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A COURSE OF
MODERN ANALYSIS
CAMBRIDGE UNIVERSITY PRESS
C. F. CLAY, Manager
itonlion : FETTEE LANE, E.G.
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A COURSE OF
MODEEN ANALYSIS
AN INTRODUCTION TO THE GENERAL THEORY OF
INFINITE PROCESSES AND OF ANALYTIC FUNCTIONS;
WITH AN ACCOUNT OF THE PRINCIPAL
TRANSCENDENTAL FUNCTIONS
SECOND EDITION, COMPLETELY REVISED
BY
E. T. WHITTAKER, D.Sc, F.R.S.
PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF EDINBURGH
AND
G. N. WATSON, M.A.
FELLOW t/F TRINITY COLLEGE, CAMBRIDGE, AND ASSISTANT PROFESSOR
OF PURE MATHEMATICS AT UNIVERSITY COLLEGE, LONDON
CAMBRIDGE:
AT THE UNIVERSITY PRESS
1915
First Edition 1902
Second Edition 1915
PEEFACE
When the first edition of my Course of Modem Analysis became
exhausted, and the Syndics of the Press invited me to prepare a second
edition, I determined to introduce many new features into the work. The
pressure of other duties prevented me for some time from carrying out this
plan, and it seemed as if the appearance of the new edition might be
indefinitely postponed. At this juncture, my friend and former pupil,
Mr G. N. Watson, offered to share the work of preparation ; and, with his
cooperation, it has now been completed.
The appearance of several treatises on the Theory of Convergence, such
as Mr Hardy's Course of Pure Mathematics and, more particularly,
Dr Bromwich's Theory of Infinite Series, led us to consider the desirability
of omitting the first four chapters of this work ; but we finally decided to
retain all that was necessary for subsequent developments, in order to make
the book complete in itself. The concise account which will be found in
these chapters is by no means exhaustive, although we believe it to be fairly
complete. For the discussion of Infinite Series on their own merits, we may
refer to the work of Dr Bromwich.
The new chapters on Riemann Integration, on Integral Equations, and
on the Riemann Zeta-Function, are entirely due to Mr Watson : he has
revised and improved the new chapters which I had myself drafted and he
has enlarged or partly rewritten much of the matter which appeared in the
original work. It is therefore fitting that our names should stand together
on the title-page.
Grateful acknowledgment must be made to Mr W. H. A. Lawrence, B.A.,
and Mr C. E. Winn, B.A., Scholars of Trinity College, who with great
kindness and care have read the proof-sheets, to Miss Wrinch, Scholar of
Girton College, who assisted in preparing the index, and to Mr Littlewood,
who read the early chapters in manuscript and made many helpful criticisms.
Thanks are due also to many readers of the first edition who supplied
corrections to it ; and to the staff of the University Press for much courtesy
and consideration during the progress of the printing.
E. T. WHITTAKER.
Juli/ 1915
CONTENTS
PART I. THE PROCESSES OF ANALYSIS
CHAPTER PAGE
I Complex Niimb'ers 3
II The Theory of Convergence 11
III Continuous Functions and Uniform Convergence . • . . . . 41
IV The Theory of Riemann Integration 61
V The fundamental properties of Analytic Functions ; Taylor's, Laurent's,
and Liouville's Theorems 82
VI The Theory of Residues ; application to the evaluation of Definite Integrals 111
VII The expansion of functions in Infinite Series 125
VIII Asymptotic Expansions and Summable Series 150
IX Fourier Series 160
X Linear Differential Equations 188
XI Integral Equations 205
PART II. THE TRANSCENDENTAL FUNCTIONS
XII The Gamma Function ....
XIII The Zeta Function of Riemann
XIV The Hypergeometric Function .
XV Legendre Functions .....
XVI The Confluent Hypergeometric Function
XVII Bessel Functions
XVIII The Equations of Mathematical Physics
XIX Mathieu Functions
XX Elliptic Functions. General theorems and the
XXI The Theta Functions ....
XXII The Jacobian Elliptic Functions
Appendix
List of Authors quoted
General Index
Weierstrassian
Functions
229
259
275
296
331
349
379
397
422
455
484
529
541
545
[Note. The decimal system of paragraphing, introduced by Pe-ano, is adopted in this
work. The integral part of the decimal represents the number of the chapter and the
fractional parts are arranged in each chapter in order of magnitude. Thus, e.g., on
pp. 181, 182, § 9-632 precedes § 97 becausis 9-632 < 97.]
ADDITIONS AND COREECTIONS
Page 13, footnote, Page 16, line 29, Page 27, footnote and Page 29, footnote, for
' Algdbraique ' read ' Algdbrique.'
Page 15, line 2. It is not quite obvious that, if a sequence {x^ has one limit-point I, then I
is the limit of the sequence. To prove this, observe that, if, for some positive value of
e, no number % could be found such that | ^„- J | <e whenever n>no, then, given any
suf&x ?ir, we can always find a g7-eater auffix n^+i, such that \Xn — l\^f when n = rir + i>
the sequence ^Byj-^n,?.-- then has a limit-point (§ 2'21) other than I; and this is
evidently inconsistent with {x^} having I as its only limit-point.
Page 36, line 7. The introduction into Analysis of Infinite Determinants is due to
E. Fiirstenau, Darstellung der reellen Wurzeln algehraischer Gleichungen durch Deter-
minanten der Coeffizienten. (Marburg, 1860.)
Page 87, line 30, for ' Raout ' read ' Ravut.'
Page 147, example 13. Demonstrations of Wronski's expansion have been given by
Cayley, Quarterly Journal, xii. (1873), and Transon, Nouv. Ann. de Math. xiii. (1874).
00 I » 1
Page 186, example 15, /or 2 ~ ^^ sin (6?i - 3) .^; read 2 -ii\n{<o7i-2,) x.
n=l OW — o ,1=1 ZTi — 1
Page 186, example 16. The value of f{x) in the range (g7r, tt) should be -^n, not ^rr.
/z4-^\1n /^ 4. 1 \ 4"'
Page 320, line 2, for (~^\ read f^j •
Page 370, line 1, for ''for all values of x' read '■vdth limited total fluctuation.^ If this
condition is satisfied, /' {x) may be written as the difference of two bounded increasing
functions ; the formula defining F{x) then shews that F{x) is the difference of two
such functions, and so F{x) has limited total fluctuation, and it is permissible to
apply Fourier's theorem. See also Chapman, Quarterly Journal, XLiii.
PAET I
THE PROCESSES OF ANALYSIS
W. M. A,
CHAPTER I
COMPLEX NUMBERS
I'l. Rational Numbers.
