the text and plot the corresponding lines.
3. Plot the corresponding orthogonal systems of curves in these
(/S) 10 = 1 + z^, (7) w — cos z.
(«)«, = -
4. Study the correspondence between z and w near the critical points:
(a) 10 = z*, (/3) 10 = 1 — z^, (7) w = sin z.
5. Upon the lo-surface for lo = z^ plot the points corresponding to z = 1, 1 + i,
2 1, — I + I VSi, — i, —\ V3 — J i, — i, l — \i. And inthe z-plane plot the
points corresponding to m> = V2 + \/2i, i, — 4, — ^ — J VSi, 1 — i, whether in
the upper or lower sheet.
6. Construct the w-surf ace for these functions :
(a) W = Z8, (/S) W = Z-2, (7) M> = 1 + ZS (5) 10 = (z - 1)«.
In (/3) the singular point z = 0 should be joined by a cut to z = oo.
7. Construct the Riemann surfaces for these functions :
(a) w = z*-2 z2, (/3) 10 = - z* -}- 4z, (7) w = 2z» - 6z«,
(«) w = z-\-
{^)w = z^ + -
(n 1^ =
2» +
VS;
V3z2 + 1
185. Integrals and their inversion. Consider the function
=/'
w
> « = hi Wj
w
hi~^s:,
defined by an integral, and let the methods of the theory of functions
be applied to the study of the function and its inverse. If w describes
a path surrounding the origin, the integral need not vanish; for the
COMPLEX VARIABLE
49T
integrand is not analytic at i£f . 0. Let a eot Vje drawn frtm «. . 0 to
1/^ = - 00. The integral is then a single rained fuoetkin of w pmridMl
the path of integration does not cross the cut Moraovor, H k amljtfo
except at m; = 0, where the derivative, whieh is the InlsffrBod l/m,
(Miises to be continuous. Let the uN-plane as cut be maiiped oo tbe
xr-plane by allowing w to trace the path lahede/ghil, bj
value of X sufficiently to
draw the image, and by
applying the principles of
con formal representation.
W hen tv starts from w = 1
and traces 1 a, « starts from
z = 0 and becomes nega-
tively very largo. When w
turns to the left to trace a A,
z will turn also through 90*
to the left. As the integrand along ah is id^, z roust be rhanginy braa
amount which is pure imaginary and must reach B when w
When tr traces hc^ both w and dw are negative and « must be
by real positive quantities, that is, z must trace BC. When w i
rdefg the same reasoning as for the path ab will show that a
CDEFG. The remainder of the path may be completed by the
It is now clear that the whole tr-plane lying between the inl
and infinite circles and bounded by the two edges of the cut is
on a strip of width 2 iri bounded upon the right and left by two infi-
nitely distant vertical lines. If w had made a oomplele turn in the posi-
tive direction alx)ut xv — 0 and returned to its starting point, a woold
have received the increment 2 irL That is to say, the values of a whieh
correspond to the same point w reaclxHl by a direct path and bj a pttlh
which makes k turns alx)ut ir = 0 will differ by 2 kwL Henee whan w
is regarded inversely as a function of «, the function will be periodie
with the period 2 7rt. It has been seen from the correspoodapee of
cdefg to CDEFG that w becomes infinite when a moves off iiMMiiMj
to the right in the strip, and from the correspondence of BAIH with
haih that w becomes 0 when x moves oflf to the 1 ''' * ' iiee w ■«! be
a rational function of e'. As w neither beoomi •• nor vanishM
for any finite point of the strip, it must reduce merely to Cs*» with k
integral. As w has no smaller |)eriod than 2 wt, it follows that * « 1.
To determine C, compare the derivative dw/da ■■ C*" at « « 0 with its
reciprocal dz/dw = i£r-» at the corresponding point ir « 1; then C • 1
The inverse function In"** is therefore ooropletely determined aa ^.
498
THEORY OF FUNCTIONS
Id like manner consider the integral
w dw
Jpw du
0 r+"
2 =/(«;), w = 4>{z)=f-Hz).
BAKJ
Here the points w = ± i must be eliminated from the to-plane and the plane ren-
dered simply connected by the proper cuts, say, as in the figure. The tracing of
the figure may be left to the reader. The
chief difficulty may be to show that the
integrals along oa and be are so nearly equal
that C lies close to the real axis; no com-
putation is really necessary inasmuch as the
integral along oc' would be real and hence
C must lie on the axis. The image of the
cut lo-plane is a strip of width tt. Circuits
around either -f- i or — i add tr to z, and
hence ly as a function of z has the period tt.
At the ends of the strip, w approaches the
Unite values + i and — i. The function
w = (p(z) has a simple zero when z = 0 and
has no other zero in the strip. At the two points z = ± ^ tt, the function w becomes
infinite, but only one of these points should be considered as in the strip. As the
function has only one zero, the point z = ^ tt must be a pole of the first order.
The function is.therefore completely determined except for a constant factor which
may be fixed by examining the derivative of the function at the origin. Thus
-plane
w—jylane
e^iz^l i e»2-|- e-
tan 2, z = tan-iio.
186. As a third example consider the integral
dw
'£
VT
z=f(w), «; = ./.(^)=/-X^). (8)
Here the integrand is double valued in w and consequently there is
liable to be confusion of the two values in attempting to follow a path
in the w-'p\a,ne. Hence a two-leaved surface for the integrand will be
constructed and the path of integration will be considered to be on the
surface. Then to each point of the path there will correspond only one
value of the integrand, although to each value of w there correspond
two superimposed points in the two sheets of the surface.
As the radical Vl — w^ vanishes at lo = ± 1 and takes on only the single value 0
instead of two equal and opposite values, the points w = ±1 are branch points on
the surface and they are the only finite branch points. Spread two sheets over the
lo-plane, mark the branch points lo = ± 1, and draw the junction line between them
and continue it (provisionally) to lo = oo. At to = — 1 the function Vl — ib^ may
be written Vl -|- w F{w)^ where E denotes a function which does not vanish at
u> = — 1. Hence in the neighborhood of w = — 1 the surface looks like that for
Vio near lo = 0. This may be accomplished by making the connections across the
4M
H
♦I
■If a
lok*
COMPLEX VARIABLE
junction line. At the point to s 4- 1 the MirfiM noiC eat
manner. Ttiis will be ao provided thmt the
never cut ; if the sheeu luul been era^eoDMeUd aloof 1 «^«MbilfeM«
hncn separate, though craved, over 1, and the brmneh poiot «mM
have disappeared. It is noteworthy that If m deeerlbMafaMfe
circuit including both branch points, the valueaof Vl— i/ ai«
not interchanged ; the circuit doeee in each ebeei wfthmU dm*. — ^
ing into the other. This could be nrprnaMil bj sayinf that w ■ <d
is not a branch point of the function.
Now let w trace out various paths on the surface in the ai
face on the 2-plane by aid of the integral (8). To avoid any itHtolhki la the «af
of double or multiple values for t which might arise If is fimtd aboat a hmaeh
point to = ± 1, let the surface be marked in eaeb sheet over the aito of tmlk tnm
— 00 to + 1. Let each of the four half planes be traated aepaiaioly. Let m
at to = 0 in the upper half plane of the upper sheet and let the vahM ol Vl— ^
at thiH point be -f 1 ; the values of Vl — m^ near w b 0 la ir
+ 1 and will be sharply distinguished from the valoee near — 1 whiehaivsni
to corre8i)ond to points in 1\ H. Aa lo traces oo, the Integial t IneisoaH fi
a detiiiite positive number a. The value of the integnd from a to 6 b
Inasmuch as to = 1 is a branch point where two sheeu connect. It Is
assume that as to passes 1 and leaves it on the right, s will tttfn Ihiii^h
straight angle. In other words the integral from 6 to e Is aaturaUy
a large pure imaginary affected
with a positive sign. (This fact
may easily be checked by exam-
ining the change in Vl — to*
when to describes a small circle
about to = 1. In fact if the E-
function Vl + to be discarded
and if 1 — to be written as re*',
then Vrei** is that value of the
radical which is positive when
1 — 10 is positive. Now when lo
describes the small' semicircle,
0 changes from 0° to — 180° and hence the value of the radical alo«f le hseoBHS
— iVr and the integrand is a positive pure imaginary.) Hsnes whsa m UacM
be, z traces BC. At c there is a right-angle turn to the left, and ss ths valoe el
the integral over the infinite quadrant cc' is | v, the point s will move back throafh
the distance \ ir. That the point C thus reached must lie on the pore
axis is seen by noting that the integral Uken directly aloaf ec^ vrould be pus I
nary. This shows that a=\w witliout any necessity of eoai|Hitlaf Ihs faMafial
over the interval oa. The rest of the map of I may be filled lo at ones byiqrBBstfy.
To map the rest of the to-eurface is now relatively simple. For Tlet « ttaes
cc"d' ; then z will start at C and trace CIT = w. Whso « ooasss to aloag Ubs lower
side of the cut dV in the upper sheet I', the valoe of tbs teimiiad to
the value when this line de regarded as belonging to ths op^ ball
scribed, for the line is not a junction line of the snrfscs. Tbs iraes of t to thon
fore D'E\ When to traces /o' it must be remembered that I* )olns oa lo II aad
hence that the values of the integrand are ths nsgatlvs of tboss ahMg /b. Tbto
500 THEORY OF FUNCTIONS
makes z describe the segment F'Cf = — a = — ^ tt. The turn at E'F' checks with
the straight angle at the branch point — 1. It is fuither noteworthy that when w
returns to o' on I', z does not return to 0 but takes the value ir. This is no contra-
diction ; the one-to-one correspondence which is being established by the integral
is between points on the lo-surf ace and points in a certain region of the z-plane, and
as there are two points on the surface to each value of w;, there will be two points
z to each w. Thus far the sheet I has been mapped on the z-plane. To map II let
the point w start at o' and drop into the lower sheet and then trace in this sheet
the path which lies directly under the path it has traced in I. The integrand now
takes on values which are the negatives of those it had previously, and the image
on the 2-plane is readily sketched in. The figure is self-explanatory. Thus the
complete surface is mapped on a strip of width 2 ir.
To treat the different values which z may have for the same value of ly, and in
particular to determine the periods of ly as the inverse function of z, it is necessary
to study the value of the integral along different sorts of paths on the surface.
Paths on the surface may be divided into two classes, closed paths and those not
closed. A closed path is one which returns to the same point on the surface from
which it started ; it is not sufficient that it return to the same value of w. Of paths
which are not closed on the surface, those which close in lo, that is, which return
to a point superimposed upon the starting point but in a different sheet, are the
most important. These paths, on the particular surface here studied, may be fur-
ther classified. A path which closes on the surface may either include neither
branch point, or may include both branch points or may wind twice around one
of the points. A path which closes in w but not on the surface may wind once
about one of the branch points. Each of these types will be discussed.
If a closed path contains neither branch point, there is no danger of confusing
the two values of the function, the projection of the path on the lo-plane gives a
region over which the integrand may be considered as single valued and analytic,
and hence the value of the circuit integral is 0. If the path surrounds both branch
points, there is again no danger of confusing the values of the function, but the
projection of the path on the lo-plane gives a region at two points of which, namely,
the branch points, the integrand ceases to be analytic. The inference is that the
value of the integral may not be zero and in fact will not be zero unless the in-
tegral around a circuit shrunk close up to the branch points or expanded out to
infinity is zero. The integral around cc'dc"c is here equal to 2 7r; the value of the
integral around any path which incloses both branch
points once and only once is therefore 2 tt or — 2 ir ac-
cording as the path lies in the upper or lower sheet ; if
the path surrounded the points k times, the value of
the integral would be Ihn. It thus appears that w re-
garded as a function of z has a period 2 tt. If a path
closes in lo but not on the surface, let the point where it
crosses the junction line be held fast (figure) while the path is shrunk down to
wbaa'b'w. The value of the integral will not change during this shrinking of the
path, for the new and old paths may together be regarded as closed and of the
first case considered. -Along the paths wba and a'b'w the integrand has opposite
signs, but so has dw ; around the small circuit the value of the integral is infini-
tesimal. Hence the value of the integral around the path which closes in w is 21
or — 2 1 if I is the value from the point a where the path crosses the junction line
COMPLEX VARIAHLB
Ml
to the point u>. The aame concliudon would follow If Um pMh wmm
Ktirink down around the other bnwcb poinL Thus far Ite firaJWItki for t gnn
Kpundiiig to any given ioares + SHruidSMw-i« BsppoM iMil j UmI ft nifc
turns twice around one of the branch polnu and cilnwiOB iIm iwfact. Bf i
ing the path, a new equivalent path la fornod along wUeh CIm
term for term except for the small doobla eirmilt anmnd ± 1
value of the integral ii infinltealmal. Henea Um valuoa 1 4- t*v and taw— a
the only values z can have for any given value of w If • be a puftkrkr
value. TluH makes two and only two values of t la Moh tUip for mdk value ot
and the function is of the second order.
It thus appears that lo, as a function of i, baa tht pMlod tv. Is sli^
becomes infinite at both ends of the strip, baa no di^iaiaHtki wIlMu Um sU% tad
has two simple zeros at z = 0 and z = v. Hence w la a raUooal fnurtjou ol ^ wllk
the numerator e*^— 1 and the denominator c*^ -f 1. In fact
= C - = - r »iiaj
e« + e-<» ir^ + r-««
The function, as in the previous cases, has been wholly deiamlaad by Iba
methods of the theory of functions without even eonpuUuf a.
One more function will be studied in brief. Lei
— ^^^. «>o,
» (a — w) \w
t=/{w), »*♦(«) =/-»(!).
Here the Kiemann surface has a branch point at w = 0 and in addlUou Umm te i
singular point tc = u of the integrand which must be cut oui of both riMala. 1
the surface be drawn with a junction line from ie = Oloifs — « aud wiUi a <
in each sheet from w = a to to = co. The
map on the z-plane now becomes as indi-
cated in the figure. The different values
of z for the same value of to are readily
seen to arise when w turns about the
point 10 = a in either sheet or when a
path closes in to but not on the surface.
These values of z are z + 2kiri/Va and
2mm/Va. — z. Hence to as a function of
z has the period 2 iria~ i , has a aero at
z = 0 and a pole at z = xt'/ Vo, and approachea Um finite Talne w * « a» boUi eads
of the strip. It must be noted, however, thai the lero and pole are bocb ueca^
sarily double, for to any ordinary value of w oorreepood two valuee ol a la
strip. The function is therefore again of the
w-mmftm
= a^''^"^^' = aUnh«iaV^ ««-ltanb-«%p
(e-^- + l)
'/(•)
The succeas of this method of determining the fnnciioo i
integral, or the inverse w =/->(«) = ♦(«). *»<»• •»"
with which the integral may be used to map the
z-plane, and second upon the simplicity of the map, wbldi wae aaeb aa to
cate that the inverse function was a single valued pertedk fttaeck>« li
byau
502 THEORY OF FUNCTIONS
realized that if an attempt were made to apply the methods to integrands which
appear equally simple, say to
2 = r Va* — w^dw, z= i (a — w) dw/y/w,
the method would lead only with great difficulty, if at all, to the relation between
z and w ; for the functional relation between z and lo is indeed not simple. There
is, however, one class of integrals of great importance, namely,
.= r ^ ""'
•^ V(u> - a^)(w - ag) • • • (u> - a„)
for which this treatment is suggestive and useful.
EXERCISES
1. Discuss by the method of the theory of functions these integrals and inverses :
, , r^" dw ,^, r^ 2dw , , r"' dw
(,)r-^^, (oT-^. (0/"—^^=.
»'«> ,« -v/jrt2 _ /j2 »'0 -v/9! «i« _ 9/»2 »/l
'" w Vm>2 _ ^2 «^o V2 aw) - 11)2 «>'i (m, 4- 1) Vw^ _ i
The results may be checked in each case by actual integration.
_ p^ dw /* '" dw
2. Discuss / and / (§ 182, and Ex. 10, p. 489).
^'^ Viy(l-M))(H-ty) *^o VI-m;*
CHAPTER \1X
ELLIPTIC FUNCTIONS AND INTEGRALS
187. Legendre's integral I and its inversion.
dw
CoMidar
-r
V(l-tr«)(l-jrtr«)
0<k<l.
(0
The Riemann surface for the integrand* has braneh poiols at w « j^ 1
and ± 1/k and is of two sheets. Junction lines may be dfmwti between
+ 1, -f 1/A: and - 1, - 1/k. For very large values of v, the nMlied
V(l — iv^ (1 — khv^ is approximately ± ku^ and henoe them is bo
danger of confusing the values of the function. Across the Jnaelion
lines the surface may be connected as indicated, so that in Um nelgk>
borhood of w = ±1 and ir = ± 1/k it looks like the sorfsoe for vS.
Let -h 1 b» \hi' \'\\\\w (»f th.' iiit-egrand at tr = 0 in the upper
Further let
1
A'
X
dw
•'-r
dm
(I)
V(l - wr«) (1 - jfeV^ J^ V(l-.ie^(l-ikV)
Let the changes of the integral be followed so as to map the snrCaee
on the ;s-plane. As tv moves from o to a, the integral (I)
by A', and z moves
S_D
OB
1
^ 9^
2
from Oto A. As w
continues straight
on,« makes a right-
angle turn and in-
creases by pure
imaginary incre-
ments to the total
amount iK' when
w reaches b. As w
continues there is
another right-angle turn in «, the integrand again
z moves down to C. (That z reaches C follows from the
• The reader unfamiliar with Riemaan mirfieM (| IS4) msj ]
(I) aud (2) by Ex. 9, p. 47r> and may take (1) and otter i
508
#— piflme
w-mafoet
real, and
thntthn
504 THEORY OF FUNCTIONS
integral along an infinite quadrant is infinitesimal and that the direct
integral from 0 to ioo would be pure imaginary like dw.) If w is allowed
to continue, it is clear that the map of I will be a rectangle 2 Khy K^
on the «-plane. The image of all four half planes of the surface is as
indicated. The conclusion is reasonably apparent that w as the inverse
function of z is doubly periodic with periods 4 K and 2 iK\
The periodicity may be examined more carefully by considering different possi-
bilities for paths upon the surface. A path surrounding the pairs of branch points
1 and Jfc-i or — 1 and — A;-i will close on the surface, but as the integrand has oppo-
site signs on opposite sides of the junction lines, the value of the integral is 2iK\
A path surrounding — 1, -}- 1 will also close ; the small circuit integrals about — 1
or -I- 1 vanish and the integral along the whole path, in view of the opposite values
of the integrand along /a in I and II, is twice the integral from/ to a or is AK.
Any path which closes on the surface may be resolved into certain multiples of
these paths. In addition to paths which close on the surface, paths which close in
10 may be considered. Such paths may be resolved into those already mentioned
and paths running directly between 0 and w in the two sheets. All possible values
of z for any w are therefore 4 mK -|- 2 niK^ ± z. The function w (z) has the periods
4 K and 2 iK\ is an odd function of z as lo (— 2) = lo (2), and is of the second order.
The details of the discussion of various paths is left to the reader.
Let w =f(z). The function /(;*;) vanishes, as may be seen by the
map, at the two points z = 0, 2 K ot the rectangle of periods, and at
no other points. These zeros of w are simple, as f(z) does not vanish.
The function is therefore of the second order. There are poles at
z = iK\ 2K -ir iK\ which must be simple poles. Finally f{K) = 1. The
position of the zeros and poles determines the function except for a con-
stant multiplier, and that will be fixed by f{K) = 1 ; the function is
wholly determined. The function f{z) may now be identified with sn z
of § 177 and in particular with the special case for which K and K' are
so related that the multiplier g = 1.
For the quotient of the theta functions has simple zeros at 0, 2 Kj
where the numerator vanishes, and simple poles at iK', 2 K -{- iK\ where
the denominator vanishes ; the quotient is 1 at « = A" ; and the deriva-
tive of sn « at « = 0 is // en 0 dn 0 = «7 = 1, whereas /'(O) = 1 is also 1.
The imposition of the condition g = 1 was seen to impose a relation
between K, K', k, k\ q by virtue of which only one of the five remained
independent. The definition of K and /C' as definite integrals also makes
them functions K{k) and K\k) of k. But
iK\k)^£
ELLIPTIC FUNCTIONS
dw
V(l-u^(l-*V)
if 1/; = (1 - A:V)* and A:* + *•■ - 1. Henoe it appear* UmH Kmmyh^
computed from k' as K* from ib. This is rery useful in
A*" is near 1 and jfc" near 0. Thus let
/■
K
and compare with (37) of p. 472. Now either k ot k* ^ graater tbaa 0.7,
and hence either q or y' may be obtained to five plaoee with only Otts
term in its expansion and with a relative error of only about 0.01 per
cent. Moreover either q or q* will be less than 1/20 and henee a
term 1 -h 2 y or 1 -f 2 y' gives A" or /C' to four plaoet.
188. As in the relation between the Riemann surface and the
the whole real axis of x corresponds periodically to the part of the i«al
axis of w between — 1 and -f 1, the function sn *, for real «, Is leaL
The graph of ^ = sn a; has roots at x = 2 m A', maxima or minima altrr-
nately at (2 m -)- 1) A', inflections inclined at the angle 45* al the rooCei
and in genei-al looks like y = sin (irr/2 A*). Kiamined mat
sn^A' = (l + Ar')~^ > 2~i = 8in)w; it is seen that the 0Wf« n x
ordinates numerically greater than sin (w3r/2 A'). Aa
en X = Vl — sn* x, dn x = VT— J^snTJ (5)
the curves y = en x, y = dn x, may readily be sketched in. ii may he
noted that as sn (x + A) t^ en x, the curves for sn x and en x cannot
be superposed as in the case of the trigonometric funetiooa.
The segment 0, t'A" of the pure imaginary axis for m eormpoods to
the whole upper half of the pure imaginary axis for w. Heoee n Ir
with X real is pure imaginary and — t sn ur is real and poeMve for
0 ^ X < A' and becomes infinite for x = A". Henoe — isn ir looks in
general like tan (7rx/2 A"). Hy (5) it is seen that the enrres for y • co u-.
y = dn IX look much like sec (wx/2 A*) and that en ix lies above dn ir.
These functions are real for pure imaginary values.
It was seen that when k and A:' interchanged, K and K* also \xAm-
changed. It is therefore natural to look for a relatioii beta ten liMalli^
tic functions sn («, *), en («, Ar), dn (», k) fomed with tha aodatat k
506 THEORY OF FUNCTIONS
and the functions sn («, k'), en (z, k'), dn (z, k') formed with the com-
plementary modulus k' It will be shown that
1 1
en (z, k') ^ '^ en (iz, k')
A^(i^ ,. dn(^,A:) _ dn (iz, k') ^
Consider sn (iz, k). This function is periodic with the periods 4 K and
2 i/C' if i;s; be the variable, and hence with periods 4 iK and 2 A"' if s; be
the variable. With z as variable it has zeros at 0, 2 iK, and poles at
K\ 2 lA" -f K'. These are precisely the positions of the zeros and poles
of the quotient H(z, q')/H^(z, q'), where the theta functions are con-
structed with q' instead of q. As this quotient and sn (iz, k) are of the
second order and have the same periods,
/• 7x r. H(z,q') ^ sn (z,k')
sn (iz, k)= C ) ^ ; = C. — t^-tr '
^i(^j ? ) ' «n (z, k')
The constant C^ may be determined as C^= i by comparing the deriva-
tives of the two sides at ;?; = 0. The other five relations may be proved
in the same way or by transformation.
The theta series converge with extreme rapidity if q is tolerably
small, but if q is somewhat larger, they converge rather poorly. The
relations just obtained allow the series with 5- to be replaced by series
with q' and one of these quantities is surely less than 1/20.
In fact if V = 7ric/2 K and v' = 7rx/2 K', then
/ h\ — Z^ 2 sin V — 2 ^''^ sin 3 V -f- 2 ^^ sin 5 V
sn (x, f^)-:^ l-2^cos2v + 2y*cos4v-2j«eos6v + -" ,g.
_ 1 sinh v' — 5''^sinh 3v' -f- ^'^sinh 5v' — •• •
Vik cosh v' + q*^ cosh 3 v' H- q'^ cosh 5v' -\
The second series has the disadvantage that the hyperbolic functions
increase rapidly, and hence if the convergence is to be as good as for
the first series, the value of q' must be considerably less than that of
q, that is, K' must be considerably less than K. This can readily be
arranged for work to four or five places. For
a _ A" / Snx S vx\
- BIT
q"* = e
-BIT— / *'* _*ff\
'* - - ^', cosh 5 v' = i [e^' + e «^7 , O^x^ K'
where owing to the periodicity of the functions it is never necessary
to take X > K'. The term in q'^ is therefore less than J q'^K If the term
ELLIPTIC FUNCTIONS ^
in y» is to be equally negligible with thai id /,
2/=Jy'l with log7log/«»«,
from which q' is determined as about 9' » .02 and f ac abool f m M;
the neglected term is about 0.0000000 and ia Ymn\y MMOgh to ffftm
8ix-])la(!e work except through the multiplication of enuiB. TIm val^
of k corresi)onding to this critical value of 7 is about k ■■ 0.8ft.
Another form of the integral under consideratioo is
Jf ♦ d$ r9 4^
8in«^ = y = snx, ^=ramar, 00s ^ s Vl — aii*« ■■ eac,
A<^ = Vl - A:y = VI - ^8in«4 - dn x, Jt'-l-P,
a; = 8n->(y, ^-) = cn->(Vl-y«, k) = dn-«(Vl - Ay, jt).
The angle ^ is called the amplitude of x ; the fonolaoM sn x, co jr,
dn X are the sine-amplitude^ cosine-amplitude^ dttltn nnjtjiimdt of x« The
half periods are then
Jo Vl-A^sin'^ \2
and are known as the complete elliptic integraf< -'
189. The elliptic functions and integrals «
that call for a numerical answer. Here Jt* is given and tl»«
integral A' or the value of the elliptic functions or of the el
gi-al /'X<^, k) are desired for some assigned argument Tht
K and F(<^, k) in terms of sin-*Jk are found in tables (B
pp. 117-119), and may be obtained therefrom. The table. ..^. .-
used by inversion to find the values of the function an x, en x, da jr
when X is given ; for sn x = sn F(^, k) = sin 4» •»<! if * — F is givan*
4^ may be found in the table, and then sn x = sin 4. It ii
easy to compute the desired values directly, owing to the
rapidity of the convergence of the series. Thus
>^ = e.(0), ^j?^ = e(0). 1^V^-|(«.(0) + •(•».
508 THEORY OF FUNCTIONS
The elliptic functions are computed from (6) or analogous series.
To compute the value of the elliptic integral F (<^, k), note that if
_ dno; _ 1 -f 2 ^ cos 2 V 4- 2 y^ cos 4 V H
^^^"" Vik''~l-2ycos2v + 2y*cos4v + - •' ^^"^
/I \ cot X — 1 ^ cos 2 V + o-^ cos 6 J/ H
tan I 7 TT — X = ■ ^ . ^ = 2q ^ , f. . -. — ; ;
\4 / cot X -H 1 ^ 1 + 2 y* cos 4 V H
and tana7r-X) = 2ycos2v or tanQ7r-X)= , ^^T^^ (10')
^* ^ ^ ^* ^ 1 + 2 «^* cos 4 V ^ ^
are two approximate equations from which cos 2 v ma^ be obtained ;
the first neglects q*^ and is generally sufficient, but the second neglects
only q^. If k^ is near 1, the proper approximations are
1 dn(a;, A:)^dn(ia;, A;')^l + 2y'cosh2v' + '--
V^cn(a:,A:) V^ 1 - 2.y'cosh 2 v' + • •- ^^
tana7r-X) = 2y'cosh2v' or tan (J tt - X) = ^^0^^|^^ (11')
Here q*^ cosh 8 v' < y '* is neglected in the second, but </'* cosh 4 v' < q*'^
in the first, which is not always sufficient for four-place work. Of course
if ^ with sn a; = sin <^ or if y = sn a; is given, dn x = Vl — k"^ sn^ x and
en a; = Vl — sn^ x are readily computed.
/•« dS
As an example take J . .= and find X, sn f ^, F(J tt, |). As fc^ = |
and Vfc' > 0.9, the first term of (37), p. 472, gives q accurately to five places.
Compute in the form : (Lg = logjo)
Lgfc'^ = 9.87606 Lg (l - VJk') = 8.84136 Lg 2 ir = 0.7982
Lg Vfc' = 9.96876 Lg (l + VF) = 0.28669 2 Lg (l + VP) = 0.6714
VF = 9.93060 Lg27 = 8.66667 Lg A' = 0.2268
1 - VP = 0.06940 2 g = 0.03696 X = 1 .686
1 + VP = 1.93060 q = 0.01797 Check with table.
sn^.B:^2^^^"^'^~^'^^"'^'^'" = 2'^^^
8 VJk l-2</cos|7r+... .^^ 1 + g *
2 V6\/7 iLg 6 = 0.38908 Lg sn | £" = 9.9460
8 ~ 1.01797 i Lg 9 = 9.66366 sn | ET = 0.8810.
-Lg 1.018 = 9.99226
^=Jt A0 = dnx = V\-\ sin* J tt = Vl - ^ sin J it VT+fsinJir.
ELLIPTIC FUNCTI0N8
|iin|r = 0.19184
l'i8ini«' = 0.80eM
1-f i 8in I «-= 1.10184
} Lg (1 - I 8iniir)=: 9.06888
|Lg(l + |Biniv) = 0.08809
- Lg V*' = 0.08184
Lg cot X = 0.08814
As a second example conaider a pendolum of ItagUi m
arc of 300°. Find the period, the time when tb* paadnliim la
{>o8ition after dropping for a third of the time requirad for Um «
Let z^ + 2/^ = 2 ay be the equation of the path and A s «(l 4 | Vl) tba
height. When y = h, the energy ia wholly potential and aqoala wi§k; aa
the general value of the potential energy. Tha kinetle ttMrgj la
XmWWW
ignMmijtm
|v-XBi«8r8r'
U'-tJM
Lf tmB«8.4SM0
-Lffl8i«T.14IT
Lgt«-S.iM67
Lc<«tJM8
Lgooa8»-i8J087S
««MM8
tr«4rir
CiMek vlik laMa.
180< = IT (48.80)
is the equation of motion by the principle of energy. Heoee
''o V2^V(A-y)(2ay-y«) >» •'• V(l -io^(l -l«i^* i' ««'
are the integrated results. The quarter period, from highaif tf»
K y/a/g\ the horizontal position is y = a, at which t la derired ; and tiM
for Vy/ai = | ii is the third thing required.
1— VT K' —8 Lee'
Jk2 = 0.93301, 2<^ = i — ^* jr = ~— logy^« ^ ^X,-
Lg ik« = 0.90988 Lg (l - Vi) = 8.88668 l« 8 ■ 0.8010
LgVfc = 9.99247 - Lg (l + Vi) = ». 70878 Lgif*-* - 0.8784
Vi = 0.98280 -Lga = 0.60807 >IcJr«tJ8B8
l_Vfc = 0.01720 I^T'rr 7.68728 - 8 Lg (l 4- Vl) - t-IOi*
1+ \/Jk= 1.98280 7' = 0.00484 LglTaMMa
Hence JT = 2.768 and the complete periodic time la 4 IT Va/J.
y = a, ic« = ?. cnio= Vl-o/A, dn if « Vl - IH/A.
J,^=J/i;;i = cotX, tan(lir-x)=8^oodi8/. J-^-^^f'
VA: cnw \3 \4 / ^ ^« «
Lg*« = 9.0e988 X = 48«»8«'18- 8/-I.8U
Lg4 = 0.60206 |w-X = l«88'4r' Lg8^-8J8M
-Lg8 = 9.68888 Lg tan » 8.48808 - Lg« f '-« - M888
Lg cot* X = 0.09488 LgSg' = 0.08886 Lgir-».«1t
Lg cot X = 0.08870 Lg coah 8 »' = 0 4077R ^ Vi 5 "
510 THEORY OF FUNCTIONS
Hence the time for y = aist = 0.3333 K Va/g = ^ whole time of ascent.
= h sn2 k-K k-- /si"^^ ^^/3 K' - Q'^ sinh Trg/JT V
^ ^"\a3 \g~ k \cosh 7rif/3 iC' + g'^ cosh irK/K')
=20* A-i-.-;-7%--.oy,,^ A>-i-,-i-,v
iLg7' = 9.21241 g'i = 0.1631 ^^^^ /5.9645\2
- ^ Lg7' = 0.78759 7'"^ = 6.1319 ~ \6.2993/ '
This gives y = 1.732 a, which is very near the top at ^ = 1.866 a. In fact starting
at 30° from the vertical the pendulum reaches 43° in a third and 90° in another
third of the total time of descent. As sn l_K is (1 + k')~^ it is easy to calculate
the position of the pendulum at half the total time of descent.
EXERCISES
1. Discuss these integrals by the method of mapping :
/» W duo 1y
(a) z= I , a > & > 0, 10 = 6 sn a^, k = -,
^0 V{a^-w^){b^-w^) a
, w = sn^{~z,ki z = 2sn-i(V^, A:),
0 Vi« (1 -w){l- k^w) \2 /
{y)z= r ^^ , ^^ = ^ILi^ = tn(^,A:), z = tn-^{w,k).
*^o V(l + Mj2) (1 4- r2uj2) cn(z, A:)
2. Establish these Maclaurin developments with the aid of § 177 :
(a) sn 2 = z - (1 + A:2) — + (1 + 14 A:2 + fc4) ^ ^
3 ! 5 !
(/3) cnz = l-|^ + (l+4A;2)|^-(l + 44A:2+16A:4)|^ + ...,
(7) dn 2 = 1 - A:2|^ + ^2(4 4. ^.2^^^ _ j^^^iq ^ uk'^ ■\- k^)~ + . . . .
3. Prove C' ^ ^^ =1 ^ ^ ^^ ^ > l, sin^^ = Psin^^.
•^0 Vl-i2 8ina0 ^ •^o Vl-f-2sin2^
4. Carry out the computations in these cases :
(a) f^ ^^ t/>find TT, sn?^, fZ-tt, — Y
^0 Vl-0.1sin2(9 3 ' \^8 ' VlO/
(/3) C . ^^ z^tofindJT, snijT, f/Itt, -4=Y
^^ Vl-0.9sin2(9 3 ' 1^3 ' VlO/
5. A pendulum oscillates through an angle of (or) 180°, (/3) 90°, (7) 340°. Find
the periodic time, the position at i = | IT, and the time at which the pendulum
makes an angle of 30° with the vertical.
ELLIPTIC FUNXT10N8 5||
6. With the aid of Ex.8 and the aw ol the lemknto r* » i««r»t^ Aii«
the arc from ^ = 0to^ = 80^, andtbe middla poiM ol Ike »r
7. A bead moves around a vertical eiide. The velodty at the lo^ b to l^
velocity at the bottom ait 1 : n. Kxprtee the aolutioa in lenM of HHftk
8. In Ex. 7 compute Uie periodic time if a s S, S, or 10.
9. Ne^MectinK gravity, solve the problem of the it»pli« lopa. Tlike Om
Itori/oiital througii the ends of the rope, and the y^uie vwtleal !hrnMh ea
Kenu'uiber that "centrifugal force** variea as the dIaUaee f fomlbe ask el
The first iiiid Kecond integrations give
-^-^. ..^<^.(>^.^.
V(6«-y«)«-a«
d0
10. Express f , a > 1, in teme of elUpUc
•^ va — coe^
11. A ladder stands on a smooth floor and reeU at an aafle of MP
Bmooth wall. Discuss the descent of the ladder after its releaee f itND llUi |
Find the time which elapses before the ladder leaves the wall.
12. A rod is placed in a smooth hemispherical bowl and roaihw froai the ha^
toin of the bowl to the edge. Find the time of oeciUation when Um rod ii i
190. Legendre's Integrals II and III. The treatment of
Jo Vr~;? J« V(l-ir^(l-irte^ ^ '
by the iiiethoil of cgnforinal mapping to determine the fuMtioo and iti
inverse does not give satisfactory results, for the map of the Rigwana
surface on the ;s-plane is not a simple region. Hut tlie integral amj be
treated by a change of variable and be reduced to the integifml of aa
elliptic function.- For with t/; = sn u, h s gn~' w,
1 V(1-«>«)(1-*V) J» (U)
-"£""
The problem thus becomes that of integrating sn* u. To effect tbe fai-
tegration, sn^ ?/ will be expressed as a derivative.
The function sn^ m is doubly periodic with periods 2K, 2iK*, and
with a pole of the second order at u = iK'. But now
0(m 4- 2 A') = e(«), ©(w 4- 2iA")= - ?-•«'*'•(")
log(=)(„4.2A-) = loge(»0, logre4-2iA-)-loge(M)-^i«-ki«(-f>
512 THEORY OF FUNCTIONS
It then appears that the second derivative of log 0(t/) also has the
periods 2 K, 2 iK\ Introduce the zeta function
Z(«) = flog®(«) = ^, ZXu) = f^- (13)
^ ^ du ^ ^ ^ ®(u) ^ ^ du 0(w) ^ ^
The expansion of ®\u) shows that ®\u) = 0 at w = mK. About u = iK*
the expansions of 7t\u) and sn* u are
z'00 = -(^rr^^ + "o + -, ^^'- = F^(^7^^ + ^o + .--.
Hence k^ sn^ u = - Z'(u) + Z'(0), Z'(0) = 0"(O)/0(O),
and k^ f srv^udu = - Z(u) -^ uZ'(0),
(1 - P sn'^ w) du = u(l- Z'(0)) + Z (?e). (14)
X
The derivation of the expansions of Z'(m) and sn^ u about m = iB"' are easy.
G(u) = CTr(l - q^^'+^e^^"), loge(M) = 2) log(l - ^an+ig^Z") + log c
log e (m) = log \l — qe ^ "j + function analytic near u = iK\
G'(u) _ iTnyd ^ _ iirq
BTO + . . . = 9 + (I
d e'(u) _ - 1
/(u) = e>^" =/(iiiro + (u - iK')r{iK^ + . . . = ^ + (u _ t^')!^g +
G'(u)
e(u)
u
+ 1
+ •
' ' »
sn
(u +
1^0 =
1
~ k
1
snu
du e(u) {u-iK')^
stiHu + iiTO = — — ,
^ ' A:2sn2u
/(u) = SUM = u/'(0) + \u^r\Q) + . . . = M + cm8 + . . .,
In a similar manner may be treated the integral
Jo K-a) V(1-w;^(1-Fm7'^ Jo sn^^-a ^ ^
Let a be so chosen that sn** a = a. The integral becomes
Jr" du 1 r 2 sn g en « dn g
g sn" w — sn^ a 2 sn a en a dn a j sn'* w — sn'* a ' ^ ^
ELLIPTIC FUNCTIONS 511
The integrand is a function with periodi 2K,2UC'%ad wHk
I)oles Sit u = ± a. To find the retidoM at
li,„ ."^». -lin, 1 ^ t\
«A*a sn'u — sn'a •«*« 2 8nM onudnii 2ta«ea«dBc
The coeiiicient of (u 7 a)-* in expanding aboot ± « it thwifan ± L
Such a function may be written down. In Cm!
2 sn g en g dn g H'(u — g) __ //*(> 4- g)
8n«M-gn«g "* //(u - g) "" //(ii + a) "*" ^
= Z,(y-g)-2,(,i + g)+C,
if Zj = 7/7/f. The verification is as above. To deterraioe C let « « 6l
Then c = -^^Ii^^ + 2Z,(«), but .n«— i.^.
J «^i cnttdni*.,. ^..
and -r- log sn tt = « Z,(w) — Z(ii).
du ° snti *^ ' ^ '
Hence C reduces to 2 Z (a) and the integral is
X sn««-sn»a = 2»nacLdna ['°8 fjfr^ + 2 -«(•)]• (»«)
The integrals here treated by the substitution lo = nil mod Umm
integrals of elliptic functions are but special caMS of the Integrmtioo of way
function R(w, V VT) of w and the radical of the biquadratio IT s (I - i^l -
The use of the substitution is analogous to the nae of w s iln ii la eoawtlaf mi
integral of R(w, Vl - m^ into an integral of trigonometrie fnneCloML Aaf la*
tional function R(w, Vw) may be written, by nUionaltaatioa, as
B U VW) = ^(^)-^^(^)^^ - B{w)-¥R(m)y/W
where R means not always the same function. The lnt«fial d Bim^vW) h
thus reduced to the Integral of /?,(») which Is a raliooal fiaeCSott, pli» tto lai»>
gral of wIi^iv}»)/VW which by the flobttltntioii m^ « a rtdaets lo aa
R (w, V(l - i/)(l - **tt) and may be oonaldered aa belongiag lo tkmmtM
plus finally
By the method of partial fractions A, may be raeolvad aad
are the types of Integrals which most be oralaatMl U> flnlih tlie Ity^ea «* !*•
given R (tr, V^. An IntegraUon by parU (B. O. Pelnw, »©. I*!) ifc*^ t*^ •«
514 THEORY OF FUNCTIONS
the first type n may be lowered if positive and raised if negative until the integral
is expressed in terms of the integrals of sn^x and sn'^x = 1, of which the first is
integrated above. The second type for any value of n may be obtained from the
integral f or n = 1 given above by differentiating with respect to a under_the sign
of integration. Hence the whole problem of the integration of R{w, y/w) may
be regarded as solved.
191. With the substitution w = sin <f>, the integral II becomes
E (if,, k)= f y/l-k^sin' Ode = C 2/lszJ^ dw (^^x
= w (1 - Z'(0)) H- Z {u), u = F{4>, k).
In particular E (J tt, k) is called the complete integral of the second kind
and is generally denoted by E. When <^ = ^ tt, the integral u = F(if,, k)
becomes the complete integral K. Then
E = K(1- Z'(0)) -{-Z(K) = K(1- Z'(0)), (18)
and E (<f>, k) = EF{4>, k)/K + Z {u). (19)
The problem of computing E (<f>, k) thus reduces to that of computing
Kj Ej F(<f>, k) — u, and Z(w). The methods of obtaining K and F(<^, k)
have been given. The series for Z{u) converges rapidly. The value
of E may be found by computing K(l — Z'(0)).
For the convenience of logarithmic computation note that
K -^W- 0(O)"N2i^A:' K^^^ 4^+y^ )
or ^ - £: = J tt/ -V^ • (2 ir/K)"^ ^ (1 _ 4 ^» + • • •). (20)
Ai 'wf \ - ®' W — ^ g"^ sin 2 y - 2 9^^ sin 4 y H
Also L(u)-^^^^- ^ l-2ycos2v-f2y*cos4y-... ^""^^
where v — 'iru/2 K. These series neglect only terms in q^, which will
barely affect the fifth place when k ^ sin 82° or k"^ ^ 0.98. The series
as written therefore cover most of the cases arising in practice. For in-
stance in the problem which gives the name to the elliptic functions
and integrals, the problem of finding the arc of the ellipse « = a sin <^,
y z=b cos <^,
ds = Va* cos* ^ + b^ sin* <\>dff> = a Vl — e* sin* <f>d<f> ;
the eccentricity e may be as high as 0.99 without invalidating the
approximate formulas. An example follows.
Let it be required to determine the length of the quadrant of an ellipse of
eccentricity e = 0.9 and also the length of the portion over half the semiaxis
major. Here the series in q' converge better than those in 9, but as the proper
ELLIPTIC FUNCTIONS SU
expreaalon to replace Z(m) has noC been found, U will be _
the series in q and uke an addiUonal term or two. As 4 ■ 0.0, ^ • (1.10.
LgA:^ = 0.27876 Lg(l - VP) « 9.«8I» «4lf.«Mim
LgVP = 9.81069 I«(l + V^)=0.»017 L<1««1JN11
\^ = 0.660S8 dlff. =9J110t Lci»r»t«MiM0
1-.VP = 0.88978 I«S=r0.a010S tenBl*0.
1 + VF= 1.66022 Lg term 1=9.01000 lemt-
Lg«7 = 9.0101 Lg2» = 0.79M L« | r/v^ . f JIM
8Lgg = 7.0303 -2L«(l+ VP)=9.6»7 |lofSv/X«0j
4Lgv = 6.0404 Lg(l + 2flr«)= 0.0001 !««*•.<
</* = 0 0011 lgjr = 0.8680 L«(1-4«^«9.«M1
(?* = 0.0001 jr = 2.280 i«(jr-jr)»o.<Mi0.
Hence K — E = 1.109 and E •= 1.171. Tbe qoadrant la 1.171c. The y^frrf c
responding to z = | a is given by <ft = 80®. Then dn F = x/i — ff fftfiT
LgdnF=9.9509 i»-X = 8*»8ir eoa2»«O.7a0
LgVP = 9.8107 L«tan = 9.1768 Uenee 4»naer9r
LgcotX = 0.1312 L«29 = 0.8111 1 + 2f*eoe4rB 1JI«
X = 36°28i' Lgoos2r = 9.8647 2»ai«rir.