The idea of a set of numbers is derived in the first instance from the
consideration of the set of positive* integral numbers, or positive integers ;
that is to say, the numbers 1, 2, 3, 4, — Positive integers have many
properties, which will be found in treatises on the Theory of Integral
Numbers ; but at a very early stage in the development of mathematics
it was found that the operations of Subtraction and Division could only be
performed among them subject to inconvenient restrictions ; and consequently,
in elementary Arithmetic, classes of numbers are constructed such that the
operations of subtraction and division can always be performed among them.
To obtain a class of numbers among which the operation of subtraction
can be performed without restraint we construct the class of integers, which
consists of the class of positive^ integers (+ 1, + 2, + 3, ...) and of the class
of negative integers (— 1, — 2, — 3, ...) and the number 0.
To obtain a class of numbers among which the operations both of sub-
traction and of division can be performed freely:J:, we construct the class of
rational numbers. Symbols which denote members of this class are |, 3,
u, —-7-.
We have thus introduced three classes of numbers, (i) the signless integers,
(ii) the integers, (iii) the rational numbers.
It is not part of the scheme of this work to discuss the construction of
the class of integers or the logical foundations of the theory of rational
numbers §.
The extension of the idea of number, which has just been described, was not effected
without some opposition from the more conservative mathematicians. In the latter half
* Strictly speaking, a more appropriate epithet would be, not positive, but signless.
t In the strict sense.
t With the exception of division by the rational number 0.
§ Such a discussion, defining a rational number as an ordered number-pair of integers in a
similar manner to that in which a complex number is defined in § 1-3 as an ordered number-pair
of real numbers, will be found in Hobson's Functions of a Real Variable, §§ 1 — 12.
1—2
4 THE PROCESSES OF ANALYSIS [CHAP. I
of the 18tli century, Maseres (1731—1824) and Frend (1757—1841) published works
on Algebra, Trigonometry, etc., in which the use of negative numbers was disallowed,
although Descartes had used them unrestrictedly more than a hundred years before.
A rational number x may be represented to the eye in the following
manner :
If, on a straight line, we take an origin 0 and a fixed segment OP^
(Pi being on the right of 0), we can measure from 0 a length OPx such that
the ratio OPx/OPi is equal to cc; the point Px is taken on the right or left of
0 according as the number x is positive or negative. We may regard either
the point Px or the displacement OPx (which will be written OPx) as repre-
senting the number x.
All the rational numbers can thus be represented by points on the line,
but the converse is not true. For if we measure off on the line a length OQ
equal to the diagonal of a square of which OP, is one side, it can be proved
that Q does not correspond to any rational number.
Points on the line which do not represent rational numbers may be said to represent
irrational numbers ; thus the point Q is said to represent the irrational number
v/2 = l"414213 But while such an explanation of the existence of irrational numbers
satisfied the mathematicians of the eighteenth century and may still be sufficient for
those whose interest lies in the applications of mathematics rather than in the logical
upbuilding of the theory, yet from the logical standpoint it is improper to introduce
geometrical intuitions to supply deficiencies in arithmetical arguments ; and it was
shewn by Dedekind in 1858 that the theory of irrational numbers can be established on
a purely arithmetical basis without any appeal to geometry.
1'2. Dedekind' s* theory of irrational numbers.
The geometrical property of points on a line which suggested the starting
point of the arithmetical theory of irrationals was that, if all points of a line
are separated into two classes such that every point of the first class is on
the right of every point of the second class, there exists one and only one
point at which the line is thus severed.
Following up this idea, Dedekind considered rules by which a separation
or section of all rational numbers into two classes can be made, these classes
(which will be called the X-class and the P-class, or the left class and the
right class) being such that they possess the following properties :
(i) At least one member of each class exists.
(ii) Every member of the X-class is less than every member of the
P-class.
It is obvious that such a section is made by any rational number x ; and
X is either the greatest number of the Z-class or the least number of the
* The theory, though elaborated in 1858, was not published before the appearance of Dede-
kind's tract, Stetigkeit und irrationale Zahlen, Brunswick, 1872. Other theories are due to
Weierstrass (see von Dantscher, Die Weierstrass' sche Theorie der irrationalen Zahlen) and Cantor
(see Math. Ann. Bd. v.).
1"2] COMPLEX NUMBERS 5
J?-class. But sections can be made in which no rational number x plays this
part. Thus, since there is no rational number* whose square is 2, it is easy
to see that we may form a section in which the iJ-class consists of the positive
rational numbers whose squares exceed 2, and the Z-class consists of all
other rational numbers.
Then this section is such that the i2-class has no least member and the
X-class has no greatest member ; for, if x be any positive rational fraction,
, «(a^ + 6) ,, Ixi^-af) , , „ (x'-iy , ,
^^^ y^^^^T' '^^^ y-''=~^^^ ^^^ 2/^-2=(Wiy- '' ^' y
and 2 are in order of magnitude ; and therefore given any member x of the
Z-class, we can always find a greater member of the Z-class, or given any
member x' of the jR-cIass, we can always find a smaller member of the
jR-class, such numbers being, for instance, y and y , where y' is the same
function of x' as y of x.
If a section is made in which the i^-class has a least member \i4 2, or if the
Z-class has a greatest member ^1, the section determines a rational-real
number, which it is convenient to denote by the samef symbol A2 or A^.
If a section is made, such that the jR-class has no least member and the
Z-class has no greatest member, the section determines an irrational-real
numher\.
If X, y are real numbers (defined by sections) we say that x is greater
than y if the X-class defining x contains at least two§ members of the ii-class
defining y.
Let a, yS, ... be real numbers and let A-y, B^, ... be any members of the
corresponding Z-classes while J.2, ^g, . . . are any members of the corresponding
jK-classes. The classes of which A^, A^, ... are respectively members will be
denoted by the symbols (^1), (A^), ....
Then the sum (written a + y8) of two real numbers a and 8 is defined as
the real number (rational or irrational) which is determined by the Z-class
(Ai + B^) and the i2-class (^2 + ^2)-
It is, of course, necessary to prove that these classes determine a section of the rational
numbers. It is evident that Ai + Bi< A2 + B2 and that at least one member of each of the
classes (Ai + Bi), {A2 + B^) exists. It remains to prove that there is, at most, one rational
* For if pjq be such a number, this fraction being in its lowest terms, it may be seen that
{2q-p)l{p-q) is another such number, and 0<p-q<:q, so that 2^/2 is not in its lowest terms.
The contradiction implies that such a rational number does not exist.
t This causes no confusion in practice.
X B. A. W. Eussell defines the class of real numbers as actually being the class of all L-classes ;
the class of real numbers whose L-classes have a greatest member corresponds to the class of
rational numbers, and though the rational-real number x which corresponds to a rational number
X is conceptually distinct from it, no confusion arises from denoting both by the same symbol.