Now 180 F = A' (42.92). The compuUtion for F^ Z, E{\ r) la
Lg A' = 0.3680 Lg 2 »/Jr = 0.4402 L« JT/JT s O.7100
Lg 42.92 = 1.6320 Lg9=: 9.0101 LfF« 0.710
-Lg 180 = 7.7447 Lg aln 2 r = 9.8881 EF/K m^JKtm
LgF=9.7353 - Lg(l- 29 0oe 2 r) = 0.0706 %m%Mm^
F=0.5436 LgZ = 0.S680 Jr(|«) « •.IMi.
The value of Z marked • is corrected for the term ~ 29*aln 4 r. The peit el the
quadrant over the flrst half of the axia is therefore 0.6048 a and 0.688 e ofer the
second half. To insure complete four-figure accuracy In the raeolt, ive
should have been carried in the work, but the Taluee
table except for one or two units in the last
If
I
EXERCISES
1. Prove the following relttioiM (or Z(u) and Z,(ii).
Z(-u) = -Z(u), Z(u-|-2K)=Z(v), Z(ii-f tUr)>l(li)-to/r.
z,(«, = Aiogfl(.) = fg. «.(« + ur'>=«(.)-^.
-i_ = -z;(») + z(0). /*..-«.(.)+rt-«.
z,(«)-z(.) = lfc,»« = 5!^. z,TO— .
616 THEORY OF FUNCTIONS
2. An elliptic function with periods 2 IT, 2 iK' and simple poles at a^, ag, • • • , a„
with residues c^, c^, • • • , f„, 2c = 0, may be written
f{u) = CiZi(u - a,) + C2Zi(u - ttj) + • • • + c„Z^(u - a«) + const.
« Jk* an a en a dn a sn^ u 1 _ . . 1„, , » , „,. v
^- — I ZT-i 2 = z^{u-a)--Z{u-\-a)-\- Z'(a),
1 — ik* sn^ a sn* u 2 2
r" sn*udn 1, Qla — u) . „,, ^
l^snacnainal ^_^,_,_^ = - log ^1-^ + „Z (a).
4. („) f _^^ = x„z'(0) - V-xz(V-xu) - Vx ""^^"dnVxu ^ p
•^ sn^ VXu sn VXu
/Tt;./ • 1 /7 \ /rcnVXudnVxu , ^
= Xu — V\E{<p = sin-isn VXu) — VX := + C,
sn Vxu
/«v rk'Hu r , « , -oSnucnu „,^ . ■ . -„snMcnu
(iS) / = I dn^Mdu — A:2 — ; = J? (0 = sin-i sn u) — A:* ,
^^' J dn^u J dnu ^^ ' dnu
r ^"'^<^^ = u - 2 JE:(0 = sin-i sn u) + ^""^ (1-2 dn^ u).
^'' J sn2udn2u ^ ' snudnu^ '
5. Find the length of the quadrant and of the portion of it cut off by the latus
rectum in ellipses of eccentricity e = 0.1, 0.5, 0.75, 0.95.
6. If e is the eccentricity of the hyperbola x^/a^ — y^/b^ = 1, show that
6* /»« sec2 0d0 , ae ^ ,1
8 = — I — — , where — 2/ = tan 0, A; = - ,
aeJo Vl-fc2sin2 0 6^ e
62
= — F(0, A:) — aeE (0, A;) + ae tan 0 V 1 — A;2 sin2 0.
ae
7. Find the arc of the hyperbola cut off by the latus rectum if e = 1.2, 2, 3.
8. Show that the length of the jumping rope (Ex. 9, p. 611) is
a(k'K/\^2-\- V2E/k').
9. A flexible trough is filled with water. Find the expression of the shape of
a cross section of the trough in terms of F(0, A:) and E{<p, k).
10. If an ellipsoid has the axes a>b>c, find the area of one octant.
11. Compute the area of the ellipsoid with axes 8, 2, 1.
12. A hole of radius 6 is bored through a cylinder of radius a>b centrally and
perpendicularly to the axis. Find the volume cut out.
18. Find the area of a right elliptic cone, and compute the area if the altitude
is 8 and the semiaxes of the base are 1| and 1.
ELLIPTIC FUNCTIONS ^17
192. Weierstrasa's integral and iu ioTtnta. In ^
Keiuiul theory of doubly periodic funoiioot (f 182), Um two
limctions /7(u), p\u) were construnied and dbcuMed. li
dw
■^^==_=. «-,(.). ..,(0).
dw
-r
where the fixed limit oo has been added to the intagnd to
and « = 0 correspond and where the roots have been called « t iU v
Conversely this integral eould be studied in detail by the anthod oC
mapping ; but the method to be followed is to make only
of the con formal map sufficient to give a hint as to how the
li{z) may be expressed in terms of the functions sns and ens. The
discussion will be restricted to the
T
V^^^
case which arises in practice, namely, m J^
when g^2iXi(i g^ are real quantities. f
There are two cases to consider, one
when all three roots are real, the other when one is real and tiM
two are conjugate imaginary. The root «, will be taken as tha
real root, and e^ as the smallest root if all thre** ara rKal. Note tliai the
sum of the three is zero.
In the case of three real roots the Kiemauu surfaoe majr be dimwn
with junction lines e^, «,, and e^ x. The details of the map may readily
be filled in, but the observation is sufficient that theie are only two
essentially different paths closed on the sorfaoe, namely, aboot %^ i^
(which by deformation is equivalent to one aboat a,, oe) and abool %^ «,
(which is equivalent to one about 0,, — oo). The integral about •^•^*^
real and will be denoted by 2 <u,, that about ^ «| it pure imafinaij aitd
will \y^ denoted by 2 (a,. If the function /»(«) be ooostnifltwl aa in f 182
with (i> = 2 <ttj, w' = 2 M, the function will have as always a dooble pole
at « = 0. As the periods are real and pure imaginary, it is nalaiml to
try to express p {z) in terms of sn «. As /»(«) depends on two eoMtaati
9 It 9ti whereas sn % depends on only the one *, the fonotion pi*) will
be expressed in terms of sn ( V3u, k\ where the two comtants A, k are
to be determined so as to fulfill the identity p^^Aj^^ f^ — f^ In
particular try
518
THEORY OF FUNCTIONS
This form surely gives a double pole- at ;s = 0 with the expansion Xj^
The determination is relegated to the small text. The result is
i>(s) = e,+
1^ =
<i,
ui„V\ = iK\
(23)
X = e^ — ^2 > 0,
In the case of one real and two conjugate imaginary roots, the
Hiemann surface may be drawn in a similar manner. There are again
two independent closed paths, one about e.^, e^ and another about Cg, e^
Let the integrals about these paths be respectively 2 co^ and 2 m^. That
2 coj is real may be seen by deforming the path until it consists of a
very distant portion along which the integral is infinitesimal and a path
in and out along 6^,00, which gives a real value to the integral. As
2 0)^ is not known to be pure imaginary and may indeed be shown to be
complex, it is natural to try to express p (z) in terms of en z of which
one period is real and the other complex. Try
/ \ A , l + cn(2 VJi^, k)
^^^ '^l-cn(2V;.^,A:)
This form surely gives a double pole at « = 0 with the expansion l/z\
The determination is relegated to the small text. The result is
p(z) = e^ + fi
l + cn(2 V^, k)
l-cn(2 V/i«, k)'
-=i-:-5<'.
To verify these determinations, substitute in p'^ = ip^ — g^p — g^.
p{z) = A-\-
sn«(Vxz, A;)
p'{z) =-
sn8(Vx2, k)
^u^^=},(K + {K').
cn(Vxz, Jk)dn(\/Xz, A:)
(23')
^^,(l-sn^)(l^fc^sn^)^ /^3^
8n« \
sn«/ sn2
sn^ sn*
Equate coefficients of corresponding powers of sn^. Hence the equations
4 4* - g.^A - {/g = 0, i\^k* = 12^2 - g,X, - X(l + A;^) = 3^.
ELLIPTIC FUNCTIONS 51t
The first Bhowg that ^ !■ a root e. Lei il « i,. *<*•-*■%%♦ %%*%V
by virtue of the relation e, •»• i^ -f <b « 0. The eolutioci U imaedlale ae ghm
To verify the second determlnaUon, the MifaeUUiUoo b dmllar.
p(«) = ^+^
4-cn»Vit
|r'<f)»-
4|J«4b
where <= <1 + ciO/(l - en). The IdenUty p'* = ij^ - g^p- 9^^ ihmwttof
4m« [^• + 2(1- 2A:«) <• + «] = 4(^« + 8^V + S^^ + ,iV)-i^-fW -f..
HciL' If I A =e^. The solution then appears at onoe from the form
m'^ = 3 e^* + c,c, + e,c, + e,c, = (e, - e,)(e, - e^. |i(l - SK) » 1^1/1.
The expression of the function p in terms of the fnnetiocii ftlMAdj
studied permits the determination of the value of the funetioii, mad hr
inversion i)ermits the solution of the equation />(«) s e. The ftoetki
p(z) may readily be expressed directly in terms of the tbeli astiM.
In fact the ])eriodic ])ro))erties of the function and the oonmpODdimg
properties of the quotients of theta series allow such a
2<-)
V 2
^et<P<e,'^
J^ A. ••
£^
"■ e,>p>et
9
^''^^<P<
^ . /ep>p>g|j €t<p<m
0 -axp'<0 o<p'«o 2(0, 0-*oo<p(ro 0<pkjai t^
to be made from the work of 1 175, provided the series be alkyvvd eoaa-
plex values for q. But for practical purposes it is desirable to have the
expression in terms of real quantities only, and this is the reaaoo for a
different expression in the two different cases here treated.*
The values of z for which />(«) is real may be read off from (SS) and
(23') or from the correspondence between the tc^sorfiiee and the <
They are indicated on the figures. The functions ji and /' may bt 1
to express parametrically the curve
y« = 4x«-ir^-i7, by y =-/>'(«). »-/(«)•
♦ It is. howrver, poxsihif, if dt»«irpd, to trasafone the fivee emWc 4 •■ - f||V - §t
two roinplex nH)ts into ;i Hiiniinr ruhlr with all thrse raoCs rMl awl UMMa««M Ihs
cate forms. The trauHformutioii is not ici%en hers.
620 THEORY OF FUNCTIONS
The figures indicate in the two cases the shape of the curves and the
range of values of the pai-ameter. As the function p is of the second
order, the equation p{z) = c has just two roots in the parallelogram,
and as p (z) is an even function, they will be of the form z = a and
«=2o>j4-2a)2 — a and be symmetri-
cally situated with respect to the cen-
ter of the figure except in case a lies z=<^i^u
on the sides of the parallelogram so ^^J^^|yifeLL_ _J}^^j±l
that 2 o>j -h 2 <o.^ - a would lie on one «=«i+Wj' -
of the excluded sides. The value of
the odd function p^ at these two points
is equal and opposite. This corresponds precisely to the fact that to
one value x = c oi x there are two equal and opposite values of y on
the curve y^ — Aoi? — g^ — g^. Conversely to each point of the parallelo-
gram corresponds one point of the curve and to points symmetrically
situated with respect to the center correspond points of the curve sym-
metrically situated with respect to the ic-axis. Unless z is such as to
make both p («) and jy'(^) ^'^^^> ^^^ point on the curve will be imaginary.
193. The curve y^ = ix^ — g^x — g^ may be studied by means of the properties
of doubly periodic functions. For instance
Ax-\- By -\- C = Ap'iz) + ^p{z) +0 = 0
is the condition that the parameter z should be such that its representative point
shall lie on the line Ax ■\- By ■\- G — d. But the function Ap\z) + Bp {z) + C is
doubly periodic with a pole of the third order ; the function is therefore of the
third order and there are just three points Zj, z^^ Zg in the parallelogram for which
the function vanishes. These values of z correspond to the three intersections of
the line with the cubic curve. Now the roots of the doubly periodic function sat-
isfy the relation
Zi + Zg + Zg — 3 X 0 = 2 wiiWi + 2 m<ja>2.
It may be observed that neither in^ nor m^ can be as great as 3. If conversely Zj, Zj, Zg
are three values of z which satisfy the relation Zj + Zj + Zg = 2?niWj + 2m^(a^, the
three corresponding points of the cubic will lie on a line. For if Zg be the point in
which a line through Zj, Zj cuts the curve,
Zi 4- Za + Zg = 2 mjWj + 2 m'^w^, z^ — z^=2 (m^ — m\) Wj + 2 (m^ — nQ Wj,
and hence Zg, Zg are identical except for the addition of periods and must therefore
be the same point on the parallelogram.
One application of this condition is to find the tangents to the curve from any
point of the curve. Let z be the point from which and z' that to which the tangent
is drawn. The condition then is 2 + 2z' = 2m^<a^ + 2mjWj, and hence
are the four different possibilities for z' corresponding to m^ = m, = 0 ; m^ = 1,
m, = 0 ; tiij = 0, wij = 1 ; wij = 1, m, = 1. To give other values to m^ or m, would
ELLIPTIC FUNCTIOKB Ml
merely reprodaoe one of the four poiaU eseepi for the aadilloa nt ■itm pm^uitL
Hence there are four Ungenu to U»e curve from ahj potot of Um esfveu IW
(|ueKtiun of the reality of tbeee UngenU may readily be I ruled, ■>«y|int g dMsHM
a real point of the curve. If the point lice on the intelle poftloa, • < f < t ml, «ki
tliM first two pointi) x' will aleo eaUafy the eoodltloM 0 < !"< t», cieepc f«r llm
1 Mble audition of 'iwp Heooe there are alwaye two real tai^eamio ite ewve
iroin any point of the infinite branch. In eaee the room «,, «p i^ are all leal, ike
hiKt two pointji z' will correspond to real polou of the oval potlkm aai all lav
tangents are real ; in the case of two imaginary room iheae valvea el r gN» Im^
inary points of the curve and there are only two real tangMHe. II Iht Ihfee i«em
are real and z corresponds to a point of the oval, t ia of the form «^ <f ■ aatf all
four values of z* are complex,
and none of the tangents can be real. The iHsmeilon to eomplem.
As an inflection point is a point at which a line may cut a cunre la tluee
cident point8, the condition 8 r = 2m,t#, + S m^m, holds for the parameier f el <
points. The possible different combinations for t are ulmi :
z = 0 j«, |«,
I**! l-I+f-« 1-1 + l««
l-i f-i+l-i l-i + f-S-
Of these nine inflections only the three in the first oolonw are teal.
two inflections are given a thirtl can be found so that t, 4- fg 4* tg to a
period, and hence the inflections lie three by three on twelve Uoea.
If p and p' be substituted in Ax* + Bxy + Cy* •(- Ac -f iPy + ^. the rsenU u a
doubly periodic function of order 6 with a pole of the 6th order at the Ofigla.
The function then has 6 zeros in the parallelogram eonoeeted by the lelalioa
z, + r, + r, + «4 + tj + rg = >■!,•*, + tM|Mt«
and this is the condition which connects tlie parameters of the 6 poiats la
the cubic is cut by the conic ^z> + Itxy + Cy* -^ Dx -^ Ejf -^ F m t, Oae
tion of interest is to the discussion of the conies which amy be I
three points z,, z^, z,. Tlie condition then reduoee to i, •!- 1, 4- Sg a a»,W| 4- "VS*
If m^, m, are 0 or any even numbers, thto condltloe ezpreseee
three points lie on a line and is therefore of little IntereeL The
apart from the addition of complete periods, are
Zi + «s + «« = -P *i + «« + «t = -t» »i + «i + ^*»i + «%-
In any of the three cases two pointa may be choeea at raadom oa the
the tliird point is then fixed. Hence there are three coatoe whieh aia
the cubic at any two assigned poinU and at some other |
of interest is to the conies which have contact of the 6th older with the caMe.
The condition is then Oz = 2mj*#, 4- Sm,Mg. As m,, 04 amy have aaj el the •
values from 0 to 5, there are 86 poinU on the eubie at which a eoale m^f haw
conuct of the 5th onler. Among theee points, however, are the
obtained by giving m,, »», even values, and theee are ol Uttle li
conic reduces to the inflectional tangent taken twice. There reamla t7
which a conic may have contact of the 5th older with the
522 THEORY OF FUNCTIONS
EXERCISES
1. The function f («) is defined by the equation
-r(2)=P(2) or f(2) = - fp{z)dz = ---c^z^^
V Z O
. 4, p. 616, that the value
f(0) = -e^z+ V\E{<p, k)+V\-
z 3
Show by Ex. 4, p. 616, that the value of f in the two cases is
en Vx« dn s/Xz
V5^;
sn VXz
m = - (^ + e,)z + 2 V^E{<p, k) + V; ^^^z (2dn2 -^^ _ i),
sn Vfiz dn v /tz
where X = e^ — e^, k^ = (Cg — e2)/{e^ — e^), <p = sin-i sn V\2,
and n = V{e^ — e^){ei — 63), k^ = ^ — S e^/i /x, 4> = sin-i sn VJiz.
2. In case the three roots are real show that p (z) — e, is a square.
sn v^2 sn Vxz sn Vx
What happens in case there is only one real root ?
3. Letp(2 ; ^25 9s) denote the function p corresponding to the radical
■V4p»-g^p-g^.
Compute p (i ; 1, 0), p (i ; 0, i), p (| ; 13, 6). Solve p{z; 1,0) = 2, p {z ; 0, ^ = 3,
p{z; 13, 6) =10.
4. If 6 of the 9 points in which a cubic cuts y^ = 4:X^ — g^x — g^ are on a conic,
the other three are in a straight line.
5. If a conic has contact of the second order with the cubic at two points, the
points of contact lie on a line through one of the inflections.
6. How many of the points at which a conic may have contact of the 5th order
with the cubic are real ? Locate the points at least roughly.
7. If a conic cuts the cubic in four fixed and two variable points, the line join-
ing the latter two passes through a fixed point of the cubic.
8. Consider the space curve x = sn «, y = en «, z = dn t. Show that to each
point of the rectangle 4 E" by 4 iK' corresponds one point of the curve and con-
versely. Show that the curve is the intersection of the cylinders x^ ■\- y^ = \ and
kH^ -f- 2* == 1. Show that a plane cuts the curve in 4 points and determine the
relation between the parameters of the points.
9. How many osculating planes may be drawn to the curve of Ex. 8 from any
point on it? At how many points may a plane have contact of the 3d order with
the curve and where are the points ?
10. In case the roots are real show that f(2) has the form
"' VX
ELLIPTIC FUNCnOKB 513
Henco lo^ r (r) = f f{t) d« = 1 ?1 »• + log a(Vig) ^ Q
'^ I M|
11. By general methodB like thoM of 1 100 prove Umu
and f * L|«l<*±f> + ,!?«.
12. Let the functiona 0 be defined by
with 9 = e **! . Show that the tf-eeries conTerge If w, Ic real and «^ la pwv taa^
nary or complex with \\» imaginary part poaltlTe. Sbow wuam (tatrmlly llMift %h»
Reries converge if tlu* angle from t#| to m, is poaltlTe and \tm tlMui 100^.
"■- '<---:*?• '•«='''--:^-
Prove (r(2 + 2wj) = - c*V« + «»i)r(«) and similar relations for #^*).
U. Let 2n, = -^^i-^ . or ^,««t-Vi = T*
Prove <r (z + 2«j) = — «''•«(• + ••«)#(«) and similar relations for #«<f).
15. Show that ff(- z) = — c(z) and develop #<«) as
16. With the determination of f ^ as in Ex. 16 prove that
OS oz'
by sliowin^ that p(«) as here defined Is doubly periodic with periods fl«|. t«^
with a pole 1/r^ of the second order at s = 0 and with no eoHMUii tatai te Hi
development. State why this Identifies p(r) wiih the function of tba ten.
CHAPTER XX
FUNCTIONS OF REAL VARIABLES
194. Partial differential equations of physics. In the solution of
physical problems partial differential equations of higher order, partic-
ularly the second, frequently arise. With very few exceptions these
equations are linear, and if they are solved at all, are solved by assum-
ing the solution as a product of functions each of which contains only
one of the variables. The determination of such a solution offers only
a particular solution of the problem, but the combination of different
particular solutions often suffices to give a suitably general solution.
For instance
is Laplace's equation in rectangular and polar coordinates. For a solu-
tion in rectangular coordinates the assumption V= X(x) Y{ij) would be
made, and the assumption V = R (r) $(<^) for a solution in polar coor-
dinates. The equations would then become
Now each equation as written is a sum of functions of a single variable.
But a function of x cannot equal a function of ?/ and a function of ?•
cannot equal a function of <f> unless the functions are constant and have
the same value. Hence
or (2')
These are ordinary equations of the second order and may be solved
as such. The second case will be treated in detail.
The solution corresponding to any value of m is
* = a„, cos m<f> -h b„^ sin w<^. A* = A ,„/•"• -f I),^)'- "•
and F = 7?* = (A^r^ -f- B,^r-'^)(a^ cos m<f> -f *« sin ?«</>)
624
REAL VARIABLES 515
That any number of solutionB oorretpoDding to dUbraol thttt oT m
may l)e added together to give another lolutioii it doe to the limmrkm
of the given equation (f 96). It may be that a tingla tarn viU adka
as a solution of a given problem. Rut it may be aaeii bi fMmtl tlali
A solution for V may be found in the form of a Foorier aeriM vhkli
shall give V any assigned values on a unit ctrele and tithar ba wmrtt-
gent for all values within the circle or be ooprafgaut te aO vmlaaa
outside the circle. In fact let /(^) Iw the Yalnea of K on tha imli dfcla.
Expand /(<^) into its Fourier series
m
Then V=ia, + '^i^(a^coBm^ + t^Emm4) (T)
m
will be a solution of the equation which rednoea to/(4) on tba dida
and, as it is a power series in r, converges at every point within ttm
circle. In like manner a solution convergent outside the etivla ia
m
The infinite series for V have been called solutions of LaplsM*s <
matter of fact they have not been proved to be solutions. The
by taking any inunber of terms of the series would larely be s
limit of that Riim wlien the series becomes infinite is not thereby prmred to bt assla-
tion even if the series is convergent. For tlieoreticsl purposes it would be
to give the proof, but the matter will be puMd over here ss bsving s
bearing on the practical solution of many problems. For in prscUes lbs
f{<f>) on the circle could not be exactly Icnown and could tbsfsfoca
represented by a finite and in general not very large nunber of Unm ef tbs 4t^
velopment of /(0), and these terms would give only a flnifcs series for Iba
function V.