§ If the classes had only one member in common, that member might be the greatest
member of the L-ciass of x and the least member of the iJ-class of ij.
6 THE PROCESSES OF ANALYSIS [CHAP. I
number which is greater than every ^dj + 5i and less than every A2 + B2; suppose, if possible,
that there are two, x and y {y >x). Let ai be a member of {A 1) and let a2 be a member
of (Jg) ; and let ^be the integer next greater than (a2~^i)/{i Cy~-^)}- Take the last of
the numbers ai + -^ {a^ — di), (where m = 0, 1, ... N), which belongs to (^1) and the first of
them which belongs to (J 2) ; let these two numbers be Cj, c^. Then
C2-Ci = ^.(a2-«i)<i(y--»)-
Choose c?i, c?2 in a similar manner from the classes defining ^; then
C'i + d^-Ci-di<y-x.
But c^ + d^^y, Ci + di^x, and therefore C2 + c?2 — <'i-<^i^2/~'^ j '^e have therefore
arrived at a contradiction by supposing that two rational numbers x, y exist belonging
neither to {A + B) nor to (a + 6).
If every rational number belongs either to the class {A■^^\-B■|) or to the class {A2-\-B^,
then the classes {Ai+Bi), {A^+B^) define an irrational number. If one rational number x
exists belonging to neither class, then the Z-class formed by x and (^i + ^i) and the
i2-class (Jj + ^g) define the rational-real number x. In either case, the number defined
is called the sum a + /3.
The difference a — jS of two real numbers is defined by the Z-class {Ai—B-^ and the
iE-class {Ac^ — Bi).
The product of two positive real numbers a, /3 is defined by the ^-class (-42-^2)
and the Z-class of all other rational numbers.
The reader will see without difficulty how to define the product of negative real num-
bers and the quotient of two real numbers ; and further, it may be shewn that real
numbers may be combined in accordance with the associative, distributive and commuta-
tive laws.
The aggregate of rational-real and irrational-real numbers is called the
aggregate of real numbers ; for brevity, rational-real numbers and irrational-
real numbers are called rational and irrational numbers respectively.
1"3. Complex numbers.
We have seen that a real number may be visualised as a displacement
along a definite straight line. If, however, P and Q are any two points in a
plane, the displacement PQ needs two real numbers for its specification ; for
instance, the differences of the coordinates of P and Q referred to fixed
rectangular axes. If the coordinates of P be (^, 77) and those of Q (^ -f a;, 77 -f 3/),
the displacement PQ may be described by the symbol \x, y\ We are thus
led to consider the association of real numbers, in ordered* pairs. The natural
definition of the sum of two displacements \x, y\ [x, y'] is the displacement
which is the result of the successive applications of the two displacements ;
it is therefore convenient to define the sum of two number-pairs by the
equation
[x, y] + [x, y']=[x + x',y + y'\
* The order of the two terms distinguishes the ordered number-pair [x, y'\ from the ordered
number-pair [y, x\.
1-3] COMPLEX NUMBERS 7
The product of a number-pair and a real number x is then naturally
defined by the equation
x' X \x, y\ = \x'x, x'y\ ■
We are at liberty to define the product of two number-pairs in any
convenient manner; but the only definition, which does not give rise to
results that are merely trivial, is that symbolised by the equation
[x, y] X [x, y'] = [xx - yy', xy + xy\
It is then evident that
\x, 0] X \x\ 2/'] = \xx', xy'] = x x [x, y']
and [0, y] x [x, y] = [- yy', x'y] = yx[- y, x\
The geometrical interpretation of these results is that the effect of
multiplying by the displacement \x, 0] is the same as that of multiplying by
the real number x ; but the effect of multiplying a displacement by [0, y]
is to multiply it by a real number y and turn it through a right angle.
It is convenient to denote the number-pair [x, y] by the compound
symbol x + iy ; and a number-pair is now conveniently called (after Gauss)
a complex number ; in the fundamental operations of Arithmetic, the complex
number x + iO may be replaced by the real number x and, defining i to mean
0 + il, we have i^ = [0, 1] x [0, 1] = [— 1, 0] ; and so i^ may be replaced by — 1.
The reader will easily convince himself that the definitions of addition
and multiplication of number-pairs have been so framed that we may perform
the ordinary operations of algebra with complex numbers in exactly the same
way as with real numbers, treating the symbol i as a number and replacing
the product ii by — 1 wherever it occurs.
Thus he will verify that, if a, b, c are complex numbers, we have
a + b = b + a,
ab = ba,
(a + b) + c = a + (b + c),
ab . c = a . be,
a{b + c) = ab + ac,
and if ab is zero, then either a or 6 is zero.
It is found that algebraical operations, direct or inverse, when applied to
complex numbers, do not suggest numbers of any fresh type ; the complex
number will therefore for our purposes be taken as the most general type
of number.
The introduction of the complex number has led to many important developments in
mathematics. Functions which, when real variables only are considered, appear as
essentially distinct, are seen to be connected when complex variables are introduced :
8 THE PROCESSES OF ANALYSIS [CHAP. I
thus the circular functions are found to be expressible in terms of exponential functions
of a complex argument, by the equations
cos^ = -(e''=^ + e~"), sin^=-^(e*^-e-"').
2 2^
Again, many of the most important theorems of modern analysis are not true if the
ntimbers concerned are restricted to be real ; thus, the theorem that every algebraic
equation of degree n has n roots is true in general only when regarded as a theorem
concerning complex numbers.
Hamilton's quaternions furnish an example of a still further extension of the idea
of number. A quaternion
is formed from four real numbers w, oo, y, z, and four number-units 1, i, j, k, in the same
way that the ordinary complex number x + iy might be regarded as being formed from
two real numbers x, y, and two number-units 1, i. Quaternions however do not obey
the commutative law of multiplication.
1'4. The modulus of a complex number.
Let X + iy be a complex number, x and y being real numbers. Then
the positive square root of x"^ + y^ is called the modulus of {x + iy), and is
written
\x -\-iy\.
Let us consider the complex number which is the sum of two given
complex numbers, x -f iy and u + iv. We have
{x -f iy) -f (w + iv) = {x -\- u) -\- i {y + v).
The modulus of the sum of the two numbers is therefore
[{x + uY + (y + vY]^,
or {{x^ + 2/0 + {u^' + v^) + 2 (xu + yv)]^.
But
{\x+iy\ + \u + iv\Y= {(x"^ + y-)^ + (u"^ + v^^p
= (^2 + 2/0 + (u^ + V) + 2{x'' + 2/0* (u' + V)^
= (x^ -f 2/0 + ("" + ^0 + 2 {(xu + yv)- + (xv - yuf]^,
and this latter expression is greater than (or at least equal to)
{x"^ + 2/0 + {u^ + V') + 2 {xu -f yv).