In some problems it is better to keep the partieolar iolotloiia
rate, discuss each possible particular solution, and then imafjna Ibaai
compounded physically. Thus in the motion of a dmmbcad, tha aMMi
general solution obtainable is not so instructive as tba partiealar i
corresponding to particular notes ; and in the motion of tha
the ocean it is preferable to discuss individual types of warei
})ound them according to the law of superposition of ataall
(p. 226). For example if
526 THEORY OF FUNCTIONS
be taken as the equation of motion of a rectangular drumhead,
-{
sin axj ^^ _ fsin ^x, ^ __ fsin c V«^J-_^
Lcos pxj Icos c -yJa^ -\- f^t
cos ax.
are particular solutions which may be combined in any way desired
As the edges of the drumhead are supposed to be fixed at all times,
« = 0 if » = 0, a; = a, y = 0, y = ^, t = anything,
where the dimensions of the head are a by h. Then the solution
„,,^ • 'rrnrx . niry \m^ , in}
« = Xrr=sm sm-T^cosc7r\^-^+7^^ (4)
is a possible type of vibration satisfying the given conditions at the
perimeter of the head for any integral values of m, n. The solution is
periodic in t and represents a particular note which may be omitted.
A sum of such expressions multiplied by any constants would also be
a solution and would represent a possible mode of motion, but would
not be periodic in t and would represent no note.
195. For three dimensions Laplace's equation becomes
in polar coordinates. Substitute V = R (r)®($)^(<f>) ; then
^ sill' 0 d<t>'
Here the first term involves r alone and no other term involves r
Hence the first term must be a constant, say, n(n-\- 1). Then
£
dr
Next consider the last term after multiplying through by sin^ 6. It ap
pears that ^-i$" is a constant, say, — m^. Hence
$" = — rn}^^ $ = a„ cos m<f> -f- ^„, sin m<^.
Moreover the equation for 0 now reduces to the simple form
1 d / dR\ 1 d / . ^d®\ 1 d-'P^r.
Rdr\ dr)'^® sin 0 dS \^'" do)'^ ^ sir^ ^ -^-^^
V y) - ^(^ + 1)^ = ^j R = Ar^-{- Br-
dcosd
^<i-«-'*)^j+h»+^)-CT-j®=o-
The problem is now separated into that of the integration of three
differential equations of which the first two are readily integrable. The
third equation is a generalization of Legendre's (Exs. 13-17, p. 252),
REAL VARIABLES 5S7
and in case n, m are positive integera the Mlotloii onj b* mtitwmmtml in
terms of polynomials /'^. (cos ^ in oos A Any exprsssiuii
2 (^.1- + /*.r— »)(a, COS iii4 4- A. sin iii4)i»^. (eos #)
is therefore a solution of Laplace's equation, and it may be
by combining such solutions into infinite series, a solntioa tmy be
obtained which hikes on any desired valoes on the unit spbers aad
converges for all points within or outside.
Of particular simplicity and importance is the ease in wbieb V is tm^
posed indei)endent of ^ so that m s 0 and the equation for • is sohibis
in terms of Legeudre's polynomials /*, (cos ^ if n is intcgrmL As the
potential V of any distribution of matter attmeUng aeoofding to the i»>
verse square of the distance satisfies Laplace's equation at all points
exterior to the mass (§ 201), the potential of any mass synmeirie with
respect to revolution al)out the polar axis # k 0 may be txprsaaad if
its expression for }X)ints on the axis is known. For inslanee, the
tial of a mass M distributed along a circular wire of radios « is
V =
VoM^ "" 1 M/a _ 1 «_• L3 «^ _ 1»3»5 ^
a\r 2t*'^24i* 2 4 6/''^
at a point distant r from the center of the wire along a perpendievlar
to the plane of the wire. The two series
»•<«».
V =
a^o 2««^«^2.4a*^* 2.4.6ii*'*^ '
are then precisely of the form S.!.!*/'., X-i.r— •!». admissible fcr
solutions of Laplace's equation and reduce to the known value of V
along the axis d = 0 since />.(!) = 1. They give the values of T al all
])oints of s|mce. ^ ^^__
To this point the method of combining solutions of the given dtftiw
ential equations was to atld them into a finiti? or infinite serin. Uto
also possible to combine them by integration and to obtain a mAntim
as a definite integml instwwi of as an infinite series. It should be nctod
in this case, too, that a limit of a sum lias repboed a sum ami ihalH
would theoretically be necessary to demonstrate that the tfa»H of the
sum was really a solution of the given equation. It wtfl l» •••eif^
at this point to illustrate the method without any rigorous altemp* to
528 THEORY OF FUNCTIONS
justify it. Consider (2') in rectangular coordinates. The solutions for
X, Fare
JC" Y"
— = — m^, — = m*, X = a„ cos 7nx-{-h^ sin w^, Y=A ^e*"" + B^e' '"^,
where Y may be expressed in terms of hyperbolic functions. Now
I e~ "'" [a (m) cos mx -f i (m) sin ma] «?m
(6)
= lim V e~''S^[« (m,) cos mfc + i(m,.) sin m,«] AWf
is the limit of a sum of terms each of which is a solution of the given
equation ; for a (m,) and b (?/i,) are constants for any given value m = m,-,
no matter what functions a (m) and b (m) are of m. It may be assumed
that F is a solution of the given equation. Another solution could be
found by replacing e""'^ by e"''^.
It is sometimes possible to determine a (m), b (m) so that V shall
reduce to assigned values on certain lines. In fact (p. 466)
f(x) = - I j f(k) cos m(\-x) d\dm. (7)
Hence if the limits for m be 0 and oo and if the choice
a (m) = — I /(X) cos rnXoLXj b (m) = — / /(A.) sin m\d\
U — 00 %J — 00
is taken for a{m)^ b(ni), the expression (6) for V becomes
-iX"X
+ «
e" "'^/(A) cos m (X - ic) <5?X6?w (8)
and reduces to f(x) when y = 0. Hence a solution F is found which
takes on any assigned values f(x) along the x-axis. This solution clearly
becomes zero when y becomes infinite. When f{x) is given it is some-
times possible to perform one or more of the integrations and thus
simplify the expression for V.
For instance if
f{x) = 1 when a; > 0 and f{x) = 0 when x < 0,
the integral from — oo to 0 drops out and
F=- I I e-'"*'- 1 -cos m(\ — x)dXdm = - | i e- "•«' cos m (\ — .x) dmdX
irJo Jo tvJq Jo
IT Jo v'' + (X-i)2 ir\2 vl ir X
REAL VARIABLES 5|^
It raay readily be shown that wbra y > 0 tiM fttml of ikt ofd«r «| la^aatkm
iH permiffiible ; but as F i« detarmlnad eoaipl«ulj, It b dmfktr lo ^^*^ ,^
value a« found in the equaUon and tee tluti V^ ^ K;; » «, Mtf l« -trrifc Ito fact
that F reduces to /(x) when y = 0. It OMj ptriuipa te npiiiiiiiu to itoto ftet
the proved correctness of an answer does not ibow %h$ JwUicMloa ol ikt mtm by
which tiiat answer Is found ; but on tlie otlMr ?r*nd ts *^i«ff mtm wtf^utel
golely to obtain the answer, there b no pnictloy oatd of JoaUfvimr i^aM U i^
answer is clearly right.
1. Find the indicated particular solutions of
dt fijt^ ry« \coscaat, leosc;^,
2. Determine the solutions of Laplace*s equation In tiM piMM thai mtv » ■ |
for 0 < 0 < IT and r = — lforir<^<Srona unit
3. If r = |ir — 0| on the unit circle, find the expansion for F.
4. Show that V = Za^sinmvx/l • cos cm«</< Is the solution of Em. 1 {0) wMcfc
vanishes at x = 0 and x = L Determine the coeflideoU a^ m tiMii for f a • llM
value of V shall l)e an assigned function /(z). This la Um prnWem of Iko tklhi
string started from any assigned configuration.
5. If the string of Ex. 4 is started with any assigntd volodty ti /n - / ^x) wiMa
t = 0, show that the solution is ZOm aln mwx/l • dn cmmi/t nad bmUM iIm propor <
mination of the constants a^.
6. If the drumhead is started with the shape x =/{i^ y), sliow
abJo Jo a o
7. In hydrodynamics it is shown that^sf-^M^ItUM ^WmrntM o^na-
tion for the surface of the sea in an estuary or on a hsneh of brMillli k m
h measured perpendicularly to the x-axis which hi supposed to nw ssswaid. F1a4
(a) y = AJq{1cx) cos n/, lr« = nVy*, (^ y = ^•'•(^ vif) oos al, * ■ i»»/^a.
as particular solutions of the eqoatlon wiion (a) tiM dopUi is
breadth is proportional to the dislanoe out lo ssa, and wiMn (^ tiM kvMiUl b aal^
form but the depth is mx. Discuss the shape of tho waves that MSj Umb itoai sa
the surface of the estuary or
530 THEORY OF FUNCTIONS
8. If a series of parallel waves on an ocean of constant depth h is cut perpen-
dicularly by the xy-plane with the axes horizontal and vertical so that y = — his
the ocean bed, the equations for the velocity potential <p are known to be
Find and combine particular solutions to show that <f> may have the form
<f> = A cosh k{y ■\- h) 00s (kx — nf), n^ = gk tanh kh.
9. Obtain the solutions or types of solutions for these equations.
, , c'^V c^V IdV , 1 dH'' ^ . ^. rcos7n0^ _ ,, ,
^ ' dz^ df^ r dr r^ d<p^ Lsm m<f>j '"^ "
d^V 1 dV 1 d^V x-N
(/3) —Y + - -r- + -; -r-T + ^ = <^. ^^' Zj (Omcosm^ + bmSinmAp)Jm{r),
cr^ r cr r^ c<p ^^
ex2 ay2 ez2
(a„,« cos 7n0 + 6n,m sin 7n^),
10. Find the potential of a homogeneous circular disk as (Ex. 22, p. 68 ;
Ex. 23, p. 332)
y_2MV\a l.la» 1 • 1 ■ 3 a'^ 1 . 1 . 3 ■ 5 a^ -|
2 3fr r„ lr2^ 1 • 1 r* , 1 • 1 • 3 r« _, "I
= -^L'^a^^ + 2^^^-^4^^* + 2-:4^^^«-"J' ^<^'
where the negative sign before P^ holds f or ^ < ^ tt and the positive tor 6 > \ v.
11. Find the potential of a homogeneous hemispherical shell.
12. Find the potential of {a) a homogeneous hemisphere at all points outside
the hemisphere, and (/3) a homogeneous circular cylinder at all external points.
Q x^ — d^
13. Assume -^ cos-i is the potential at a point of the axis of a conduct-
2 a x^ + a^
ing disk of radius a charged with Q units of electricity. Find the potential anywhere.
196. Harmonic functions; general theorems. A function which
satisfies Laplace's equation V'^ -\- Vy^ = 0 or F^^ -f- V^^ + V'^^ = 0, whether
in the plane or in space, is called a harmonic fum^t'wn. It is assumed
that the first and second partial derivatives of a harmonic function are
continuous except at specified points called singular points. There are
many similarities between harmonic functions in the plane and har-
monic functions in space, and some differences. The fundamental theo-
rem is that : If a function is harmonic and has no singularities upon
or within a simple closed curve (or surface), the line integral of its nor-
mal derivative along the curve (respectively , surface) vanishes ; and con-
versely if a function V(Xf y), or F(a;, y, «), has continuous first and second
REAL VARIABLES 5tl
partial derivativeM and the line imUgral (or Mur/aee imUfrmi) tfny
closed curve (or surface) in a rtgian vmmUkm^ ik$ /kmtimn i» km
For by Green's Formula, in the respective cuee of pUiM %M
(Ex.10, p. 349),
Now if the function is harmonic, the right-hand fide
must the left; and conversely if the left-hand tide Taniah« for all
closed curves (or surfaces), the right-hand side miiat vaaU far evetj
region, and henee the integrand must vanish.
If in particular the curve or surface be taken as a eirde or spbora at
radius a and polar coordinates be taken at the oeDt«r» the hohmI d»»
rivative becomes d V/dr and the result is
1 ^'•"* = " "' 1 1 a, •"•*'*'♦-».
where the constant a or a* has been discarded from the eleneDt of air
adtft or the element of surface a* sin $d$d^. If these eqnatkms be iale-
grated with respect to r from 0 to a, the integrals may be evalnaled by
reversing the order of integration. Thus
["•1 ar''*=iia7''«'* -/<•'.- »-.>'♦•
and C VM=yof ^♦t or l^.-^^ti (!•)
Jo Jo
where V^ is the value of )' ou the circle of radios a and K, is the valat
at the center and l\ is the average value along the periOMter of tbe
circle. Similar analysis would hold in space. The result states the
important theorem: The average value of a karfnomk ^meiiam •mr e
circle (or sphere) is equal to the value ai the eemter.
This theorem has immediate corolhtries of impodance. A
function which has no singularities within a refien etumsi
mum or minimum at any point within the region. For if the
were a maximum at any point, that point could be snmMUMled by *
circle or sphere so small tliat the value of the function at every point
of the contour would lie less than at the assumed maiimttm and
the average value on the contour could not 1ms ih« value at the
532 THEORY OF FUNCTIONS
A harmonic function which has no singularities within a region and is
constant on the boundary is constant throughout the region. For the
maximum and minimum values must be on the boundary, and if these
have the same value, the function must have that same value through-
out the included region. Two harmonic functions which have identical
values upon a closed contour and have no singularities within, are iden-
tical throughout the included region. For their difference is harmonic
and has the constant value 0 on the boundary and hence throughout
the region. These theorems are equally true if the region is allowed to
grow until it is infinite, provided the values which the function takes
on at infinity are taken into consideration. Thus, if two harmonic
functions have no singularities in a certain infinite region, take on the
same values at all points of the boundary of the region, and approach
the same values as the point (a;, y) or {x, y, z) in any manner recedes
indefinitely in the region, the two functions are identical.
If Green's Formula be applied to a product Ud V/dn, then
Jo ^^ Jo dx "^ dy
-jju ( F- + O dxdy +ff(^'^K + KK) dxdy,
or CudS'VV = jUv^VVdv 4- fvU>VVdv (11)
in the plane or in space. In this relation let V be harmonic without
singularities within and upon the contour, and let U = V. The first inte-
gral on the right vanishes and the second is necessarily positive unless
the relations F^ = F^ = 0 or F^ = F^ = F^ = 0, which is equivalent
to V F = 0, are fulfilled at all points of the included region. Suppose
further that the normal derivative dV/dn is zero over the entire bound-
ary. The integral on the left will then vanish and that on the right
must vanish. Hence F contains none of the variables and is constant.
If the normal derivative of a function harmonic and devoid of singular-
ities at all points on and within a given contour vanishes identically
upon the contour, the function is constant. As a corollary : If two
functions are harmonic and devoid of singularities upon and within a
given contour, and if their normal derivatives are identically equal
upon the contour, the functions differ at most by an additive constant.
In other words, a harmonic function without singularities not only Is
ietermlned by its values on a contour but also (except for an additive
wnstant) by the values of its normal derivative upon a contour.
REAL VARUBLE8
Laplac6*s equation arisM dIracUjr upon Um ilnnMat of mtm MUmm te
physicN in mathcmatirai form. In Um lint pttit *<i»twMfr th» iov tt ImsI m tg
electricity in a conducting bodj. The pbjiM law la UmI Imm ia«a tkiM ika
direction of most rapid decrMUN of tMBpamtaio T, and Ckac iko iimat «l iW lb«
is proportional to the rato of doertaM. Aa — VT glvot Um dltvcUoa a*d a^od.
tude of the nio«t rapid decreaM of temperatora, Um 6ow of Imm mmj W fwfimamai
by - kVT, where fc ii a constant. The rate of flow In any dli«eUo« te Um mm^
nent of this vector in that dIrecUon. The rat« of flow tLenm aay ^mmtarj b
therefore the integral along the boundary of tho nonml doHiraUvo of T. Sow ilM
flow is said tobetteadylf there \» no Inrrwioi ordoowMtol IwiwIUUaiy (
boundary, that is >,
ik/dS.vr^O or riabamoBk.
Hence the problem of the distribuUon of the tonporatore la a
a steady flow of heat is the problem of IntcffmUng LapUeo't iiiaatliwi la IUm
manner, the laws of the flow of electricity being IdonUeal wlUi tkoM for Ite flow
of heat except that the potential V replaces the iMBpetmture 7, tka pfoblMi of Iks
distribution of {M>tential in a body supporUng a sloady flow of sloctflrity iHB §im
be that of solving' Laplace*s equaUon.
Another problem which gives rise to Laplaos*s oqoaUoa Is UmI of Iki InalBlloaal
motion of an incompresKible fluid. If T is Uie velocity of Ibo flald, Um aMiloa b
called irrotationai when Vxy = 0, that is, when the Une Inlsgral of Um isliwilf
about any closed curve is zero. In this case the negaUve of tiM llao lalsgfal hmm
a fixed limit to a variable limit deflnes a function 4(x, y, f)
potential, and the velocity may be expressed as t =— V4. As Um
pressible, the flow across any closed boundary Is necessarily toio.
fdS»V* = 0 or rv.V*ds = 0 or ?.?♦«•,
and the velocity potential ^ is a harmonic function. BoUi
stated without vector noUtion by carrying out the ideas Involvod wllli tte aid si
ordinary coordinates. The problems may also be solvod for Um plaao bHMatf of
for space in a precisely analogous manner.
197. The conception of the flow of electricity will be adrmiitafiow
ill discussing the singularities of harmonic functkmi and a mort §€••
ei-al conception of steady flow. Suppose
an electrode is set down on a sheet of zinc
of which the })erimeter is grounded. The
equipotential lines and the lines of flow
which are orthogonal to them may be
sketched in. Electricity passes steadily
from the electrode to the rim of the sheet
and off to the ground. Across any circuit
which does not surround the electrode the
flow of electricity is zero as the flow is steady, but aeros any tiMtmM
surrounding the electrode there will be a certain dellaili Bow; tha
circuit integral of the normal derivative of the potential V ufmad mA
634 THEORY OF FUNCTIONS
a circuit is not zero. This may be compared with the fact that the
circuit integral of a function of a complex variable is not necessarily
zero about a singularity, although it is zero if the circuit contains no
singularity. Or the electrode may not be considered as corresponding
to a singularity but to a portion cut out from the sheet so that the
sheet is no longer simply connected, and the comparison would then
be with a circuit which could not be shrunk to nothing. Concerning
this latter interpretation little need be said ; the facts are readily seen.
It is the former conception which is interesting.
For mathematical purposes the electrode will be idealized by assum-
ing its diameter to shrink down to a point. It is physically clear that
the smaller the electrode, the higher must be the potential at the elec-
trode to force a given flow of electricity into the plate. Indeed it may
be seen that V must become infinite as — C log r, where r is the distance
from the point electrode. For note in the first place that log r is a solu-
tion of Laplace's equation in the plane ; and let U = V -\- C log r or
V = U — C log r, where f7 is a harmonic function which remains finite
at the electrode. The flow across any small circle concentric with the
electrode is r^'^dV r^"" dU
and is finite. The constant C is called the strength of the source situ-
ated at the point electrode. A similar discussion for space would show
that the potential in the neighborhood of a source would become infinite
as C/r. The particular solutions — log r and 1/r of Laplace's equation
in the respective cases may be called the fundamental solutions.
The physical analogy will also suggest a method of obtaining higher singular-
ities by combining fundamental singularities. For suppose that a powerful positive
electrode is placed near an equally powerful negative electrode, that is, suppose a
strong source and a strong sink near together. The greater part of the flow will be
nearly in a straight line from the source to the sink, but some part of it will spread
out over the sheet. The value of V obtained by adding together the two values for
source and sink is
F = _ J Clog (r2 + /2 _ 2 W COS0) -1- J Clog (r^ + /^ + 2 rl cos 0)
= _-Clog^^l--co80 + -j + -Clogh-h ycos0-l--j
= cos 0 -I- higher powers = — cos 0 + • • • .
r r
Thus if the strength C be allowed to become infinite as the distance 2 1 becomes
zero, and if M denote the limit of the product 2 iC, the limiting form of V is
Mr-^ CO8 0 and is itself a solution of the equation, becoming infinite more strongly
than — logr. In space the corresponding solution would be JVfr-^ cos <f>.
REAL VARIABLES MS
It was seeu tliat a hannooio fum;tion which had noi
within a given coutour was determiDed by iu ralim on tlM«ak
iletcrinined except for an additive oonsUuii bj Um nlw* a| Hi
derivative upon the contour. If now thera be iotaallj within the (
certain singularities at which the function beoooM
particular solutions K,, K,, • • , the function (/ « K — K, — P^ — . . . ii
iiionic without singularities and may be detennioed ■• before.
the values of F^, \\y -or their normal dertTatiTee nay be
known upon the contour inasmuch as theee afe definite paitlealir ioli^
tions. Hence it appears, as before, that ths hanm&mie /kmeii^m V is dMm^
rnmed by its values on the boundary oftke regism or (sMtpi/sr mm mdJitism
constant) by the values of its normal dsrivaUM •» fiU houmimrp, jinwidtd
the slntjularities are specified in position and their mode efhseemim§ is^h^
it 6 is given in each ease as some particular solution o/LaplmN^s sgmmiimsL
Consider again the conducting sheet with ita perimeter groiUMied aad
with a single electrode of strength unity at some interior point of tW
sheet. The potential thus set up has the propertiee that : 1* the polen>
tial is zero along the i)erimeter because the perimeter ii gnmndedt Twi
the position P of the electrode the potential becomee infinite ae — lof r ;
and 3** at any other point of the sheet the potential is regular and nU
isiies Laplace's equation. This particular distribution of potential b
denoted by G{P) and is called the Green Function of the aheei rekltte
to /'. In space the Green Function of a region would still «ttsfy I'and
3**, but in 2° the fundamental solution — log r woold have to be
by the corresponding fundamental solution 1/r. It aboold be
that the Green Function is really a function
G{P) = 6- (a, A; x, y) or G{P) » G{a, *,«»;», Jf, a)
of four or six Variables if the position P(a, b) or /*(«, *, e) of the
trode is considered as variable. The function is eonaidwed as known
only when it is known for any position of P,
If now the symmetrical form of Green's Formola
- /T(i/Ar - r\u^.lrdy -\-f(u ^ - ^^^ " ^' ^^
where A denotes the sum of the second derinitiTea, be appiiea t« lor
entire sheet with the exception of a small circle eoofientrie with P and
if the choice u = G and r « K be made, then at G and V are hnrmonie
the double integral drops out and
536 THEORY OF FUNCTIONS
Now let the radius r of the small circle approach 0. Under the assump-
tion that V is devoid of singularities and that G becomes infinite as
— log r, the middle integral approaches 0 because its integrand does,
and the final integral approaches 2 7rF(P). Hence
This formula expresses the values of V at any interior point of the sheet
in terms of the values of V upon the contour and of the normal deriva-
tive of G along the contour. It appears, therefore, that the determination
of the value of a harmonic function devoid of singularities within and
upon a contour may he made in terms of the values on the contour pro-
vided the Green Function of the region is known. Hence the particular
importance of the problem of determining the Green Function for a
given region. This theorem is analogous to Cauchy's Integral (§ 126).
EXERCISES
1. Show that any linear function ax •\- by + cz + d = 0 m harmonic. Find the
conditions that a quadratic function be harmonic.
2. Show that the real and imaginary parts of any function of a complex vari-
able are each harmonic functions of (x, y).
3. Why is the sum or difference of any two harmonic functions multiplied by
any constants itself harmonic ? Is the power of a harmonic function harmonic ?
4. Show that the product UV of two harmonic functions is harmonic when
and only when t/^F^ -f LT^y^ = 0 or VU»W = 0. In this case the two functions
are called conjugate or orthogonal. What is the significance of this condition
geometrically ?
5. Prove the average value theorem for space as for the plane.
6. Show for the plane that if V is harmonic, then
U= I -r-dsz= I —-dy — —-dx
J dn J dx dy
is independent of the path and is the conjugate or orthogonal function to F, and
that U is devoid of singularities over any region over which V is devoid of them.
Show that F + ilT is a function of z = x -\- iy.
7. State the problems of the steady flow of heat or electricity in terms of ordi-
nary co&rdinates for the case of the plane.
8. Discuss for space the problem of the source, showing that C/r gives a finite
flow 4 ttC, where C is called the strength of the source. Note the presence of the
factor 4 IT in the place of 2 x as found in two dimensions.
9. Derive the solution Mr- *coa 0 for the source-sink combination in space.
REAL VARIABLES Ut
10. DiscuM the problem of the nuUI magMt or Um tlMUle ilniMn te niu if
Kx. U. Note that ae the attraction b invenwly ■• tiM aqoMV of Um
)>otential of the force satlsflea Laplace'* equation la ^■•rt.
11. Let equal infinite lourcet and doka be k»eMe4 alutmielj ai Um ^
of an infinitemmal square. Find Um comtpondlng |i^»*frihr iHittmi lot Im
case of the plane, and (/9) in the eaae of ipneo. What *«*— Wi^^^yn ol mfBito^Hn
this reprcHent if the point of view of Ex. 10 be Ukta, and for what tpagnrrt It 0m
(•(jinbination used ?
12. Kxpii'Hs V{P) In terms of (7(P) and the boundary valoM of K la ifaca.
13 . I f an analytic function has no singularlUea wiUda or oa a i
Iiiti-^Mal Kive8 the value at any interior point. If Umts aia viUda UM
tuin poles, what nuiHt bt> known in addition to the booadary valiMs io<
the function ? Compare with the analogous tbeorma for Mriiwntfl
14. Why were the solutions In f 194 as seriea Um only pw lldi
provided they were really solutions? Is there any difflculty In BMkfti^ Um i
inference relative to the problem of the potential of a circular wiiv la | IMf
15. Let G{P) and G{Q) be the Green FuncUons for the si
to two different points P and Q. Apply Green's syniBMttie
from which two small circles about Pand Q have been raaKHn
u = G{P) and v = G{Q). Hence show that (7(P) at Q to eqoal to <7<^ al F. Thb
may be written as
G(a, 6; X, 1/) = 0(x, 1/; a, 6) or &(a, 6, e; s, y, i) s 0(c, y, t; a, |«, r).
16. Test these functions for the harmonic property, daCanalBa Um ooi^^^ii
functions and the allied functions of a complex rariable:
(a) xy, {P)x^-\ v», (y) | lof («• + »•),
(S) e'sinz, («) sin z cosh y, (f) tan-*(ooixtaak|r).
198. Harmonic functions ; special theorems. For Uie pttrpoMi oC
the next puragruplis it is necessary to study the propartiaa of the g9^
metric transformation known as inversion. The definiUon of iavaiskHi
will be given so as to be applicable either to space or to the pluau
The transformation which replaoes each point P by a point P* wmk
that OP ' OP' = A;' where O is a given fixed point, k a ooostsnt, and /**
is on the line O/', is called inversion with the eeni^ O mmd the rmJimg k.
Note that if /' is thus carried into P\ then P* will be earned into P\
and hence if any geometrical configuration is carried into another, that
other will be carried into the first Points veiy near to O ars earrfad
otf to a great distance; for the point O itself the definitkm hieaht
down and 0 corresponds to no point of space. If desired, one mty add
to space a fictitious ])oint called the point at infinity and any th«i say
that the center 0 of the inversion corresponds to the point at infinity
(p. 481). A pair of points P, P* which go over into each other, and aaothv
pair Q, Q' satisfy the equation OPOP'^ OQOif.
538 THEORY OF FUNCTIONS
A curve which cuts tlie line OP at an angle t is carried into a
curve which cuts the line at the angle t' = tt — t. For by the relation
OP 'OP' = OQ ' 0Q\ the triangles OPQ^ OQ'P' are similar and
Z OPQ = Z OQ'P' = 7r-Z0-Z OP'Q'.
Now it Q^P and Q' = P', then Z 0 = 0, Z OPQ = t, Z OP'Q' :^ r and
it is seen that t = tt — t' or t' = tt — r. An immediate extension of
the argument will show that the magnitude
of the angle between two intersecting curves p
will be unchanged by the transformation; the "^ ^
transformation is therefore conformal, (In
the plane where it is possible to distinguish between positive and neg-
ative angles, the sign of the angle is reversed by the transformation.)
If polar coordinates relative to the point 0 be introduced, the equations
of the transformation are simply rr' = W' with the understanding that
the angle <^ in the plane or the angles <^, B in space are unchanged. The
locus r = k, which is a circle in the plane or a sphere in space, becomes
r' = k and is therefore unchanged. This is called the circle or the sphere
of inversion. Relative to this locus a simple construction for a pair of
inverse points P and P' may be made as indicated in the figure. The locus
7^ + A;2 ^ 2 Va^ + A;V cos <^ becomes k^ + r*^ = 2 VoF+l^r' cos <f>
and is therefore unchanged as a whole. This locus represents a circle
or a sphere of radius a orthogonal to the circle or sphere of inversion.
A construction may now be made for finding an inversion which cai'-
ries a given circle into itself and ^5,
the center P of the circle into any
assigned point P' of the circle ; the
construction holds for space by re-
volving the figure about the line OP.
To find what figure a line in the plane or a plane in space becomes
on inversion, let the polar axis <f> = 0 ov 0 = 0 he taken perpendicular
to the line or plane as the case may be. Then
r = ^ sec <^, r' sec <l> = k^/p or r=p sec 0, r' sec $ = k^/p
are the equations of the line or plane and the inverse locus. The locus
is seen to be a circle or sphere through the center of inversion. This
may also be seen directly by applying the geometric definition of in-
version. In a similar manner, or analytically, it may be shown that
any circle in the plane or any sphere in space inverts into a circle or
into a sphere, unless it passes through the center of inversion and
becomes a line or a plane.
REAL VARIABLES Stt
If d be the distance of P from the ditO* or iplMM of InvMkHmW 4I«mm» ^
P from the center ia A - d, the dlMuMW o( P* fitM Um tmfat !• *«Aft • A. Mtf
from the circle or sphere it is (f = dk/{k - tf). Now If tbo iiidias t Is fwrtalM
in comparison with d, the rmtio k/{k - d) «• "•Miy 1 Md r b mmiif mmL Ud.
U k\n allowed to become infinite so thst the etncar of lavonioa ramda liiil^^^v
and the circle or sphere of inTendon approadMt a Um or pImml tW dfaZIa?
approaches d as a limit. As the tnuMfomuloM widdi npliii mdk aatei kr m
point equidiHtant from a given line or plaoo and psfpwMllfliilj OfpailM •^^^
point i8 the ordinary Inversion or relloctioo In tho Uoo or f^m mA m Is
in opticM, it appears that reflection in a lino or piano nay bo nnidsd m ll
ing case of inversion in a circle or sphere.
The importance of inversion in the stodj of li^*nKmii? ftmrlioM IIm
in two theorems applicable respectively to the pUoe and to
First, if V is harmonic over any region of ths plans ami {ftkmi
be inverted in any circle, the function \"{l*^wm V(P) fonmsJ hp
ing the same value at P' in the new region as ike fitmetm had «l |A#
point P which inverted into P' it aUo harmonie, SMond, (^ T if kmr^
monic over any region in space, and if thai rsgwn he Jnwwfisd m a t^kms
of radius ky the function F^P*) = kV{P)/r' formed hp assi^img mt I*
the value the function had at P muUipUed hp k and dimiiad kw ths di^
lance OP' = r' of P' from the center of inversion is aiso kmrmemie. Tbo
significance of these theorems lies in the fact that if ooo distrtbatioii
of potential is known, another may be derived from it bjr iovtnios;
and conversely it is often possible to determine a distribotioii of polvB.
tial by inverting an unknown case into one that is known. Tho pcoof
of the theorems consists merely in making the chaogea of Tariable
r = kyr' or r' = k^r, ^' = 4, r-l
in the polar forms of Laplace's equation (Exs, 21, 22, p. 112).
The method of using inversion to determine distribotJon of p^twflal la rismu>
statics is often called the method of eleetrie ims^. As a ohiigi t localod at a
point exerts on other p^jint charges a force proportional to tiM lavont a^aaiw of
^he distance, the potential due to e Is as l//», where p Is tbo dlHiars fi«B Ibo
charge (with the proper uniu it may be taken as r/p), and ilhUos Laplaco*^
equation. The potential due to any number of point chargos Is iIm saai of iW
individual potentials due to the charges. Thus far the tbsory la
same as if the charges were attracting particles of matter. In sisctiidty,
the question of the distribution of potential is fortbor eonplkalo
in tlie neighborhood of the charge* certain condueting MffaoHL For I* a
ing surface in an electrostatic field must everywhere bo at a ooartaai poloalAal or
there would be a component force along the surf aoo and tba risuliiiiljy s^sm H
would move, and 2*^ there is the phononenon of lodocod rioUiklty wtwwby a
variable surface charge is induced apon tlio conductor bj otbor okaifM la iW
neighborhood. If the potential V{P) dne to any dlatHbotloa of ifcaigM bo
inverted in any sphere, the new potenUal l* kVtPi/r^. As tbo pm— list TfH
540
THEORY OF FUNCTIONS
becomes infinite as e/p at the point charges e, the potential kV{P)/r' will become
infinite at the inverted positions of the charges. As the ratio ds':ds of the in-
verted and original elements of length is r^/k\ the potential kV{P)/r' will become
infinite as k/r" • e/p' ■ r"^/)^^ that is, as r'e/kp'. Hence it appears that the charge e
inverts into a charge e' = r'e/k ; the charge — e' is called the electric image of e.
As the new potential is kV{P)/r' instead of V{P), it appears that an equipoten-
tial surface V = const, will not invert into an equipotential surface V'{P') = const,
unless T = 0 or r' is constant. But if to the inverted system there be added the
charge e = — kV a,t the center O of inversion, the inverted equipotential surface
becomes a surface of zero potential.
With these preliminaries, consider the question of the distribution of potential
due to an external charge e at a distance r from the center of a conducting spheri-
cal surface of radius k which has been grounded so as to be maintained at zero
potential. If the system be inverted with respect to the sphere of radius A;, the
potential of the spherical surface remains zero and the charge e goes over into a
charge e' = r'e/k at the inverse point. Now if p, p' are the distances from e, e" to
the sphere, it is a fact of elementary geometry that p : p' = const. = 7^ :k. Hence
the potential
p' \p kp'l kpp'
F =
due to the charge e and to its image — e', actually vanishes upon the sphere ; and
as it is harmonic and has only the singularity e/p outside the sphere (which is the
same as the singularity due to e), this value of V throughout all space must be
precisely the value due to the charge and the grounded sphere. The distribution
of potential in the given system is therefore determined. The potential outside
the sphere is as if the sphere were removed and the two charges e, — e' left alone.
By Gauss's Integral (Ex. 8, p. 348) the charge within any region may be evaluated
by a surface integral around the region. This integral over a surface surrounding
the sphere is the same as if over a surface shrunk down around the charge — e',
and hence the total charge induced on the sphere is — e' = — r^e/k.
199. Inversion will transform the average value theorem
^iP) = -hrf^^^''' '"'° nP') = 2^jr'V'#, (14)
a form applicable to determine the value of V at any point of a circle
in terms of the value upon the circumference. For suppose the circle
with center at P and with the set
of radii spaced at angles c?</>, as
implied in the computation of the
average value, be inverted upon an
orthogonal circle so chosen that P
shall go over into P\ The given
circle goes over into itself and the series of lines goes over into a series
of circles through P' and the center 0 of inversion. (The figures are
drawn separately instead of superposed.) From the conformal property
REAL VARIABLES 4||
the angles between the cirole« of the mHm m equal to Um «m|m b».
tween the radii, and the cirolat cut the giren fAnh otthflmd^iMt
as the radii did Let V* along the area 1', 2*, 3'. . . . ba ^qmSZr^Z
the corresponding area 1, 2, 3, . . • and let y(P) . I'Xi^ ., tmrnnd^
the theorem on inversion of harmonie foneiioiia. Then tJbe two hJ^
grals are equal element for element and their Taliiea V(F\ aad r/^
are equal. Hence the desired form follows from the gir«i fom m
stated. (It may be observed that d^ and d^, atriellj tlMikiac k«^
opposite signs, but in determining the arerage valiie F*(/*% J4 k uW«
positively.) The derived form of integral may be written
na a line integral along the arc of the cirole. If /^ is at the disuooe r
from the center, and if a be the radius, the center of tOTetakNi O b il
the distance a*/r from the center of the circle, and the value of i ia
seen to be A:' = (a* - f^a*/f*. Then, if Q and Q' be poiiila on tha eM%
Now d\lf/(y may be obtained, beoaose of the equality of d^ and d^ aad
ds^ may be written as ar/^'. Hence
Finally the primes may be dropped from V* and /»', the poaition of f^
may be expressed in terms of its oodrdinates (r, 4), and
is the expression of V in t«rms of its boundarj valwa.
The integral (15) is called PoisBon^M Ini^groL It ahould be solid puw
ticularly that the form of Poisson's Integral first obtained by iawrikis
represents the average value of V along the dieumferanoey profidid thaft
avei-age be computed for each point by conaidering the valoaa aloif tlw
circumference as distributed relative to the angle ^ as indepaodaot vmii-
able. That V as defined by the integral aetnally approadMt tha valw oa
the circumference when the point approaehea the eiremlMMS ii aliar
from the figure, which shows that all except an infinilviaHd fkaelioa ol
the orthogonal circles cut the oirole within infiniteaimal limili wbea Um
point is infinitely near to the droamferaiioa. Fotoaon^ latagral wmj ba
542 THEORY OF FUNCTIONS
obtained in another way. For if P and P' are now two inverse points
relative to the circle, the equation of the circle may be written as
p/p' = const. = r/a, and G{P)— — log p -}- log p' -f log (r/a) (16)
is then the Green Function of the circular sheet because it vanishes along
the circumference, is harmonic owing to the fact that the logarithm of the
distance from a point is a solution of Laplace's equation, and becomes
infinite at P as — log p. Hence
It is not difficult to reduce this form of the integral to (15).
If a harmonic function is defined in a region abutting upon a segment
of a straight line or an arc of a circle, and if the function vanishes along
the segment or arc, the function may be extended across the segment
or arc by assigning to the inverse point P' the value F(P') =— V{P)j
which is the negative of the value at P; the conjugate function
/dV rdV dV
takes on the same values at P and P'. It will be sufficient to prove
this theorem in the case of the straight line because, by the theorem on
inversion, the arc may be inverted into a line by taking the center of
inversion at any point of the arc or the arc produced. As the Laplace
operator D^ -f D^ is independent of the axes (Ex. 25, p. 112), the line
may be taken as the ar-axis without restricting the conclusion.
Now the extended function V{P^ satisfies Laplace's equation since
Therefore ViP") is harmonic. By the definition V{P') = — V{P) and the assumption
that V vanishes along the segment it appears that the function V on the two sides
of the line pieces on to itself in a continuous manner, and it remains merely to show
that it pieces on to itself in a harmonic manner, that is, that the function V and
its extension form a function harmonic at points of the line. This follows from
Poisson's Integral applied to a circle centered on the line. For let
Hlx, y)= f ^Fd^ ; then H(x, 0) = 0
Jo
because V takes on equal and opposite values on the upper and lower semicircum-
ferences. Hence H = V{P) = V{P') = 0 along the axis. But H = V{P) along the
upper arc and II = V{P') along the lower arc because Poisson's Integral takes on
the boundary values as a limit when the point approaches the boundary. Now as
H is harmonic and agrees with V{P) upon the whole perimeter of the upper semi-
circle it must be identical with V(P) throughout that semicircle. In like manner
REAL VARIABLES Sa
It is identical with r(P^ throoghoai the low — liiiiji. Aa Um ^-rinar r(A
and V(P^ are identical with the alogto bannoBle fvaoUoa H, ttef avi ■!«•
together harmonically acroM tb« axb. The IImmwb to tbtti iMiiiiiHiij iMMC
The Btatement about the conjugal* f aoelloB mj b« T«HAtd by iMkU^ Uw tel^^
along paths 8ynimetric with respeet to Um oil.
200. // a funeiion w »/(«) « it -f <v o/ a wm^^ 9mrimkU kmmmm
rail ultmg the segment of a line or ths aro of a ehrtU^ ike f^metimn mtmm
he extfnded analytically across the segmeni or art by asti^imf to CA#
inrrrst', point P' the value wssu — io eoiymyais to thai «| P, Tbb it
merely a corollary of the preceding theorem. For if w be twal^ Um
harmonic function v vanishes on the line and may be
and opposite values on the opposite sides of the line;
function u then takes on equal values on the opposite ridee of Um
line. The case of the circular arc would again f'iii"i¥ fr..... :r.«.<..«4«|||
as before.
The method employed to identify functiona in f § 186-187
map the halves of the t/;-plane, or nither the aeveral repetittoiM of
halves which were required to complete the map of the M«iirl!Me, on a
region of the sr-plane. By virtue of the theorem just obtidned the eoi^
verse process may often be carried out and the fanetaon wwmf{a)
which maps a given region of the «-plane upon the half of the
may be obtained. The method will apply only to regiooe of the
which are bounded by rectilinear segments and circular aroi ; for it is
only for such that the theorems on inversion and the theofem oo Um
extension of harmonic functions have been prOTed. To identify the
function it is necessary to extend the given region of the e-phMM by
inversions across its boundaries until the tc^«ttrfiMe it oompleled. The
method is not satisfactory if the soooessiTe extentiont of the region fai
the s-plane result in overlapping.
The method will be applied to determining the function («) whieh
maps the first quadrant of the unit circle in the e-|ihuM npon the vpptr
half of the MT-plane, and 09) which mai)e a 3aHKr-9(r triangle npon Um
upper half of the ir-plane. Sup-
pose the sector ABC mapped on
the M?-half-plane so that the perim-
eter ABC corresponds to the
real axis al>c. When the perime-
ter is described in the order written and the interior it on Um WII»
the real axis must, by the principle of conformality, be deneribed in
such an order that the upper half-phine which is to oorrtepond lo the
intAM-ior shall also lie on the left The points e, *, #
544
THEORY OF FUNCTIONS
Ay Bj C. At these points the correspondence required is such that the
conformality must break down. As angles are doubled, each of the
points A, By C must be a critical point of the first order for w =/(«)
and a, b, c must be branch points. To map the triangle, similar con-
siderations apply except that whereas C is a critical point of the first
order, the points A'y B' are critical of orders 5, 2 respectively. Each
case may now be treated separately in detail.
Let it be assumed that the three vertices A, B, C of the sector go into the
points* w = 0, 1, CO. As tlie perimeter of the sector is mapped on the real axis,
the function w=f{z) takes on real vakies for points z along the perimeter.
Hence if the sector be inverted over any of its sides, the point P" which corre-
sponds to P may be given a value conjugate to w at
P, and the image of P' in the ir-plane is symmetrical
to the image of P with respect to the real axis. The
three regions 1', 2', 3' of the z-plane correspond to
the lower half of the iw-plane ; and the perimeters
of these regions correspond also to the real axis.
These regions may now be inverted across their
boundaries and give rise to the regions 2, 3, 4 which
must correspond to the upper half of the lo-plane.
Finally by inversion from one of these regions the
region 4' may be obtained as corresponding to the
lower half of the lo-plane. In this manner the inver-
sion has been carried on until the entire 2-plane is covered. Moreover there is no
overlapping of the regions and the figure may be inverted in any of its lines with-
out producing any overlapping ; it will merely invert into itself. If a Riemann sur-
face were to be constructed over the lo-plane, it would clearly require four sheets.
The surface could be connected up by studying the correspondence ; but this is not
necessary. Note merely that the function f{z) becomes infinite at C when z = i
by hypothesis and at C" when z = —i by inversion ; and at no other point. The
values ± i will therefore be taken as poles of /(z) and as poles of the second order
because angles are doubled. Note again that the function /(z) vanishes at A when
z r= 0 by hypothesis and at z = oo by inversion. These will be assumed to be zeros of
the second order because the points are critical points at which angles are doubled.
The function
w =f{z) = Cz^{z-i)-^z + i)-^ = Cz^{z^ -\- 1)-*
has the above zeros and poles and must be identical with the desired function when
the constant C is properly chosen. As the correspondence is such that /(I) = 1 by
hypothesis, the constant C is 4. The determination of the function is complete as
given.
Consider next the case of the triangle. The same process of inversion and re-
peated inversion may be followed, and never results in overlapping except as one
• It may be observed that the linear transformation (710 -h S) lo' = aw -f- /9 (Ex. 15,
p. 157) has three arbitrary constants or: /3: 7: 5, and that by such a transformation any
three points of the w-plane may be carried into any three points of the to'-plaae. It is
therefore a proper and trivial restriction to assume that 0, 1, 00 are the points of the
to-plane which correspond to A, B, C.