We have therefore
\ X -\- iy \ -\- \ u + iv \'^ \ {x -\- iy) + {u + iv) \ ,
i.e. the modulus of the sum of two complex numbers cannot be greater than the
sum of their moduli; and it follows by induction that the modulus of the sum
of any number of complex numbers cannot be greater than the sum of their
moduli.
1-4, 1*5] COMPLEX NUMBERS 9
Let us consider next the complex number which is the product of two
given complex numbers, x + iy and u-\-iv] we have
{x + iy) (u + iv) = {xu — yv) + i (xv + yu),
and therefore
I {x + iy) (u + iv) I = {{xu — yvy + (xv + yuY}^
= {{^ + 2/') (w' + «')]*
= \x + iy\\u + iv\.
The modulus of the product of two complex numbers (and hence, by in-
duction, of any number of complex numbers) is therefore equal to the product
of their moduli.
1"5. The Argand diagram.
We have seen that complex numbers may be represented in a geometrical
diagram by taking rectangular axes Ox, Oy in a plane. Then a point P
whose coordinates referred to these axes are x, y may be regarded as
representing the complex number x + iy. In this way, to every point of
the plane there corresponds some one complex number ; and, conversely, to
every possible complex number there corresponds one, and only one, point
of the plane.
The complex number x + iy may be denoted by a single letter* z. The
point P is then called the representative point of the number z; we shall
also speak of the number z as being the ajffix of the point P.
If we denote {x^ 4- y^Y by r and choose 6 so that r cos 6 = x, r sin 0 = y,
then r and 0 are clearly the radius vector and vectorial angle of the point P,
referred to the origin 0 and axis Ox.
The representation of complex numbers thus afforded is often called the
Argand diagramf.
By the definition already given, it is evident that r is the modulus of z.
The angle 0 is called the argument, or amplitude, of z.
We write 0 = arg z.
From geometrical considerations, it appears that (although the modulus of a complex
number is unique) the argument is not unique J ; if ^ be a value of the argument, the
other values of the argument are comprised in the expression 2nn + d where n is any
integer, not zero. The priiidpal value of arg 3 is that which satisfies the inequality
— 7r<arg2^7r.
* It is convenient to call x and y the real and imaginary parts of z respectively. We fre-
quently write x = R{z), y = I{z).
t J. R. Argand published it in 1806 ; it had however previously been used by Gauss, and
by Caspar Wessel, who discussed it in a memoir presented to the Danish Academy in 1797 and
published by that Society in 1798-9.
X See the Appendix.
10 THE PROCESSES OF ANALYSIS [CHAP. I
If Pj and Pg are the representative points corresponding to values z-^
and z^ respectively of z, then the point which represents the value ^j 4 z^ is
clearly the terminus of a line drawn from Pj, equal and parallel to that
which joins the origin to Pj.
To find the point which represents the complex number z-^Z2, where z-^ and
Z2 are two given complex numbers, we notice that if
•s'l = r^ (cos ^1 -F i sin 6^,
Z2, = Tg (cos ^2 + * sin ^2)
then, by multiplication,
z^z^ = nrg {cos (^1 + ^2) + * sin (^1 + ^2)}-
The point which represents the number z-^^z^ has therefore a radius vector
measured by the product of the radii vectores of Pj and Pg, and a vectorial
angle equal to the sum of the vectorial angles of Pj and Pg.
REFERENCES.
The logical foundations of the theory of number.
A. N. Whitehead and B. A. W. Russell, Principia Mathematica.
On irrational numbers.
R. Dedekind, Stetigkeit und irrationale Zahlen. (Brunswick, 1872.)
V. VON Dantscher, Vorlesungen ueber die Weierstrass'sche Theorie der irrationalen
Zahlen.
E. W. HoBSON, Functions of a Real Variable, Ch. i.
T. J. I'a. Bromwich, Theory of Infinite Series., Appendix i.
On complex numbers.
H. Hankel, Theorie der complexen Zahlen-systeme. (Leipzig, 1867.)
O. Stolz, Vorlesungen iiber allgemeine Arithmetik, li. (Leipzig, 1886.)
G. H. Hardy, A course of Pure Mathematics., Ch. in.
Miscellaneous Examples.
1. Shew that the representative points of the complex numbers l+4t, 2 + 7i, 3 + 10i,
are collinear.
2. Shew that a parabola can be drawn to pass through the representative points of
the complex numbers
i + i, 4 + 4i, 6 + 9i, 8 + 16z, 10 + 25i.
3. Determine the nth roots of unity by aid of the Argand diagram ; and shew that the
number of primitive roots (roots the powers of each of which give all the roots) is the
number of integers (including unity) less than n and prime to it.
Prove that if 5,, ^2) ^3) ••• be the arguments of the primitive roots, 2 cosp^ = 0 when
n
jo is a positive integer less than ~r where a, b, c, ... k are the different constituent
CtOC » , » tC
primes of n ; and that.
the constituent primes
primes of n ; and that, when P = -, 7, 2 co>i p6= ~-^ — ,, where fi is the number of
(Math. Trip. 1895.)
CHAPTER II
THE THEORY OF CONVERGENCE
2'1. The definition* of the limit of a sequence.
Let Zi, z^, Zz, ... be an unending sequence of numbers, real or complex.
Then, if a number I exists such that, corresponding to every p.ositivef
number e, no matter how small, a number Wq can be found, such that
I ^'n — ^ I < e
for all values of n greater than Wq, the sequence {z^ is said to tend to the limit I
as n tends to infinity.
Symbolic forms of the statement]: ' the limit of the sequence (^„), as n
tends to infinity, is I ' are :
lim Zn — I, lim Zn = l, Zn—^l as n—^ <x> .
n-*'Oo
If the sequence be such that, given an arbitrary number N (no matter
how large), we can find n^ such that | ^^^ | > iV for all values of n greater than
rio, we say that '\zn\ tends to infinity as n tends to infinity,' and we write
I ^„ I — ► GO .
In the corresponding case when — Xn> N when n > n^ we say that
a;„ — > — 00 .
If a sequence of real numbers does not tend to a limit or to go or to — oo ,
the sequence is said to oscillate.
2'11. Definition of the phrase ' of the order of
If (^n) and {Zri) are two sequences such that a number n^ exists such that
I i'^nlzn) I < K whenever n > n^, where K is independent of n, we say that ^n is
* of the order of Zn> and we write §
?„ = 0 (zn) ;
* A definition equivalent to this was first given by John Wallis.
t The number zero is excluded from the class of positive numbers.
X The arrow notation is due to Leathern, Camb. Math. Tracts, no. 1.
§ This notation is due to Landau.
12 THE PROCESSES OF ANALYSIS [CHAP. II
thus Mi+1? = 0 f i
1 + W \W
If lim (^n/^n) = 0, we write ^n = o (zn).
2"2. The limit of an increasing sequence.
Let (xn) be a sequence of real numbers such that Xn+i ^ Xn for all values
of n; then the sequence tends to a limit or else tends to infinity (and so it does
not oscillate).
Let cc be any rational-real number; then either:
(i) Xn'^x for all values of n greater than some number n^, depending on
the value of x.
Or (ii) Xn< X for every value of n.
If (ii) is not the case for any value of x (no matter how large), then
Xn—^<X>.
But if values of x exist for which (ii) holds, we can divide the rational
numbers into two classes, the X-class consisting of those rational numbers x
for which (i) holds and the i^-class of those rational numbers x for which (ii)
holds. This section defines a real number a, rational or irrational.
And if 6 be an arbitrary positive number, a — ^e belongs to the Z-class
which defines a, and so we can find ?ii such that Xn^o- — ^e whenever ti > Mj ;
and a+|e is a member of the -R-class and so a;„<a4-^6. Therefore,
whenever n>ni,
\a — Xn\<€.
Therefore Xn — > a.
Corollary. A decreasing sequence tends to a limit or to — oo .
Example 1. If lim Zm = li lim z^ = l\ then lim (2m+2m') = ^+^'-
For, given e, we can find n and n' such that
(i) when m>n, \ z^ — l \ < ^e, (ii) when m>n', \ zj - ^' | < Jf-
Let %i be the greater of n and n' ; then, when ??i > rii ,
I (2m + 2„/) - {l + l') 1 ^ I {Z,n-l) I + I (Zm'-l') I ,
<f ;
and this is the condition that lim {zm + Zm') = l + l'.
Example 2. Prove similarly that lim (2^-2,„') = ^-^', lim(2^2™') = ^^') and, if Z'=t=0,
lim(2,„/2^') = ^/^'-
Example 3. Ii 0 < x <\, x^ -*-().
For if ^=(l + a)-i, a>0 and
0 <r' r" = <^
{l + aY \+na'
by the binomial theorem for a positive integral index. And it is obvious that, given a
positive number e, we can choose Mq such that ( I ^na) ~ i < e when n>no\ and so x"^ -*► 0.
22-2-22] THE THEORY OF CONVERGENCE 13
221. The Bolzano- Weierstrass* theorem on limits of a sequence.
Let (aJn) be a sequence of real numbers such that X^Xn^p, where \, p are
independent of n. Then the sequence has at least one ' limit-point ' G ; that
is to say, there exists a number G such that, if e be an arbitrary positive
number, there are an unlimited number of terms of the sequence which satisfy
G- €< CCn< G + e.
For, choose a section in which (i) the ^-class consists of all the rational
numbers which are such that, if A be any one of them, there are only a
limited number of terms a;„ satisfying Xn> A; and (ii) the Z-class is such that
there are an unlimited number of terms Xn such that Xn>a for all members a
of the X-class.
This section defines a real number G ; and, if e be an arbitrary positive
number, G — ^e and G + ^e are members of the L and R classes respectively,
and so there are an unlimited number of terms of the sequence satisfying
G-e< G-^€^Xn^G + ^€< G+ e,
which is the result stated.
2*211. Definition of ' the greatest of the limits.'
The number G obtained in § 2-21 is called 'the greatest of the limits of
the sequence (Xn).' The sequence (xn) cannot have a limit point greater
than G; for if G' were such a limit point, and € = ^(G'—G), G' — e is a
member of the i2-class defining G, so that there are only a limited number of
terms of the sequence which satisfy Xn>G' — e. This condition is incon-
sistent with G' being a limit point. We write
G = lim Xn .
The ' least of the limits,' L, of the sequence (written lim Xn) is defined by the
equation
Iim (— Xn) — — L.
2"22. Cauchy's-f theorem on the necessary and sufficient condition for
the existence of a limit.
We shall now shew that the necessary and sufficient condition for the
existence of a limiting value of a sequence of numbers z^, z^, z-^, ... is that,
corresponding to any given positive number e, however small, it shall be
possible to find a number n such that
■"n-\-p
— Zn \ < €
for all positive integral values of p. This result is one of the most important
* This theorem, frequently ascribed to Weierstrass, was proved by Bolzano, Paradoxien des
Unendlichen (1851). It seems to have been known to Cauchy.
t Analyse Algebraique, p. 125.
14 THE PROCESSES OF ANALYSIS [CHAP. II
and fundamental theorems of analysis. It is sometimes called the Principle
of Convergence.
First, we have to shew that this condition is necessary, i.e. that it is
satisfied whenever a limit exists. Suppose then that a limit I exists ; then
(§ 2'1) corresponding to any positive number e, however small, a number n
can be chosen such that
i ^n - ^ I < ie, I Zn+p -l\<^e,
for all positive values of p ; therefore
I ^n+p -Zn\ = \ (^n+p -I) — {^n — I) \
^ I Zn+p —l\ + \Zn—l\<e,
which shews the necessity of the condition
I ^n+p ^n I "^ ^j
and thus establishes the first half of the theorem.
Secondly, we have to prove* that this condition is sufficient, i.e. that if
it is satisfied, then a limit exists.
(I) Suppose that the sequence of real numbers (ocn) satisfies Cauchy's
condition ; that is to say that, corresponding to any positive number e, a
number n can be chosen such that
I '^n+p '^n I "^ ^
for all positive integral values of ^.
Let the value of n, corresponding to the value 1 of e, be m.
Let Xi, pi be the least and greatest o( {v^, x^, ... Xm] then
\-l<Xn< pi + 1,
for all values of n ; write Xj — 1 = X,, p^ + \ = p.
Then, for all values of n, \ < Xn< p. Therefore by the theorem of § 2"21,
the sequeiice {x^} has at least one limit point G.
Further, there cannot be more than one limit point ; for if there were
two, G and H (H < G), take e < l(G — H). Then, by hypothesis, a number
n exists such that j x^+p — Xn\<€ for every positive value of p. But since G
and H are limit points, positive numbers q and r exist such that
! G — Xn+q \< e, \H — Xn+r | < 6.
Then j G — X^+q \ + \ X^+q — Xn\ + \ X^ — X^+r I + I OCn+r — H \ < ^e.
But, by § 1-4, the sum on the left is greater than or equal to \G— H\.
Therefore G - H < 4>e.
But e was chosen so that e<^{G — H).
* This proof is given by Stolz and Gmeiner, Theoretische Arithmetik, Bd. ii. (1902), p. 144.