REAL VARIABLn
region falls into abtolote oolnddeoM with mm
wliole z-plane the invenlon would ha?* to b« ,__
be obeerved that the recuogl* iodoMd bj Um iMvy
iH repeated Indeflnitely. Hence t9 * /(t) to a douU j peHotfto
function with the periods JIT, 2iK* If SJr, SA" ba tka
length and bruadth of the recUuigle. The function haa a
pole of the second order at C or x = 0 and at tha polM%
marked with circles, into which the orifiu to carriad by
the Kucceasive inversions. As there aro lis potoa at tba
second order, the function to of order twelve. Wbaa MmK
iii A or z =z iK' at A' the function Tanivhea and aaeb of
these zero8 is of the sixth order becauae angles an Inetaaaad
0-fold. Again it appears that the function to of onlar It.
It i8 very simple to write the function down In tema of
the theta functions constructed witli the periods t JT, S iK\
w=/{z)=C
/'•(*)o»(«)
H«(r) e«(«) m{z - a) e«(f - a) ir«(i - ^af <s - 0^ *
For this function is really doubly periodic. It Ysntobea lo tlM rixtb otder at JT, UT',
and has poles of the second order at the points
0, A' + iA", a = iK^\iK\ a -^ K •\. iK\ fimtK^'a, fi-^K^-UT.
As /S = 2 A' - a the reduction H*{z - /J) = //'(t + a), e,(i - ^ « B^{9 -f «) H^
be made.
«,=/(z)=C ?M^
H*(z)ef(z) Il*{z - a)iT«(« + a)e«(i - a)ef{M 4 «t)
The constant C may be determined, and the exprearion for /(() way be nd«t*4
further by means of identities; it might be expressed In tenas of sn<i, t) aad
en {z, ^-), with properly chosen k, or in terms of p(x) and ^(<). For tba piifpwss «l
computations that might be involved in carrying oat tba dataOa of Iba aaf^ il
would probably be better to leave the exprwion of /(t) la lanaa of Iba
functions, as the value of q is about 0.01.
1. Show geometrically that a plane InTerta Into a
inversion, and a line Into a circle through tba caotar of latanloii.
2. Show geometrically or analytically that in tba plasa a drala tevwts
circle and that in space a sphere Inverts Into a
3. Show that In the plane anglea are revenad la sign by lavaialott.
in space the magnitude of an angle between two eurraa to
4. If ds, dS, dv are elemenU of are, surface, and toIobm, sbow tbal
r' r^
dS'^^dS^^dS, #<«-,#r-j|A
Note that in the plane an area and lu InTorbsd area are of
the same is tnie of voluuies in spaoa.
546 THEORY OF FUNCTIONS
5, Show that the system of circles through any point and its inverse with respect
to a given circle cut that circle orthogonally. Hence show that if two points are in-
verse with respect to any circle, they are carried into points inverse with respect to
the inverted position of the circle if the circle be inverted in any manner. In par-
ticular show that if a circle be inverted with respect to an orthogonal circle, its cen-
ter is carried into the point which is inverse with respect to the center of inversion.
6. Obtain Poisson's Integral (15) from the form (16'). Note that
o o , o n / X dG cos {p, n) cos (p', n) a^ — r^
r^ = p^ -\- a^ — 2ap cos (p, n), — = ^^-^— ^ ^^ ' = — — — .
dn p p' CL^p^
7. From the equation p/p' = const. = r/a of the sphere obtain
p r p' ATraJ [a2 + r2 _ 2 ar cos (r, a)] t
the Green Function and Poisson's Integral for the sphere.
8. Obtain Poisson's Integral in space by the method of inversion.
9. Find the potential due to an insulated spherical conductor and an external
charge (by placing at the center of the sphere a charge equal to the negative of
that induced on the grounded sphere).
10. If two spheres intersect at right angles, and charges proportional to the
diameters are placed at their centers with an opposite charge proportional to the
diameter of the common circle at the center of the circle, then the potential over
the two spheres is constant. Hence determine the effect throughout external space
of two orthogonal conducting spheres maintained at a given potential.
11. A charge is placed at a distance h from an infinite conducting plane.
Determine the potential on the supposition that the plane is insulated with no
charge or maintained at zero potential.
12. Map the quadrantal sector on the upper half-plane so that the vertices
C, A, B correspond to 1, oo, 0.
13. Determine the constant C occurring in the map of the triangle on the plane.
Find the point into which the median point of the triangle is carried.
14. With various selections of correspondences of the vertices to the three points
0, 1, 00 of the w-plane, map the following configurations upon the upper half-plane :
{a) a sector of 60°, (/3) an isosceles right triangle,
(7) a sector of 45°, (5) an equilateral triangle.
201. The potential integrals. If p(x, y, z) is a function defined at
different points of a region of space, the integral
^(^' '^ ^) -JJJ V(^ - .f + (, - yf + « - .f -J r (1«)
evaluated over that region is called the potential of p at the point
(^, 1;, ^). The significance of the integral may be seen by considering
the attraction and the potential energy at the point (^, rj, ^) due to a
HEAL VARIABLB8
distribution of matter of density ^(c, y, «) in Man
If /i be a mass at (^, ,,, ^ and m a niMt at («, y, .). Um
forces exerted by m upon fi ai«
M7
r* r
and
rr=
*^.
'^^•-f
(«•)
-«M7 +
are respectively the total force on fi and the poCantkl mngr of tW
two masses. The potential energy may be oonsiderad M tiie wmk Itbi
by F or A', K, Z on /i in bringing the
mass fi from a fixed point to the
point (^, i;, Q under the action of m
at (ar, y, «) or it may lie regarded
as the function such that the nega-
tive of the derivatives of V by ar, y, x
give the forces A', >', Z, or in vector
iiotiition F = — V V. Hence if the
units be so chosen that c = 1, and if
tlie forces and potential at (^, ly, Q
be measured per unit mass by dividing by /a, the rasiilta aie (afler 4hh
regarding the arbitrary constant C)
H
r* r
Y =
Z- =
r
(ir)
(\^r\
Now if there be a region of matter of density p(jr, y, «), Uie f
l)otential energy at (^, 17, ^) measured per unit nia« there
be obtained by summation or integration and are
JJJ [(|_„)« + (,_y). + ({_,)r|J f
It therefore a]>{)ears that the potential r <ie(ined by (19) U the
of the potential energy V due to the distribution of matt
ther that in evaluating the integrab to determine X, K, Z,
the variables x, y^ t with respect to which the
formed will drop out on substituting the limits whieh determJiw the
region, and will therefore leave X^ Y, Z, U mm fnnctjoot of the
eters ^, 1;, ( which appear in the integrand. And finally
Noli tow
K-^^
""'h'
x^~
(m
• In electric nod magiMtio tiMOCy. wImm lOra npili MlM. dw
energy have the same rign.
648 THEORY OF FUNCTIONS
ai*e consequences either of differentiating f7 under the sign of integration
or of integrating the expressions (19') for .Y, F, Z expressed in terms of
the derivatives of U^ over the whole region.
Theorem. The potential integral U satisfies the equations
d^u d-'u d-'u . ^u dHj d'^u . .^.^
known respectively as Laplace^s and PoissorCs Equations, according as
the point (^, 7;, Q lies outside or within the body of density p (x, y, z).
In case ($, rj, Q lies outside the body, the proof is very simple. For
the second derivatives of U may be obtained by differentiating with
respect to i, rj, ^ under the sign of integration, and the sum of the
results is then zero. In case ($, rj, 0 ^i^s within the body, the value
for r vanishes when (^, rj, 0 coincides with (x, ?/, z) during the integra-
tion, and hence the integrals for f/, X, Y, Z become infinite integrals
for which differentiation under the sign is not permissible without jus-
tification. Suppose therefore that a small sphere of radius r concentric
with (^, >7, ^) be cut out of the body, and the contributions F' of this
sphere and F* of the remainder of the body to the force F be considered
separately. For convenience suppose the origin moved up to the point
(^, ,;, 0- Then
Vf/ = F* + F' =
C pV^dv + F'.
Now as the sphere is small and the density p is supposed continuous,
the attraction F' of the sphere at any point of its surface may be taken
as ^ Trr^pjr^, the quotient of the mass by the square of the distance to the
center, where p^ is the density at the center. The force F' then reduces
to — I TTp^T in magnitude and direction. Hence
V.F = V'VU = V.F* + V.F' = / pV. V -du-\- V.F'.
• = XpV.vJ
The integral vanishes as in the first case, and V«F' = — 4 irp^. Hence
if the suffix 0 be now dropped, V»VU = — 4 Trp, and Poisson's Equation
is proved. Gauss's Integral (p. 348) affords a similar proof.
A rigorous treatment of the potential U and the forces X, F, Z and their de-
rivatives requires the discussion of convergence and allied topics. A detailed treat-
ment will not be given, but a few of the most important facts may be pointed out.
Consider the ordinary case where the volume density p remains finite and the body
itself does not extend to infinity. The integrand p/r becomes infinite when r = 0.
But as dv is an infinitesimal of the third order around the point where r = 0, the
term pdv/r in the integral U will be infinitesimal, may be disregarded, and the
integral U converges. In like manner the integrals for X, Y, Z will converge
REAL VAK1ABLE8 S|0
because p(| - x)/r; etc., become InflDile ai r « 0 lo ealy Um
a.V/d{ were obuined by differenUalioa nadir
P U - 2) V^ would become inBnite to Um iMid
/^^'///h'*""^"^**^'
aa expreaeed In polar oo6rdinatee with orifln at r s 0, ai« ••«■ U
the derivatives of the forces and the Mcond derivaiivM ot iW
tained by difTerentiatlng under the dgn, are Ttlinlem
Consider therefore the following device :
al"r = -S/ ^=J''S'r*'°"/^Sr^
?xr axr '^tor J '^ U r J rU J 9a r
The last integral may be transformed into a surface lotefTBl so
It should be remembered, however, that If r = 0 within Ibo body, tlw
tion can only be made after cutting out the singularity r s 0,
gral must extend over the surface of the excised r^gloo as well as owr the
of the body. But in this case, as d3 is of the seoood order of loAollaslmols
is of the first order, the Integral over the Mirfaoe of tl
when r = 0 and the equation is valid for the whole region. la
It is noteworthy that the first integral gives the potential of Vp, that K (ko lal^
gral is formed for Vp just as (18) was from p. As Vp b a vector, tlM aMMBitfaa
is vector addition. It is further noteworthy that in Vp the diffi leattailda li wtUk
respect to x, y, z, whereas in V IT it is with respect to |, ^ f. Wow tfliH«Miato
(22) under the sign.' (Distinguish V as formed for |, f , f and «, y, i by Ti aaA %)
!!^= fll^-Pdv- fpco.a^ld8orV^.V^U= fv,\.V.pi.^fpV^l^
di^ J d^rdx J ^ ^{ r ' * J *r J ^r
or again V^.V^U = ^ Jv^i .V^t + J pV, ^'dS. W
This result is valid for the whole region. Now by Green*s rUMsala (Sa. !•, ^Mi|
Here the small region about r = 0 must again be exdsed and the
must extend over its surface. If the region be takon as a spbera
being exterior to the body, is directed along — *r. Tbns for Ibt
550 THEORY OF FUNCTIONS
where o is the average of p upon the surface. If now r be allowed to approach 0
and V«Vr-i be set equal to zero, Green's Formula reduces to
f^x- '^xpdv = f pV^- .dS + 4 irp,
where the volume integrals extend over the whole volume and the surface integral
extends like that of (23) over the surface of the body but not over the small sphere.
Hence (23) reduces to V.VU ^ — iirp.
Throughout this discussion it has been assumed that p and its derivatives are
continuous throughout the body. In practice it frequently happens that a body
consists really of several, say two, bodies of different nature (separated by a bound-
ing surface S^^) in each of which p and its derivatives are continuous. Let the
suffixes 1, 2 serve to distinguish the bodies. Then
The discontinuity in p along a surface S^^ does not affect a triple integral.
yU'-^f^dv^-f^ dSi, ^^ + f^dv^-f ^ dSa, 21.
Here^the first surface integral extends over the boundary of the region 1 which
includes the surface S-j^^ between the regions. For the interface S^^ the direction
of dS is from 1 into 2 in the first case, but from 2 into 1 in the second. Hence
^-/?^»-/^«-/^-^'^s-
It may be noted that the first and second surface integrals are entirely analogous
because the first may be regarded as extended over the surface separating a body
of density p from one of density 0. Now V»VU may be found, and if the proper
modifications be introduced in Green's Formula, it is seen that V»V?7 = — 4 7rp
still holds provided the point lies entirely within either body. The fact that p
comes from the average value Jo upon the surface of an infinitesimal sphere shows
that if the point lies on the interface S^^ at a regular point, V»V U = — ATr{\p^-\- \p^.
The application of Green's Formula in its symmetric form (Ex. 10, p. 349) to
the two functions r-?^ and U", and the calculation of the integral over the infini-
tesimal sphere about r = 0, gives
J \r r) J \r dn dn r)
/dU\ _/i
j ---dv=^J dS,,
-x/
dU\ (dU\
dS,^
(24)
{U,-U,)^-dS^^-AwU,
dn r
where S extends over all the surfaces of discontinuity, including the boundary of
the whole body where the density changes to 0. Now V»V U" = — 4 tt/o and if the
definitions be given that
dU\ /dU\ . rr TT A
drill \dnh ' 1 a
REAL VARIABUES Ul
then '^ = /^*' + /-** + /^^-^
M here the surface Int^graU eiteod over all aaHaees oi (
U appears more geueral than the initial fonn (18), aa4 ladead It b
for it takes into account the dlaoootlnoitiae of U and In 4«HvMlf«,
arlKe when p is an ordinary cootlnuoua f oaeCioii immwirtl^g, a ««taa» dlMHtatfi
of mutter. The two surface intefrals may be laterpreud m <
tioiiH. For suppose that along some surface there Is a mnfum 4mtkf # of
Tlien the first surface Integral repreaenu the poCaoiial oi the aaiMr In Ike i
Strictly spealcinf;, a surface distribution of matter with # uattB ef
Kurface Ik u physical impotwibility, but it b none the lew a eoavMl
cal fiction wlien liealing with thin sheets of maHer or with the dMifsof
upon a conducting surface. The surface distribatioa May be rspuded as a
ing case of volume distribution where p becomee Inflnlte aad tlw
out wliicli it \s Kpread beoomee Infinitely thin. In fact If 4a be
tlie Hheet of matter pdndS = ^dS. The ■eoond mrfaM iMiflsl aaf
regarilod a^ a limit. For Muppoee that there are two muUem liiMlilj mv
getlur upon one of wliich tliere is a surface density — r,aiMl vpoa tkeollMrami
density <r. Tlie potential due to the two equal superimpoMd eleaMMa 40 li dH
rj r, Vr, r,/ da r 4m r
Hence if adn = r, the potential Ukes the form rdr-^/dmi8, Jmi this east el
tribution of magnetism arises in the case of a magnelic Aell, that la» a
covered on one side with positive poles and on the other with aeptlte polsa. The
three integrals in (25) are known reepectitely aa Tolome pntartial, eiffaet ptttm-
tial, iind double surface potential.
202. The ]X)tentials niay be used to obtain partkwUr tBtafmk of
some differential equations. In the first place the eqiaftkiB
as its solution, when the integral is extended over the rogioo Uumifli-
out which/ is detined. To this particular eolntion for V oyiy In addid
any solution of Laplace's equation, but the particular aolittion b fcw-
quently precisely that particular solution whieh b desired. If the
functions U and f were vector functions so that U — 1'^ + J'*, + kf *,,
and f = i/j + i/;+ V«» ^« ^^^ ^ould be
where the integration denotes vector summation, as may he «•• hjr
adding the results for V.Vl\ =/,, V.Vi\ «/,, V.vr, »/, allsr K«W-
plication by i, J, k. If it is desired to indicate the vectorial
U and f , the potential U may be called a veeUir polsotkL
552 THEORY OF FUNCTIONS
In evaluating the potential and the forces at ($, rj, 0 due to an ele-
ment dm at (Xy ?/, z), it has been assumed that the action depends solely
on the distance r. Now suppose that the distribution p (x, y, z, t) is a
function of the time and that the action of the element pdv at (x, y, z)
does not make its effect felt instantly at ($, rj, ^) but is propagated
toward (i, rj, ^) from (x, y, z) at a velocity 1/a so as to arrive at the time
(t + (ir). The potential and the forces at (^, rj, ^) as calculated by (18)
will then be those there transpiring at the time t -\- ar instead of at the
time t. To obtain the effect at the time t it would therefore be necessary
to calculate the potential from the distribution p {x, y^z^t-^ ar) at the
time t — ar. The potential
(26)
where for brevity the variables x, y, z have been dropped in the second
form, is called a retarded potential as the time has been set back from
t Ui t — ar. The retarded 'potential satisfies the equation
P + ^ + ^-«'?=-^'^''(^"''^'^) °' ^ (27)
according as ($, iy, ^) lies within or outside the distribution p. There is
really no need of the alternative statements because if (f, rj, ^) is out-
side, p vanishes. Hence a solution of the equation
is U^^ rf(x,y,z,t-ar)^^^
47rJ r
The proof of the equation (27) is relatively simple. For in vector notation,
V.VI/ = V.V r eg)d„ + v.v f P(t-ar)-p(l)^^
J r J r
= -4,p + V.v/P<'-''^)-P<'>d,.
The first reduction is made by Poisson's Equation. The second expression may
be evaluated by differentiation under the sign. For it should be remarked that
p{t — ar) — p{t) vanishes when r = 0, and hence the order of the infinite in the
integrand before and after differentiation is less by unity than it was in the cor-
responding steps of § 201. Then
^^j. ,(t-ar)-,(t)^^^j |(z^)fMV+ [,(,_a.)_,(0]V,l}d„,
REAL VARIABLES 651
+ (- a)p'V|r.V|l + (- a)pV^r.V^l + (^<<- «r).^|i,Jf^f,!^
But V^=-V, and Vr = r/r and Tr- > s - r/r« and ?«Tr'>«t,
Hence ^i'''^i'' = 1» V|r#V|r-> = - r-t^ V^l' ■ tr-«
and r.rJ'P<^-^^)-p(0^,^J«»V%,-J'^aV«->r)^^W
It w;is seen (p. 345) tluit if F is a vector funrtion with iio««rly IImI
is, if VxF = 0, then F*^/r is an exact differential J^ ; and F Mij bt •!•
pressed as the gradient of ^, that is, as F a V^. This prohleoi aaj ako
be solved by i)otentials. For suppose
F = V<^, then V.F = V.V^, ^.zijldt^,. ^
It api^ears therefore that <f» may be expressed u a poteDtiaL TIUs sohi-
tion for <^ is less general than the former beoauae it depaaida oa th§
fact that the potential integral of V*F shall converge. MoMOver aa
the value of <^ thus found is only a {lartieular solution of T«F ■■ T*T^
it should be proved that for this ^ the relation F » T4 b aetnallj sat»
isiied. The proof will be given below. A similar method may now bs
employed to show that if F is a vector function with no divBtgaiiea,
that is, if V»F = 0, then F may be written as thi» curl of a vvctor
function G, that is, as F = V^G. For suppose
F = VxQ, then V«F = V-V-G = W-O - V.rO.
As G is to be determined, let it be supposed that V*0 — 0.
Then F = V^G gives G = j^(-^rfr. (Sf)
Here again the solution is valid only when the vector potential lulafial
of VxF converges, and it is further necessary to ahow that F — V"C
The conditions of convergence are, however, satisfied for the
that usually arise in physics.
To amplify the treatment of (88) and (29), let It be abowa that
^ iw J r Aw J f
By use of (22) it is poealble to psM the differeaUaUooe under tbe i^a ^J**|*
tioii and apply them to the functions V.F and VkF, iMlead ol to X/fmwmU
required by Leibniz's Rule (% 119). Then
554 THEORY OF FUNCTIONS
The surface integral extends over the surfaces of discontinuity of V«F, over a large
(infinite) surface, and over an infinitesimal sphere surrounding r = 0. It will be
assumed that V.F is such that the surface integral is infinitesimal. Now as VxF = 0,
VxVxF = 0 and VV.F = V.VF. Hence if F and its derivatives are continuous, a
reference to (24) shows that
V0 = / dv = F.
^TT J r
In like manner
VxG = — / dv / xdS = ( dv = F.
4ir J r Att J r Air J r
Questions of continuity and the significance of the vanishing of the neglected sur-
face integrals will not be further examined. The elementary facts concerning
potentials are necessary knowledge for students of physics (especially electro-
magnetism) ; the detailed discussion of the subject, whether from its physical or
mathematical side, may well be left to special treatises.
EXERCISES
1. Discuss the potential U and its derivative VU for the case of a uniform
sphere, both at external and internal points, and upon the surface.
2. Discuss the second derivatives of the potential, that is, the derivatives of the
forces, at a surface of discontinuity of density.
3. If a distribution of matter is external to a sphere, the average value of the
potential on the spherical surface is the value at the center ; if it is internal, the
average value is the value obtained by concentrating all the mass at the center.
4. What density of distribution is indicated by the potential e-*"* ? What den-
sity of distribution gives a potential proportional to itself ?
5. In a space free of matter the determination of a potential which shall take
assigned values on the boundary is equivalent to the problem of minimizing
6. For Laplace's equation in the plane and for the logarithmic potential — log r,
develop the theory of potential integrals analogously to the work of § 201 for
Laplace's equation in space and for the fundamental solution 1/r.
BOOK LIST
A short list of typical books with brief ooomieDli it givMi to aid Um
student of this text in selecting material for oolktefml iiilim or fm
more iidvanced study.
1. Some standard elementary differt>ntiiil and integnl
For reference the book with which the iitiuliMit in f«iniliar \a ^nAtMf
It may be added that if the Ktudent has had the mijifortunr u> tmiu
a teacher who has not led him to acquire an eaay fomal Jnwnrlwlgt of iIm
he will save a great deal of time in the long run If he mmkm op Um
and thoroughly ; practice on the exerciees In Gnuiville*e OtlaUw (Olaa §mi Cbai-
pany), or Osborne's Calculiu (Heath k Co.), U eepedallj rMOOdMB^ei.
2. B. O. Peirce, Table of Integrals (new edition). Oton aod Ccmpunj,
This table is frequently cited in the text and Is well-nigh ladlipMMMe lo Ifce
student for constant reference.
3. JaHNKE-EmDE, Funktumentafein mu tnrmnn wmti hmrrrm,
Teubner.
A very useful table for any one who has nomerlcal resnits to oblabi tram Ihs
analysis of advanced calculus. There is rery little duplicatloo biws— tMs laUs
and the previous one.