23] THE THEORY OF CONVERGENCE 16
Hence, by assuming the existence of two limit points, we have arrived at
a contradiction. Therefore there is but one ; that is to say, the limit of (a;„)
exists.
(II) Now let the sequence {zr^ of real or complex numbers satisfy
Cauchy's condition ; and let Zn = Xn + iyn, where Xn and y„ are real ; then for
all values of n and p
I ^nr^ ^n I ^ I ^n+p ^n\} \ yn+p ~ yn\ ^ \ ^n+p ~ ^n\'
Therefore the sequences of real numbers (xn) and (yn) satisfy Cauchy's
condition ; and so, by (I), the limits of (xn) and (yn) exist. Therefore, by
§ 2 2 example 1, the limit of (zn) exists. The result is therefore established.
2'3. Convergence of an infinite series.
Let Wi, Wj, Wj, ... Un, ... be a sequence of numbers, real or complex. Let
the sum
U1 + U2+ ...+Un
be denoted by Sn-
Then, if Sn tends to a limit S as n tends to infinity, the infinite series
^1 + ^2 + ^3 + ^*4+ •..
is said to be convergent, or to converge to the sum S. In other cases, the
infinite series is said to be divergent. When the series converges, the
expression S — Sn, which is the sum of the series
'^n+i + ^n+2 + ^n-t-3 + • • • ,
is called the remainder after n terms, and is frequently denoted by the
symbol Rn.
The sum ' w„+i + w„+2 + • . • + ^n+p
will be denoted by Sn,p.
It follows at once, by combining the above definition with the results
of the last paragraph, that the necessary and sufficient condition for the
convergence of an infinite series is that, given an arbitrary positive number e,
we can find n such that j 8n,p | < e for every positive value of p.
Since u^+i — Sn,i> i^ follows as a particular case that lim Un+i = 0 — in other
words, the nth term of a convergent series must tend to zero as n tends to
infinity. But this last condition, though necessary, is not sufficient in itself
to ensure the convergence of the series, as appears from a study of the series
11111
1+2 + 3 + 4+5+---
In this series, 'Sfn.n = ^1 + ;^2 + ,TT3 + •■• + i-
The expression on the right is diminished by writing (2n)~^ in place of
each term, and so Sn,n > i-
16 THE PROCESSES OF ANALYSIS [CHAP. II
Therefore S2n^i = l+Sj^^+ 8^,2 + S,,i + Ss,s + S,e,i6 + '•■ + S^n^^""
> 2 (»? + 3) -» 00 ;
the series is therefore divergent.
There are two general classes of problems which we are called upon to
investigate in connexion with the convergence of series :
(i) We may arrive at a series by some formal process, e.g. that of
solving a linear differential equation by a series, and then to justify the
process it will usually have to be proved that the series thus formally ob-
tained is convergent. Simple conditions for establishing convergence in
such circumstances are obtained in §§ 2"31-2"61.
(ii) Given an expression 8, it may be possible to obtain a development
n
8= S Um,-^ Rn, valid for all values of n ; and, from the definition of a limit,
m = \
00
it follows that, if we can prove that Rn -^ 0, then the series S Um converges
m=l
and its sum is 8. An example of this problem occurs in § 5'4.
Infinite series were used by Lord Brouncker in Phil. Trans. 1668, and the expressions
convergent and divergent were introduced by James Gregory, Professor of Mathematics
at Edinburgh, in the same year. But the great mathematicians of the 18th century used
infinite series freely without, for the most part, considering the question of their con-
vergence. Thus Euler gave the sum of the series
as zero, on the ground that
■ + ^ + ^ + J + ^ + '+^' + '' + (^)
« + 22 + 2H... = j^ (6)
and 1 + -+ +,.. = ^ ^(c).
The error of course arises from the fact that the series (6) converges only when | 2 | < 1,
and the series (c) converges only when | ^ | > 1, so the series (a) does not converge for any
value of z.
The modern theory of convergence may be said to date from the publication of
Gauss' Disquisitiones circa seriem infinitam \+—^x +... in 1812, and Cauchy's Analyse
1 .y
Algebraique in 1821. See Reiff, Oeschichte der unendlichen Reihen (Tubingen, 1889).
2-301. Abel's inequality*.
I ^
Let fn > fn+x > 0 for all integer values of n. Then \ 2 a„/n :$ Af, where
A is the greatest of the sums
\ax\, I ffi + a2M «'i + tt2 + a3 1, ..., i «! + tt2+ ... + a^ I-
* Crelle's Journal, Bd. i. (1826), pp. 311-339. A particular case of the theorem of § 2-31,
Corollary (i), also appears in that memoir.
2301, 2-31] THE THEORY OF CONVERGENCE 17
For, writing aj + ag + . . . + ctn = *»i. we have
m
2 anfn = «i/i + («2 - Si)/2 + («3 - h)f3 + ...+iSm- Sm--)fm
n = l
^^^ *i \J\ ~J2) "^ *2 (y 2 ~Ja) + . . • + *m-i \Jm—\ ~ Jm) "I" ^mjm-
Since /i —fiifi—fs, ■'• s-re not negative, we have, when ri= 2, 3, ... m,
1 5„_, I (/n-i -fn) ^ -4 (/„_i -/n) 5 also | S^ \fm ^ ^fm,
and so, summing and using § 1*4, we get
•^ ^njn
^Af,.
Corollary. If aj, ag* ••• ^i> ^2j ••• 'ire any numbers, real or complex,
m I fm—l "j
where A is the greatest of the sums 2 a„
, (p=l, 2, ... w).
(Hardy.)
2"31. Dirichlet's* test for convergence.
Let I S an
K, where K is independent of p. Then, if fn ^fn+i > 0
and lim/„ = Of, S «„/„ converges.
M = l
For, since lim/,j = 0, given an arbitrary positive number e, we can find m
such that fm+i < €/2K.
Then
m+q
m+q
m
^
S an
+
S an
n = \
M=l
< '2K, for all positive values of q ; so
that, by Abel's inequality, we have, for all positive values of p,
m+p 1
where A < 2K.
Therefore
' n=m+l
m+p
S a-nfn < ^Kfm+i < e ; and so, by § 2-3, S a„/„ converges.
n=m+l
Corollary (i). Abel's test for convergence. If 2 «„ converges and the sequence (m„) is
jt=i
monotonia (i.e. w„^% + i always or else M„^i<„ + i always) and | li^ ! < k, where < is
independent of n, then 2 «„?<„ converges.
For, by § 2'2, «„ tends to a limit u; let [m— ?;„[=/„. Then /,(-^0 steadily ; and
therefore 2 «„/„ converges. But, if (m,i) is an increasing sequence, fn = u-u,i, and so
2 (?( — ?t„) a,j converges ; therefore since 2 ««„ converges, 2 «„a,i converges. If (?<„) is
11=1 (t=i }i = i
a decreasing sequence /ii = m,i - «, and a similar proof holds.
* Liouville's Journal, ser. 2, t. vii. pp. 253-255. Before the publication of the 2nd edition
of Jordan's Conrs d" Analyse, Dirichlet's test and Abel's test were frequently jointly described
as tbe Diricblet-Abel test, see e.g. Pringsheim, Math. Ann. xxv. p. 423.
t lu these circumstances, we say/„-*-0 steadily.
W. M. A. 2
18 THE PROCESSES OF ANALYSIS [CHAP. II
Corollary (ii). Taking «„ = (-)"-! in Dirichlet'« test, it follows that, if /„ >/„ + ]
and lim/„ = 0, /1-/2+/3-/4+ ■•• converges.
Example 1. Shew that if 0<^<27r, 2 sin ?«5 < cosec i^ ; and deduce that, if
f^-^Q steadily, 2 /„ sin n6 converges for all real values of 6, and that 2 fn cos n6 con-
n=l »=i'
verges if 6 is not an even multiple of tt.
Example 2. Shew that, if /«-*0 steadily, 2 (--)"•/■„ cos 7i^ converges if 6 is real and
n = l
X
not an odd multiple of it and 2 ( - )"/„ sin nd converges for all real values of 6. [Write
7r + ^ for ^ in example 1.]
2'32. Absolute and conditional convergence.
In order that a series S u^ of real or complex terms may converge, it is
n = l
00
sufficient (but not necessary) that the series of moduli S ] w„ ] should
00
converge. For, if crn,p = j w^+i | + | Wn+2 j + • • • + | w„+^ | and if 2 i if^ | converges,
M=l
we can find n, corresponding to a given number e, such that an,p < e for all
oc
values of p. But \Sn,p\^a'n,p< e, and so X w„ converges.
The condition is not necessary; for writing /„ = l/n in § 2'31, corollary (ii),
we see that x — 2 "^ 3 ~" 4 "^ ••" converges, though (§ 23) the series of moduli
1+9 + 3 + 1 + .. . is known to diverge.
In this case, therefore, the divergence of the series of moduli does not
entail the divergence of the series itself
Series, which are such that the series formed by the moduli of their terms
are convergent, possess special properties of great importance, and are called
absolutely convergent series. Series which though convergent are not abso-
lutely convergent (i.e. the series themselves converge, but the series of moduli
diverge) are said to be conditionally convergent.
2'33. The geometric series, and the series S — .
The convergence of a particular series is in most cases investigated, not
by the direct consideration of the sum Sn,p, but (as will appear from the
following articles) by a comparison of the given series with some other series
which is known to be convergent or divergent. We shall now investigate
the convergence of two of the series which are most frequently used as
standards for comparison.
2-32, 2-33] THE THEORY OF CONVERGENCE 19
(I) The geometric series.
The geometric series is defined to be the series
Consider the series of moduli
for this series ,Sf„,p = | ^ | "+^ + 1 ^^ | "+2 + . . . + | ^ | "+^
1 — l^'l
Hence, if |2^|< 1, then Sn,p< \ — ^ — 1 foi' all values of p, and, by § 2*2
example 3, given any positive number e, we can find n such that
I ^ I «+i {1 - I ^ I }-i < e.
Thus, given e, we can find n such that, for all values of p, Sn, p< e. Hence,
' by § 2"22, the series
1 + 1^1 + 1^1^ + ...
is convergent so long as | ^^ | < 1, and therefore the geometric series is absolutely
convergent if\z\<l.
When 1 2^ I ^ 1, the terms of the geometric series do not tend to zero as w
tends to infinity, and the series is therefore divergent.
(II) The series 1^ + ^^ + -+^^ + -+....
n \
Consider now the series 8^= S — , where s is positive.
AIT 1. 112 1
We have __^ 4 _ < _ < _ ,
11114 1
4« ^ 5s ^ (js ^ 7s "- 4« ^ 4.S-1 '
and so on. Thus the sum of 2^ — 1 terms of the series is less than
J_ 1_ J_ J_ _1 1
p-i ~^ 2«~i 4«-i 8*~' + ••• + 2(i'-i) <«-!) 1 — 2'~' '
and so the sum of any number of terms is less than (1 — 2^"*)"^ Therefore
n
the increasing sequence X wi~* cannot tend to infinity ; therefore, by § 2'2,
>w = l
« 1
the series S — is convergent if s >1 ; and since its terms are all real and
positive, they are equal to their own moduli, and so the series of moduli of
the terms is convergent ; that is, the convergence is absolute.
2 2
20 THE PROCESSES OF ANALYSIS [CHAP. II
If s = 1, the series becomes
1111
1 + 2 + 3+4"'' •••'
which we have ah-eady shewn to be divergent ; and when 5 < 1, it is a fortiori
divergent, since the effect of diminishing s is to increase the terms of the
" 1 .
series. The series X — is therefore divergent if s ^1.
«. = ! "^
2'34. The GoTnparison Theorem.
We shall now shew that a series Ui + u^-^- u^-^ ... is absolutely con-
vergent, provided that \ Un \ is always less than C \vn\, where C is some number
independent of n, and v^ is the nth term of another series which is known to
be absolutely convergent.
For, under these conditions, we have
I 1A„+i I + I Wn+2 I + ... + I Un+p \<G [\ Vn+i j + | Vn+2 | + ... + j Vn^p\],
where n and p are any integers. But since the series ^Vn is absolutely
convergent, the series S | Vn | is convergent, and so, given e, we can find n
such that
I -y/i+i 1 + 1 V«+2 I + • • • + I ^n+p I < ejC,
for all values of p. It follows therefore that we can find n such that
I ?t«+i i + I 'fn+2 I + . . . + I Ww+p i < e,
for all values of p, i.e. the series S | Wn | is convergent. The series %Un is
therefore absolutely convergent.
Corollary. A series is absolutely couvergent if the ratio of its nih. term to the
nth. term of a series which is known to be absolutely convergent is less than some
number independent of n.
Example 1. Shew that the series
cos S + J-, cos 22 + -;, C0S32 + — ;COS42 + ...
Z- 6^ V'
is absolutely convergent for all real values of z.
cos nz
When z is real, we have | cos nz j ^ 1, and therefore
n"
<-T, . The moduli of
n-
the terms of the given series are therefore less than, or at most equal to, the corresponding
terms of the series
n 1 1 1
l + ^ + ^ + _+...,
which by § 2'33 is absolutely convergent. The given series is therefore absolutely
convergent.