4. Woods and Bailey, Course in Mathematiet, Ginn and CamptMj.
5. Byerly, Differential Calculus and JnUffml Cnltulus, Giiin aad
Company.
6 Tod HUNTER, Differential Calculus and Inicgral Calcmims. Um^
milhiii.
7. Williamson. DtfTerential Calculus ^nA JmUfral CaUmiuM, LcNif*
mans.
These are Hlanditrtl woikt* in t^«' N'l'Mi!' -!'•:•'' '^ ... i*.
As sources for a<lditional pn)blen»> ml f : i: ; i; - i .* . -•■ i*
text they will prove useful for reft i< n. .
8. C. J. DE LA Vall^Eb-Poussin, Vours dr analyst, Gautbier-ViUar*.
There are a few books which Inspire a posiUTt affsecloo lor iMr MyW s^
beauty in addition to respect for their oootsnts, and this Is OM flf Ihow !••.
My A.lvanced Calculus Is nec6«arUy ttodar eoiiiid«mbls obllgstlfli ts 4t la Vi
Pous8iii\s Cours d' analyse, because I tanght the snb|se(o«t of IhM'
years and esteem the work more highly than any of Its ttm^mm \m •mf
656 BOOK LIST
9. GouRSAT, Cours d^ analyse. Gauthier-Villars.
10. Goursat-Hedrick, Mathematical Analysis. Ginn and Company.
The latter is a translation of the first of the two volumes of the former. These,
like the preceding five works, will be useful for collateral reading.
11. Bertrand, Calcul differentiel and Calcul integral.
This older French work marks in a certain sense the acme of calculus as a
means of obtaining formal and numerical results. Methods of calculation are not
now so prominent, and methods of the theory of functions are coming more to the
fore. Whether this tendency lasts or does not, Bertrand's Calculus will remain an
inspiration to all who consult it.
12. Forsyth, Treatise on Differential Equations. Macmillan.
As a text on the solution of differential equations Forsyth's is probably the
best. It may be used for work complementary and supplementary to Chapters
VIII-X of this text.
13. PiERPONT, Theory of Functions of Real Variables. Ginn and
Company.
In some parts very advanced and difficult, but in others quite elementary and
readable, this work on rigorous analysis will be found useful in connection with
Chapter II and other theoretical portions of our text.
14. GiBBs -Wilson, Vector Analysis. Scribners.
Herein will be found a detailed and connected treatment of vector methods
mentioned here and there in this text and of fundamental importance to the
mathematical physicist.
15. B. 0. Peirce, Newtonian Potential Function. Ginn and Company.
A text on the use of the potential in a wide range of physical problems. Like
the following two works, it is adapted, and practically indispensable, to all who
study higher mathematics for the use they may make of it in practical problems.
16. Byerly, Fourier Series and Spherical Harmonics. Ginn and
Company.
Of international repute, this book presents the methods of analysis employed
in the solution of the differential equations of physics. Like the foregoing, it gives
an extended development of some questions briefly treated in our Chapter XX.
17. Whittaker, Modern Analysis. Cambridge University Press.
This is probably the only book in any language which develops and applies the
methods of the theory of functions for the purpose of deriving and studying the
formal properties of the most important functions other than elementary which
occur in analysis directed toward the needs of the applied mathematician.
18. Osgood, Lehrhu^h der FunLtionentheorie. Teubner.
For the pure mathematician this work, written with a grace comparable only
to that of de la Vall^e-Poussin's Calculus, will be as useful as it is charming.
INDEX
(»•
raterio
a» a«, 4, 46, 162
AbePs theorem on uniformity, 4S8
Abflolnte convergence, of integnUi, 867,
360 ; of series, 422, 441
Absolute value, of complex numberi,
154 ; of reals, 85; sum of, 86
Acceleration, in a line, 13; In general,
174; probleniH on, 186
Addition, of complex numbers, 164; of
oi^emtors, 151; of vectors, 164, IftI
A(ijoint e(|uation, 240
Algebra, fundamental theorem of, 160,
30(J, 482; laws of , 168
Alternating series, 30, 420, 468
am = sin-' sn, 607
Ampere's Law, 350
Amplitude, function, 607; of complex
numbers, 164; of harmonic motion,
188
Analytic continuation, 444, 648
Analytic function, 304, 435. See Func-
tions of a complex variable
Angle, as a line integral, 297, 806; at
critical ]x)ints, 401 ; between cunrec,
0 ; in space, 81 ; of a complex number,
154; solid, 347
Angular velocity, 178, 846
Approximate formulas, 60, 77, 101, 888
Approximations, 50, 106; succenive, 106.
See Coujputation
Arc, differential of, 78, 80, 131 ; of elllpte,
77, 514 ; of hyperbola, 516. .S«e Length
Area, 8, 10, 25, 67, 77; as a line intemi,
288; by double integration, 824, 880;
directed, 167; element of, 80, 181, 176,
340, 342 ; general idea, 811; of a sur-
face, 330
Areal velocity, 175
Argument of a complex number, 164
Associative law, of addition, 163, 168 ; of
multiplication, 150, 168
Asymptotic expansi.in SftC) 307, 466
Asymptotic expn - *, 888
Asymptotic lines ;i: <>lu^ 144
Asymptotic series, 800
Attraction, 81, 68, 806, 882, 848, 647;
Law of Nature, 81, 807; motion nnder,
100, 264. See Central Force and l\>-
tential
of a hannoole
■urfae^SIO
Axaa,r1fflit^or
Axiom of eooUwUij. 81
ML Ml
B.
Bemoalll*a
Benioulira MUHbMiL 411,
Benioiilirs pnlywiili, tf I
lUimif\'» fH|uatMNi, 840
BeMel's funetioiM, 848, SM
Beu function, 8TB
Binomial tbeorMB, laHt
60 : Inflniia mrtmt ^M, „
Binomial, 88
Boundary of n NffaM. 87. 8M, 811
Boundary valoea, 8M, 641
Brachinoehrone, 404
Branch of a function, of em vmrinMa,
40; of two Tariahim, 80; of a eem-
plex variable, 408
Bnuich point, 488
C fiMCylioder
Caleohuion. Sm '
tion, etc.
Calculus of variations, 400-418
Cartorian exprmilon of vectom. 187
Catenary, 78, 100; rpvolved, 401, 488
Cauehy*s Formula, 80, 40, 81
Cauchy^s Integral, 804, 477
Cauchy's Intagnl tort, 4SI, 487
Cauatic, 148
Center, iiMtanlnneow, 74, 178; of In.
VeiHOII, 9mB
Cmf ^r of fravity or mnm, motion ef itei
r«mi or lamImM, 817, 8M i el
xmmm, 188; of Jiihiii^ 80
i t.*iii(«i it'rce, 178^ 804
CMitrodo, ind or movlnc. 74
Chain, oquIUbrtaa of, IM^ 101^ 480:
motion of, 418
Change of variable. In derivMheik t%
14. 67. 08. 108, 108; in Ulliiwlill
cquatlona, 804, 881^ oa; In
16,81,64,80.888,880
Characterietk enma, 148^ 087
CharadariMte flirifv 870
667
558
INDEX
Charge, electric, 539
Charpit's method, 274
Circle, of curvature, 72 ; of convergence,
433, 437; of inversion, 538
Circuit, 89 ; equivalent, irreducible, re-
ducible, 91
Circuit integrals, 294
Circulation, 345
Clairaut's equation, 230 ; extended, 273
Closed curve, 308; area of, 289, 311;
integral about a, 295, 344, 360, 477,
536 ; Stokes's formula, 345
Closed surface, exterior normal is posi-
tive, 167, 341; Gauss's formula, 342;
Green's formula, 349, 531 ; integral over
a, 341, 536 ; vector area vanishes, 167
en, 471, 505, 518
Commutative law, 149, 165
Comparison test, for integrals, 357 ; for
series, 420
Complanarity, condition of, 169
Complementary function, 218, 243
Complete elliptic integral, 607, 614, 77
Complete equation, 240
Complete solution, 270
Complex function, 157, 292
Complex numbers, 153
Complex plane, 157, 302, 360, 433
Complex variable. See Functions of a
Components, 163, 167, 174, 301, 342, 507
Computation, 59 ; of a definite integral,
77; of Bernoulli's numbers, 447; of
elliptic functions and integrals, 475,
507, 514, 522; of logarithms, 59; of
the solution of a differential equation,
195. See Approximations, Errors, etc
Concave, up or down, 12, 143
Condensation point, 38, 40
Condition, for an exact differential, 105 ;
of complanarity, 169 ; of integrability,
265 ; of parallelism, 166 ; of perpendic-
ularity, 81, 165. See Initial
Conformal representation, 490
Conformal transformation, 132, 477, 638
Congruence of curves, 141
Conjugate functions, 536
Conjugate imaginaries, 156, 643
Connected, simply or multiply, 89
Consecutive points, 72
Conservation of energy, 301
Conservative force or system, 224, 307
Constant, Euler's, 385
Constant function, 482
Constants, of integration, 15, 183; phys-
ical, 183 ; variation of, 243
Constrained maxima and minima, 120,
404
Contact, of curves, 71 ; order of, 72 ; of
conies with cubic, 621 ; of plane and
curve, 82
Continuation, 444, 478, 642
Continuity, axiom of, 34 ; equation of,
350; generalized, 44; of functions, 41,
88, 476; of integrals, 52, 281, 368; of
series, 430 ; uniform, 42, 92, 476
Contour line or surface, 87
Convergence, absolute, 357, 422, 429;
asymptotic, 456; circle of, 433, 437;
of infinite integrals, 352 ; of products,
429; of series, 419; of suites of num-
bers, 39 ; of suites of functions, 430 ;
nonuniform, 431 ; radius of, 433 ; uni-
form, 368, 431
Coordinates, curvilinear, 131 ; cylindri-
cal, 79; polar, 14; spherical, 79
cos, cos-i, 155, 161, 393, 456
cosh, cosh-i, 5, 6, 16, 22
Cosine amplitude, 507. See en
Cosines, direction, 81, 169 ; series of, 460
cot, coth, 447, 450, 454
Critical points, 477, 491 ; order of, 491
CSC, 550, 557
Cubic curves, 519
Curl, Vx, 345, 349, 418, 553
Curvature of a curve, 82 ; as a vector,
171 ; circle and radius of, 73, 198 ;
problems on, 181
Curvature of a surface, 144 ; lines of, 146 ;
mean and total, 148 ; principal radii,
144
Curve, 308 ; area of, 311 ; intrinsic equa-
tion of, 240 ; of limited variation, 309 ;
quadrature of, 313 ; rectifiable, 311.
See Curvature, Length, Torsion, etc.,
and various special curves
Curvilinear coordinates, 131
Curvilinear integral. See Line
Cuspidal edge, 142
Cuts, 90, 302, 362, 497
Cycloid, 76, 404
Cylinder functions, 247. See Bessel
Cylindrical coordinates, 79, 328
D, symbolic use, 152, 214, 279
Darboux's Theorem, 51
Definite integrals, 24, 52; change of
variable, 54, 65 ; computation of, 77 ;
Duhamel's Theorem, 63 ; for a series,
451; infinite, 352; Osgood's Theorem,
54, 65 ; Theorem of the Mean, 26, 29,
52, 359. See Double, etc.. Functions,
Infinite, Cauchy's, etc.
Degree of differential equations, 228
Del, V, 172, 260, 343, 345, 349
Delta amplitude, 507. See dn
De Moivre's Theorem, 165
Dense set, 39, 44, 50
Density, linear, 28; surface, 316; vol-
ume, 110, 326
Dependence, functional, 129; linear, 246
Derivative, directional, 97, 172; geo-
metric properties of, 7; infinite, 46;
INDEX
loKarlflimIc, 6; nomuU, W, 1S7, 17i;
of hi^'her order, II, 67, 10S, 191 ;oi
inte^'ralH, 27, 62, 288, 870 ; of prodoeU,
11, 14,48; of iierieiiiennbyt«nn,4S0;
of vectofK, 170; onllimry, 1, 46, 168;
partial, m, W ; right or left, 46; Tl»6-
ori'in of till) Mean, 8, 10, 4A, M. iS«
(Miaii^e of variiiblu, FuiicUooSi eto.
Dt'rivt'd uiiitJH, 109
DrU'niiiiKiiiLH, functional, 190; Wron-
skiaii, 241
Devt^lopable Kurfaci-, 141, 148, 148, 270
DIffereiUH'H, 4i». 4«12
Differentiiihlr finittion, 46
Difft^rential, 17, (>1 ; txact, 106, 264, ^KX> ;
<.f arc, 70, 80, 131 ; of area, 80, 181;
of heat, 107, 2^)4; of higher order, 67,
104; of 8urfacf, 840; of volume, 81,
:«0; of work, 107, 202; pMtiml, 06,
104 ; total, U5, 08, 106, 208, 206; ireo-
tor, 171, 203, 342
Differential equations, 180, 267; degree
f»f, 228 ; onier of, 180 ; Kolution or
integration of, 180 ; complete 8olution,
270; general 8<)lnti<»n, 201, 230, 200;
infinite Bolntion, 230; particular ftnlu-
tion, 230; singular solution, 231, 271.
See Ordinary, Partial, etc.
Differential ecjuations, of electric cir-
cuit.s, 222, 220 ; of in.ohanic*, 186, 20S ;
Hamilton's, 112 ; Lagrange's, 112,224,
413; of media, 417; of physics, 624;
of strings, 185
Differential geometry, 78, 131, 148, 412
Differentiation, 1; logarithmic, 6; of
implicit functions, 117; of intends,
27, 283 ; partial, 93 ; U>tal, 96; under
the sign, 281 ; vector, 170
Dimensions, higher, SSH; physic&l, 100
Direction cosines, 81, 1H9; of a line, 81 ;
of a normal, 83 ; of a tangent, 81
Directional deriva-tive, 97, 172
I)isc(mtinuity, amount of, 41, 462 ; finite
or infinite, 479
Dissipative functi<m, 225, 807
DisUince, shortest, 404, 414
Distributive law, 151, 165
Divergence, fonnula of, 342 ; (»f an inte-
gral, 352 ; of a serien, 419 ; of ji vector,
343, 553
Double integrals, 80, 131, 313. 316, 372
Double integration, 32, 285, 310
Double linjits, 89, 430
Double i>oints, 119
Double sums. 315
Double surface ix»l4»ntial, 551
Doubly periodic functions, 417, 480,
604, 517; order of, 487. Ste p, tn,
en, dn
Duhamel's Theorem, 28, 03
Dupin's indicatrix, 146
t«S.71S...,A.4S7
••.••, 4, 160, 411. IK mi^
I4i
410
417
tliKwimrtu Uwety. lift, 4
BlMBML l&Ml, Itl. 01 i of
840; ct '
207
Kllipie,uoo|,n.6l4
KllipUe fonetloM, 471, IM. M7. Ml, 017
Elliptic Inugnta, Mi, 107, 611. ill. OIT
Energy, comamikm of« 801 ; QIbh
ikNM of, 110; kioecie, 18, IOI« lit.
178,204.418; ol a gM, 100. 94. «8 ;
of ft laminmS18; poMNiy, 107. SOC.
801, 418, 647 ; prladple of. 001 s ««tt
and, 280, 001
Bntropj, 100. 1»«
EntelopM, of eorvwi, 101^ 141. m ; ««
lined elenenu. 108; of ^kumt el*.
menta, 164, 007 ; of piaiMS, 140, 108;
of Murfaeea, 180, 140. 071
Bqaatioo, adjoint, 040; alfiktakt, 100,
806,488; BemoalU^a^OOMlO: Oilfw
aut*a,l^tlt;
dc, 240 ; Laplaea*a, f04 ; o« (
860; PoiaKm's, 648;
RiccaU*s,t60: wave, 270
Equations, Hamilloa'a. 1 18 ; LMiavftX
112, 226, 418. 8m Dlfeiwdd
Uooa, Ordinary. Paitlal, aia.
Bqulcreseeot Tariabta, 40
RqailibriiuB of ttfli^ 101^ lOH
BqulpoCMlial llM or aufaoa, 01,
KqoifalMKelroait^Ol
Error, avenge, 000; tmKiiam, #. 000;
mean Kfoaia, 000, 400; la laipt
mactlee, 000; prohaMa, 000; pnae»
bility of
ErroiB,of
EMOtial ilivriMl^. 470. 401
Biiler*aOoiMttal,W^ai
Rolar^ PMTWila, 100. 180
Euler*s nombtia, 480
Kuler't tranrfonaattoa, 440
EvaluaUon of lBli«fala. OK
811. Set
Krea f onelloa.
000:«Ma,IOI
140, IN
liii^Hii loi^oKio*
kMBiial MttaliMi. 007. 087.001
660
INDEX
Expansion, asymptotic, 890, 397, 466;
by Taylor's or Maclaurin's Formula,
67, 306; by Taylor's or Maclaurin's
Series, 436, 477 ; in ascending powers,
433, 479 ; in descending powers, 390,
897, 466, 481; in exponentials, 465,
467 ; in Legendre's polynomials, 466 ;
in trigonometric functions, 458, 466;
of solutions of differential equations,
198, 260, 626. See special functions
and Series
Exponential development, 465, 467
Exponential function. See a% e^
F, complete elliptic integral, 507, 514
F(0, A:) = sn-i sin 0, 607, 614
Factor, integrating, 207, 240, 254
Factorial, 379
Family, of curves, 136, 192, 228 ; of sur-
faces, 139, 140. See Envelope
Faraday's Law, 360
Finite discontinuity, 41, 462, 479
Flow, of electricity, 663 ; steady, 663
Fluid differentiation, 101
Fluid motion, circulation, 345 ; curl, 346 ;
divergence, 343 ; dynamical equations,
361 ; equation of continuity, 350 ; ir-
rotational, 533 ; velocity potential,
633 ; waves, 629
Fluid pressure, 28
Flux, of force, 308, 348 ; of fluid, 343
Focal point and surface, 141
Force, 13, 263; as a vector, 173, 301;
central, 175; generalized, 224; prob-
lems on, 186, 264. See Attraction
Form, indeterminate, 61, 89; perma-
nence of, 2, 478; quadratic, 116,
145
Fourier's Integral, 377, 466, 628
Fourier's series, 458, 465, 625
Fractions, partial, 20, 66. See Rational
Free maxima and minima, 120
Frenet's formulas, 84
Frontier, 34. See Boundary
Function, average value of, 333; ana-
lytic, 304; complementary, 218, 243;
complex, 157, 292; conjugate, 636;
dissipative, 226, 307 ; doubly periodic,
486 ; ^-function, 62 ; even, 30 ; Green,
635; harmonic, 630; integral, 433;
odd, 30 ; of a complex variable, 167 ;
periodic, 468, 485 ; potential, 301. See
also most of these entries themselves,
and others under Functions
Functional dependence, 129
Functional determinant, 129
Functional equation, 45, 247, 262, 887
Functional independence, 129
Functional relation, 129
Functions, series of, 480; table of ele-
mentary, 162. For special functions
see under their names or syinbols ; for
special types see below
Functions defined by functional equa-
tions, cylinder or Bessel's, 247 ; ex-
ponential, 46, 387 ; Legendre's, 262
Functions defined by integrals, contain-
ing a parameter, 281, 368, 376 ; their
continuity, 281, 369; differentiation,
283, 370 ; integration, 286, 370, 373 ;
evaluation, 284, 286, 371; Cauchy's
integral, 304 ; Fourier's integral, 377,
466 ; Poisson's integral, 541, 546 ; po-
tential integrals, 646; with variable
limit, 27, 63, 209, 256, 296, 298; by
inversion, 496, 603, 617; conjugate
function, 636, 542 ; special functions,
Bessel's, 394, 398 ; Beta and Gamma,
378; error, V, 388 ; J5: (0, A;), 514; ^(0,*:),
607 ; logarithm, 302, 306, 497 ; j9-f unc-
tion, 617; sin-i, 307, 498; sn-i, 435,
603; tan- 1, 307, 498
Functions defined by mapping, 643
Functions defined by properties, con-
stant, 482 ; doubly periodic, 486 ; ra-
tional fraction, 483; periodic or
exponential, 484
Functions defined by series, p-f unction,
487 ; Theta functions, 467
Functions of a complex variable, 158,
163; analytic, 304, 435; angle of,
159; branch point, 492; center of
gravity of poles and roots, 482 ;
Cauchy's integral, 304, 477 ; con-
formal representation, 490 ; continu-
ation of, 444, 478, 642 ; continuity,
168, 476 ; critical points, 477, 491 ; de-
fines conformal transformation, 476;
derivative of, 168, 476 ; derivatives of
all orders, 305 ; determines harmonic
functions, 536 ; determines orthogonal
trajectories, 194 ; doubly periodic, 486 ;
elementary, 162 ; essential singularity,
479, 481; expansible in series, 436;
expansion at infinity, 481 ; finite dis-
continuity, 479 ; integral, 433 ; integral
of, 300, 360 ; if constant, 482 ; if ra-
tional, 483 ; inverse function, 477 ; in-
version of, 543 ; logarithmic derivative,
482 ; multiple valued, 492 ; number of
roots and poles, 482; periodic, 485;
poles of, 480 ; principal part, 483 ; resi-
dues, 480 ; residues of logarithmic de-
rivative, 482 ; Riemann's surfaces,
493; roots of, 168, 482; singularities
of, 476, 479; Taylor's Formula, 305;
uniformly continuous, 476 ; vanishes,
168. See various special functions
and topics
Functions of one real variable, 40;
average value of, 333 ; branch of, 40 ;
Cauchy's theorem, 30, 49 ; continuous,
DTDEX
Ml
41; continuoiui over denae
^ 44.
Darlxiiix'sTlitoriiii m '(•••-'vAUvcof'
by bouittr'tfiibhtM,402; exptnudon by
Lexeiulre'a polynomial*, 406; ex|»i>.
sion by Taylor's Formula, 40, 66;
expaimion by T:iylor*B8erifla,486; «x.
presHion n.s Kourler'a Intagral, 877.
4(W; im-n-:i.sin«, 7, 46, 810, 408: in.
tinii(>, 41; iiitiiiite derivative, 46; lnt«-
grable, 52, 64, 810; integral of, 16, 84,
52; inverse of, 46; limited, 40; limit
of, 41, 44 ; lower Mum, 61 ; maxima and
iniuima, 7, 9, 10, 12, 40, 4ii, 40, 76;
multiple valuoil, 40; not drcreMlnff,
54, SlU; of limited variati.Mi, 64,800,
402 ; OHcillatioi), 40, M ; UoHf^sTbeo*
rem, 8, 40; riKht-liaiul or left-hand
derivative or limit, 41, 40, 40, 408;
single valued, 40; theorems of the
mean, 8, 25, 29, 4«, 61, 62, 869; uni-
formly continuous, 42 ; unlimited, 40;
upper sum, 51 ; variation of, 800, 401.
410. .See various special topioa and
functions
Functions of several real vn»-i«»'i«- «7;
average value of, 834, .: -h
of, 90; continuity, 88; «• ;ifs
and surfaces, 87; differentiation, 93,
117; directional derivative, 97; double
limits, 89, 430 ; expansion by Taylor's
Formula, 1 13; gradient, 172; haniionic,
530; homoireneous, 107; implicit, 177;
integral of, 815, 320, JiT), 340; Inte-
gration, 319, 327; inverw, 124: maxima
and minima, 114, 118, 120, 12.'); mini-
max, 115; multiple-valued, JHJ; nnrmal
derivative, 97 ; over various regions,
91; potential, 547; slnele-vahiiMl, 87;
solution of, 117; spa< 72;
toUil differential, 9-1 11
by, 131; Theorem •-; i,„ .m,.ii.. i*4;
uniformly continuous, 91 ; variation
of, 90
Fundamental solution, 534
Fuiulamental theorem of alf^ebra, 160,
306
Fundamental units, 100
878 ; as a pfoduct,
-rpresBion,lfe8, 460;
1 nteinids in terms
f, 888; SUriing's
Gamma function,
458 ; asyinv'"'''
beta funct
of, 380; 1-,
Formula, 380
Gas, air, 189 ; molecules of a, 808
Gauss's Fonnula, 342
Gauss's Integral, 348
gd, gd-», 0, 10, 450
General solution, 201, 880, 800
Geodesicii, 412
Oaowstfy. MmC9n%
allsprcUl Uiaiai
Oradirm. V, ltl,8il. As IM
•u. «
t»rr*n ru
Oratn's
OraMi*a
G<
GTimiloa,
.IS. an
<iM
Ki^m
Half DMlodaofUMA
Uainilloa*s •qnaOaiM, IIS
Hamilton*! prindplc, 418
lUmoiiie f nnotioM, 6iO; a
«l;am|iMBl«f(
684; Green ramHiL m^i
of, 684; lllvmlMl0l,|»;
and minimura, 881, 6M
tMTal,641, M«
giUaritica, 884
Halioold, 418
Helix, m, 401
HelmlMlU, 881
Hiirher dimamlnni
Higher order, di
anit««imaU, 84, 880
Homogendtj, phyikAl, 100
107
i, 108, 180
i. iw. 187
liydnMijFMUBki. SmWlfdd
UjperboUefiiBctloM,8.
ete.
HypeifMaMirio
Imaginary, 168, 810
Imagii>»rv iMtwcfi, 161
Impli ria. 117-188.
Imit lua, MinioMU
Indefinite iutcgrml, lid 88. as
8ii7 oTpiUli, 800
IndeCenDloaie fonna, 61 ; L*
Rule, 61 ; In two vaxlalOa^
lodknlrix, DuplaX Itf
Indices, Uw of. 180
Induction, 800, 040
Inequalities,
Inertia. Sm
Infinite, 08: ■■.-■. ..^
lnflnluderifnllv«,40
InflaHe IntMiBl,
Infinite prodnci»
180
8mUMM'
5.62
INDEX
Infinite series, 39, 419
Infinite solution, 230
Infinitesimal, 63 ; order of, 63 ; higher
order, 64 ; order higher, 356
Infinitesimal analysis, 68
Infinity, point at, 481
Inflection point, 12, 75 ; of cubic, 521
Instantaneous center, 74, 178
Integrability, condition of, 255 ; of func-
tions, 52, 368
Integral, Cauchy's, 304; containing a
parameter, 281, 305; definite, 24, 51 ;
double, 315 ; elliptic, 503 ; Fourier's,
377; Gauss's, 348; higher, 335; in-
definite, 15, 53 ; infinite, 352 ; inver-
sion of, 496; line, 288, 311, 400;
Poisson's, 541 ; potential, 546 ; sur-
face, 340 ; triple, 326. See Definite,
Functions, etc.
Integral functions, 433
Integral test, 421
Integrating factor, 207, 240, 254
Integration, 15 ; along a curve, 291, 400
by parts, 19, 307 ; by substitution, 21
constants of, 15, 183 ; double, 32, 320
of functions of a complex variable
307 ; of radicals of a biquadratic, 513
of radicals of a quadratic, 22 ; of ra-
tional fractions, 20 ; over a surface,
340 ; term by term, 430 ; under the
sign, 285, 370. See Differential equa-
tions, Ordinary, Partial, etc.
Intrinsic equation, 240
Inverse function, 45, 477 ; derivative of,
2, 14
Inverse operator, 150, 214
Inversion, 537 ; of integrals, 496
Involute, 234
Irrational numbers, 2, 36
Irreducible circuits, 91, 302, 500
Isoperimetric problem, 406
Iterated integration, 327
Jacobian, 129, 330, 336, 476
Jumping rope, 511
Junction line, 492
Kelvin, 351
Kinematics, 73, 178
Kinetic energy, of a chain, 415; of a
lamina, 318; of a medium, 416 ; of a
particle, 13, 101 ; of a rigid body, 293 ;
of systems, 112, 225, 413
Lagrange's equations, 112, 226, 413
Lagrange's variation of constants, 243
Lamina, center of gravity of, 317;
density of, 315 ; energy of, 318 ; kine-
matics of, 78, 178; mass of, 32, 316;
moment of inertia of, 32, 315, 321;
motion of, 414
Laplace's equation, 104, 110, 626, 630,
533, 548
Law, Ampere's, 350; associative, 150,
165; commutative, 149, 165; distrib-
utive, 160, 166; Faraday's, 350;
Hooke's, 187 ; of indices, 150 ; of
Nature, 307 ; parallelogram, 154, 163,
307 ; of the Mean, see Theorem
Laws, of algebra, 153; of motion, 13,
173, 264
Left-hand derivative, 46
Left-handed axes, 84, 167
Legendre's elliptic integrals, 603, 611
Legendre's equation, 252 (Ex. 13 5) ; gen-
eralized, 526
Legendre's functions, 252
Legendre's polynomials, 252, 440, 466 ;
generalized, 527
Leibniz's Rule, 284
Leibniz's Theorem, 11, 14, 48
Length of arc, 69, 78, 131, 310
Limit, 35 ; double, 89 ; of a quotient,
1, 45; of a rational fraction, 37; of a
sum, 16, 50, 291
Limited set or suite, 38
Limited variation, 54, 309, 462
Line, direction of, 81, 169; tangent,
81 ; normal, 96 ; perpendicular, 81,
165
Line integral, 288, 298, 311, 400 ; about a
closed circuit, 295, 344 ; Cauchy's, 304 ;
differential of, 291 ; for angle, 297 :
for area, 289 ; for work, 293 ; in the
complex plane, 360, 497 ; independent
of path, 298 ; on a Riemann's surface,
499, 503
Lineal element, 191, 228, 231, 261
Linear dependence or independence,
245
Linear differential equations, 240 ;
Bessel's, 248; first order, 205, 207;
Legendre's, 262 ; of physics, 524 ; par-
tial, 267, 275, 524 ; second order, 244 ;
simultaneous, 223 ; variation of con-
stants, 243 ; with constant coefficients,
214, 223, 275
Linear operators, 161
Lines of curvature, 146
log, 4, 11, 161, 302, 449, 497 ; log cos, log
sin, log tan, 450 ; — log r, 635
Logarithmic differentiation and deriv-
ative, 6; of functions of a complex
variable, 482 ; of gamma function,
382 ; of theta functions, 474, 512
Logarithms, computation of, 69
M-test, 432
Maclaurin's Formula, 67. See Taylor's
Maclaurin's Series, 436
Magnitude of complex numbers, 164
Mapping regions, 643
INDEX
MaM, 110; of lamina, 310, 8S; of rod,
28; of flolid, 836; polentiAl of A,
308, 348, 627. 8ee Center of grmvltr
Maxima and iiiinima, oomlimin«l, ISO,
404; free, 120; of functions of one vari-
able, 7, 9, 10, 12, 40, 43, 40, 75 ; of func-
tioiiHof Hfveral variahh-s, 114, 118,190,
125; of harinuiiic ftiiit:iioii«, 631; of
implicit fuiictioiiH, 118, 190, 125; of
integrals, 400. 404, 400; of teUof num-
bers, 38; relative, 120
MaxwelpK assumptiun for glMt, 800
Mayer's method, 268
Mean. See Theorem of the Mean
Mean curvature, 148
Mean error, 390
Mean square error, 390
Mean value, 333, 340
Mean velocity, 392
Mechanics. See Kquilibrium. M..ii..ii,
etc.
Medium, elastic, 418; ether, 417. See
Fluid
Meusnier's Theorem, 146
Minima. See Maxima and minima
Minimax, 115, 119
Minimum surface, 415, 418
Modulus, of complex immber, 154 ; of
elliptic functions, k, Ar', 505
Molecular velocities, 392
Mouient, 176; of momentum, 170, 204,
325
Moment of inertia, curve of minimum,
404; of a lamina, 32, 315, 824; of a
particle, 31 ; of a solid, 328, 881
Momentum, 13, 173; moment of, 170,
204, 326; principle of, 204
Mon^'e's methcMl, 270
Motion, central, 176, 204; Hamilton's
equations, 112; Hamilton's Principle,
412 ; in a plane. 204 ; La^ran^e's equa-
tions, 112, 226, 413; of a chain, 415;
of a drumbeatl, 620 ; of a dynamical
system, 413 ; of a lamina, 78, 178, 414 ;
of a medium, 41(J; of the simpl(> |hm»-
dulum, 609; of systems of i>iiriirle.H,
176; rectilinear. i>**- -">■"•<• harmonic,
188. See Fluid. itions, etc.
Multiple-valued fi; »", W, 4l»2
Multiplication, by n.mpley numbem,
155; of series, 442 ; of vector*. 164
Multiplier, 474 ; n
Multipliers, meth
411
Multiply connected rsglons, 80
Newton's Second Law of Motion, 18, 178,
186
Normal, principal, 88 ; to a closed sur-
face, 167, 341
Normal derivative, 97, 187, 172
Normal lias, 8, 00
normal plaas, 181
Numbem, Bsraoitmx M8: oaailM.
158; Kol«'«.4M; f nmils^. iTiSsS!
Tal of, 84 : InatloMl, 8, 18; Mai, M,
SBis or sbUm oI, 88
Ohsenration, •rrots ot. 8M. hmsH ••«
n»nt, 101
(Md (uuctioo, 80
Operaikm,148
0p««ti0Ml WStlMMis, 814. 988. 87^ 447
Opermior, 140, \U, 178 ; teHkMit^ m
linear, 151 ; iavmi, 1|8^ tt4; tei«^
utory, 158 ; vecior^lfctmMlaili^ 178.
980, 848, 845, 840
Order, of eritlcal point, 481 : ol 4ssH.
aUvss, 11; of diSsfmMlsk 87: «f
difTereotial eqaatiowi. 18^ 4siiUt-
periodle functkw^ 487 ;q| N^y.
infinites, 'oo ; oC pola, 488
ordiiuuy illffiirnnilsl •^fmtkm^ 888;
approiimate solotioaa, 181^ If
inic from partial, 584 ; BermmlU's. Wk
; Clalraut's, 980; exact, 887. 887;
210
804,910,
gnuli«lBe«4Nrfor,807; Uma^dkmmm
of, 101; Uiisar,ssaIiBsar;«ffeMv
degree, 998; of iaflMr«f4sr,8M:9S
lems involTin^ 178; Bieesiri^ tM;
sjsteoM of, 9n^ 880; fmriabim stfia-
rable, 908. 8te Solotlon
Ortbosooal tnJseCottos, plane, 104, 884,
OithogomU Umasf ormaikm, 108
OsciilaaQKdrel«,78
OseulaUM plane, 88, 140. 148, 171. 419
Oq{00d*iTlieoram, 84, 88.888
o-functloQ, 487, 517
Pappas*s llieorsm, 889, 848
l>aralielepipsd, volaae of, 180
l^arallelism, ooodHloo of, 188
I*arsilelofnm, law of addltlim, IK 188.
807; orpariods. 488; vaelor ama «f,
105
l>aramecer, 185; lacamls witli a, 881
l*artial deriTaUves, 88; hl«bi-r or^lvr,
102
l^artial differentials, 08, lui
l*artial diflersBtlal eqoalioM. »T , c aar^
acterlsUcs of, 98T, 9T8; ClMffli^
mecbod, 974; Uf mss el mthtm,
880; Lapla«t*s.888; lfaM«;«i;97l^
584 ; Moi«t*amtlkod,978; (
684: FolsBoa*a.848
PartlsJ dUfersBllatlom 88, MB;
of variable, 08. 108
ParUal f rsctlom, 80. 88
Puticalari
564
INDEX
Path, independency of, 298
Pedal curve, 9
Period, half, 468 ; of elliptic functions,
471, 486; of exponential function, 161;
of theta functions, 468
Periodic functions, 161, 458, 484
Permanence of form, 2, 478
Physics, differential equations of, 524
Planar element, 254, 267
Plane, normal, 81 ; tangent, 96 ; oscu-
lating, 82, 140, 145, 171, 412
Points, at infinity, 481 ; consecutive, 72 ;
inflection, 12, 76, 521 ; of condensation,
38, 40 ; sets or suites of, 380 ; singular,
119, 476
Poisson's equation, 548
Poisson's Integral, 641
Polar coordinates, 14, 79
Pole, 479; order of, 480 ; residue of, 480 ;
principal part of, 483
Polynomials, Bernoulli's, 451 ; Legen-
dre's, 252, 440, 466, 627; root of, 159,
482
Potential, 308, 332, 348, 627, 630, 539,
547 ; double surface, 651
Potential energy, 107, 224, 301, 413
Potential function, 301, 647
Potential integrals, 546 ; retarded, 612 ;
surface, 651
Pov^er series, 428, 433, 477 ; descending,
389, 397, 481
Powers of complex numbers, 161
Pressure, 28
Principal normal, 83
Principal part, 483
Principal radii and sections, 144
Principle, Hamilton's, 412 ; of energy,
264 ; of momentum, 264 ; of moment
of momentum, 264; of permanence
of form, 2, 478 ; of work and energy,
293
Probability, 387
Probable error, 389
Product, scalar, 104; vector, 165; of
complex numbers, 155; of operators,
149 ; of series, 442
Products, derivative of, 11, 14, 48; in-
finite, 429
Projection, 164, 167
Quadratic form, 115, 145
Quadrature, 313. See Integration
Quadruple integrals, 335
Quotient, limit of, 145; of differences,
30, 61 ; of differentials, 64, 67 ; of power
series, 446; of theta functions, 471
Raabe's test, 424
Ratlins, of convergence, 433, 437; of cur-
vature, 72, 82, 181; of gyration, 834;
of torsion, 83
Rates, 184
Ratio test, 422
Rational fractions, characterization of,
483 ; decomposition of, 20, 66 ; inte-
gration of, 20 ; limit of, 37
Real variable, 35. See Functions
Rearrangement of series, 441
Rectifiable curves, 311
Reduced equation, 240
Reducibility of circuits, 91
Regions, varieties of, 89
Relation, functional, 129
Relative maxima and mimima, 120
Remainder, in asymptotic expansions,
390, 398, 456; in Taylor's or Mac-
laurin's Formula, 55, 306, 398
Residues, 480, 487 ; of logarithmic de-
rivatives, 482
Resultant, 154, 178; moment, 178
Retarded potential, 552
Reversion of series, 446
Revolution, of areas, 346; of curves,
332 ; volume of, 10
Rhumb line, 84
Riccati's equation, 250
Riemann's surfaces, 493
Right-hand derivative, 46
Right-handed axes, 84, 167
Rigid body, energy of a, 293; with a
fixed point, 76
Rolle's Theorem, 8, 46
Roots, of complex numbers, 155 ; of
polynomials, 156, 159, 306, 412; of
unity, 166
Ruled surface, 140
Saddle-shaped surface, 143
Scalar product, 164, 168, 343
Scale of numbers, 33
Series, as an integral, 461 ; asymptotic,
390, 397, 456; binomial, 423, 425;
Fourier's, 415; infinite, 39, 419; ma-
nipulation of, 440 ; of complex terms,
423 ; of functions, 430 ; Taylor's and
Maclaurin's, 197, 435, 477; theta,
467. See various special functions
Set or suite, 38, 478 ; dense, 39, 44, 60
Shortest distance, 404, 412
Sigma functions, o-, ca, 523
Simple harmonic motion, 188
Simple pendulum, 509
Simply connected region, 89, 294
Simpson's Rule, 77
Simultaneous differential equations, 223,
260
sin, sin-i, 3, 11, 21, 166, 161, 307, 436,
453, 499
Sine amplitude, 507. See sn
Single-valued function. 40, 87, 296
Singular points, 119, 476
Singular solutions, 230, 271
INDEX
Singularities, of functloni of a oomplax
Tariable, 476, 47»; nf hanitfiii|<< funo*
tioriH, 584
sinli, 8lnh-i, 6, 4&i
Slope, uf a curve, 1 ; u( a function, 801
Small errors, 101
Small vibralioiiN, 224, 41&
sn, 811- », 471, 475, 608, 607, 611, 617
Solid iiU'^U', 347
Solution of (tifferenUml equatiouM, com-
plete, 27U; general, 260: iiifitiit.-. 2.10
particular, 230/524 ; hi :
Solution of implicit fun.
Speed, 178
Spherical co6rdinateM, 79
Sterling's approximation, SiH\, 458
StokehV Formula, 345, 418
Strings, equilibrium of, 185
Subnormal and subtangent, 8
Substitution. See Change of variable
SuccesHive appnixiinatlona, 196
Successive difTerencea, 49
Suite, of luimbers or pointa, 88 ; of func-
tions, 430 ; uniform conveigence, 481
Sum, limit of a, 3<{, 24, 61, 410; of a
series, 41U. St-e Addition, Definite in-
tegral, Serieii, etc.
Superposition of small vibrationa, 896,
525
Surface, area of, 67, 339; cloeed, 167,
341; curvature of, 144; developable,
141, 143, 148, 279; element of, 840;
geodesies on, 412; minimum, 4(M, 416;
normal to, 96. 341; Kiemann'a, 408;
rule<l, 140; Ungent plane, 96; types
of, 209; vector, 167; mv, 492
Surface integral, 340, 347
Symbolic methods, 172, 214, 888, 860,
275, 447
Systems, conservative, 801; dynamical,
413
Systems of differential equatlooa, 888,
260
Un, tan-i, 8, 21, 807, 460. 467. 406
Tangent line, 8, 81, 84
Tangent plane, 96, 170
tanh, Unh-i, 6, 6, 460, 601
Taylor's Formula, 55, 112, 152, 806, 477
Taylor's Series, 197, 486, 477
Taylor's Theorem, 49
Test, Cauchy'a, 421; comparison, 480;
Ka&be's, 424; raUo, 422; Weieratrass's
Af-, 432. 466
Test function, 866
Theorem of the Mean, for derivativea,
8, 10, 46, 94 ; for integrals, 86, 80, M
359
Thermodtnamlca, 106, 894
TheiA f unit i. »ns, H, H.,e, 8,, as FMrier*t
jkTits, 4«tT iis urxjductJi. 471 . ' "*
diiptlc functUms, 471, M4; |MHiyb>
Mle dMivaiivv, 474. il8 ; MdZaS
iMlf ftfffMk, 4fli -fSiliJm^mmm
4T1;
Omm,$.0^m
Tonion, 68; radius oC, «, |U
Tblal flvnratttia, 146
Total dllifiMlal, 81. 01, U
Tolal dlflbivntlal
Total dlfarratiatlo^ 86
Tratectory, 196
IKM.
TnuMformatloo, nmifntaMl. 188. 488-
KulerX440; of laTOTiloi^lt7;<
onal, 100; of a plaaa, 181 ; lo
14, 79
Trigonomtriefi
TripUlntflffraUSaa
Umbilio, 148
UndeleniiiMd cwflcisnu, lyu
UndMarmlMd BulUpUor, 180. 186, Mi.
Uniform ooollaailgr, 68, 88, 476
Uniform confaifaiica, 868, ai
Unlta. fnnilfMiul and dw«v«d, 168;
dimeiMloM of, 108
Unity, rooto of, 166
Unlimited aet or aulla, 88
Vall4e>P0ii«in, do la, 878. 6tt
Value. 6m Abtoliila, A!
Variabia, oomplMt, 167;
46; real,86. fiat Cbaag*'
Variable limiu for Intsfiali, 87, 481
Variablea, aeparabla, 178), 808. «■
Punctiona
VariaUoo, 178; of a fiUMlloii, 8. It. M;
limitod,64,800: of
VariatkNia.oalottlaaof,401 ;of I
401, 410
Vector. 164. 168; ■ecalaratloa. 174 ; awa,
167, 890; compo«mim of a, 168, 167.
174, 8g; cnrramra, pi ;_■"■■ ■'*
1 7A ; mofneoi of moaMMMa. IrB;
n. 178; tondoii, 88, ITl
73
Vcvtur atiuitioa, 164. 168
VaetordMhwUaik»n,17».880.8M.8a;
foraa.178
•otorffi
Veetor functlow, 860^ 888, 88% Mi, 841^
661
Veelor operator y. Ma Del
Vector plodoc^ 161^ 161^ 646
Vaeton, addMmi of, 166. M8: «m».
r, 168; ii liUMlliln al, 166.
pioiStof. 164,' 1% !«, 8U; pt^
keCkMM of. 164. 167. 868
566
INDEX
Velocity, 13, 173 ; angular, 346 ; areal,
175 ; of molecules, 392
Vibrations, small, 224, 626; superposi-
tion of, 226, 524
Volume, center of gravity of, 328 ; ele-
ment of, 80 ; of parallelepiped, 169 ;
of revolution, 10 ; under surfaces, 32,
317, 381 ; with parallel bases, 10
Volume integral, 341
Wave equation, 276
Waves on water, 529
Weierstrass's integral, 517
Weierstrass's 3f -test, 432
Weights, 333
Work, 107, 224, 292, 301 ; and energy,
293, 412
Wronskian determinant, 241
z-plane, 157, 302, 360, 433; mapping
the, 490, 497, 503, 517, 543
Zeta functions, Z, 512 ; f, 522
Zonal harmonies. See Legendre's poly-
nomials
,.,/>^
t^
8784 5
f
r\ j^
QA
303
W5
cop. 2
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