Example 2. Shew that the series
1111
P(2-2l) 2^(2-22) 3^2-03) 4^(2-24)
where 2„ = e»S (yi = l, 2, 3, ...)
is convergent for all values of 2, except the values z = zy^^ z^, z-j,
2*34, 2*35] THE THEORY OF CONVERGENCE » 21
The geometric representation of complex numbers is helpful in discussing a question of
this kind. Let values of the complex number z be represented on a plane; then the
numbers 2i, z^., z^, ... will give a sequence of points which lie on the circumference of the
circle whose centre is the origin and whose radius is unity ; and it can be shewn that
every point on the circle is a limiting point (§ 2'21) of the points 2„.
For these special values z„ of z, the given series does not exist, since the denomi-
nator of the nth term vanishes when z = Zn- For simplicity we do not discuss the series
for any point z situated on the circumference of the circle of radius unity.
Suppose now that l^j + l. Then for all values of n, i 2-2„ | ^ | {1 -| 2 |} ] >c~i, for
some value of c ; so the moduli of the terms of the given series are less than the corre-
sponding terms of the series
which is known to be absolutely convergent. The given series is therefore absolutely
convergent for all values of z, except those which are on the circle | 2 | = 1.
It is interesting to notice that the area in the 2-plane over which the series converges
is divided into two parts, between which there is no intercommunication, by the circle
\z\ = l.
Example 3. Shew that the series
2 sin --I-4 sin --f 8 sin — -h ... + 2" sin j^-f ...
converges absolutely for all values of 2.
Since* lim 3" sin (2/3") =2, we can find a number fr, independent of n (but depending
on 2) such that | 3" fein (2/3") \<k ; and therefore
2«szn-|<^(-
Since 2^(5) converges, the given series converges absolutely.
n=l \<5/
2"35. Gauchys convergency test'f.
00
// lim \un\'^'^< 1, S Un converges absolutely.
n — *- ao n = 1
For we can find m such that, when n^m, \Un\^'^ ^ p < I, where p is
00
independent of n. Then; when n >m, | m„ | < p'* ; and since % p" converges,
X / 00 \
it follows from § 2-34 that S Un (and therefore % iin] converges ab-
solutely.
[Note. If lim |m„|^"*>1, m„ does not tend to zero, and, by § 2-3, 2 u^ does not
converge.]
* This is evident from results proved in the Appendix.
t Oeuvres Mathematiques, ser. 1, t. viii. p. 270.
22 ^ THE PROCESSES OF ANALYSIS [CHAP. II
2'36. D'Alembert's* ratio test.
We shall now shew that a series
til + ?«2 + ^'3 + W4 + ...
is absolutely convergent, provided that for all values of n greater than some
fixed value r, the ratio j-^^ is less than p, where p is a positive number
independent of n and less than unity.
For the terms of the series
I i<r+l I + I U-,.j^2 I + i ?''r+3 I + . . .
are respectively less than the corresponding terms of the series
|m,.+J(1+/J + /32 + P=^+...),
00
which is absolutely convergent when p <\; therefore S Un (and hence
the given series) is absolutely convergent.
A particular case of this theorem is that if lim j (wm+i/ww)| = ^ < 1, the
series is absolutely convergent.
For, by the definition of a limit, we can find r such that
!^!f!i±j|_A|<l(l_/),whenJi>;-,
and then I !^i| < J (1 + ^)< 1
n
when n > r.
[Note. If lim 1 2/„ + i-7-?f„ [> 1, Un does not tend to zero, and, by § 2-3, 2 m„ does not
n=l
converge.]
Example 1. If | c | < 1, shew that the series
M=l
converges absohitely for all values of z.
[For i(„ + iK = c<'*+^)'-^'%^ = c2»+i e^-*0, as k-* oo, if | c | < 1.]
Example 2. Shew that the series
,,«-6 ,(a-b){a-2b) „ (« -6) (a- 26) (a -36) ,
converges absolutely if | s | < [ 6 [ ~ ^.
[For'^^^^^^^-'^^s-^
!6^|<1, i.e. |2!<|6j-i.]
rri ''f» + i a — nh
'- ''~!<~~ n + l ^'^~ ' *^ >i-*« ; so the condition for absolute convergence is
Opuscules, t. V. (1768), pp. 171-182.
2-36, 2-37]
THE THEORY OF CONVERGENCE
23
Example 3. Shew that the series 2 — — j- zr^^n converges absolutely if \z\<\.
n- I
[For, when |«|<1, | i»- (l + «-»)» | > (l+»-i)»-| s" | > 1 + I + -2- +... - 1 >1, so the
moduli of the terms of the series are less than the corresponding terms of the series
00
2 « |2"~i I ; but this latter series is absolutely convergent, and so the given series con-
n = l
verges absolutely.]
2'37. A general theorem on series for which lim -^^
It is obvious that if, for all values of n greater than some fixed value r,
I Un+1 1 is greater than \Un \, then the terms of the series do not tend to zero as
= 1.
tends to
/I -*- 00 , and the series is therefore divergent. On the other hand, if
is less than some number which is itself less than unity and independent
of n (when n > r), we have shewn in § 2"o6 that the series is absolutely con-
vergent. The critical case is that in which, as n increases,
the value unity. In this case a further investigation is necessary.
We shall now shew that* a series Ui+ U2 + Us+ ..., in which lim
will be absolutely convergent if a positive number c exists such that
I '^71+ 1
= 1,
lim n
n-^ CO
-1 =-1
For, compare the series ^ | m^ j with the convergent series ^v„, where
Vn = An-^-^^'
and J. is a constant ; we have
Vn
n + 1
^^*'=ri + iv'''*'=i-~v±-i%o
As n -* CO ,
n{^^-l[--l-ic,
i Vn )
and hence we can find m such that, when n > m,
By a suitable choice of the constant A, we can therefore secure that for
all values of n we shall have
\Un\< Vn.
As ^Vn is convergent, S j w„ | is also convergent, and so l,Un is absolutely
convergent.
* This is the second (D'Alembert's theorem given in § 2-36 being the first) of a hierarchy of
theorems due to De Morgan. See Chrystal, Algebra, Chap. xxvi. for an historical account of
these theorems.
24
THE PROCESSES OF ANALYSIS
[chap. II
Corollary. If K^ =1 + — + 0(-J where A^ is independent of n,
then the series is absolutely convergent if A^< — 1.
Example. Investigate the convergence of 2 «'" exp ( — ^ 2 — ] , when r> k and when
?• < ^.
2"38. Convergence of the hypergeometric series.
The theorems which have been given may be illustrated by a discussion
of the convergence of the hypergeometric series
a.h a(a+l)b(b + l) ^ a(a + l)(a + 2)b(b + l)(b+2) ^
^l.c"""^ 1.2.c(c + l) 1.2.3.c(c + l)(c + 2) ^ + '•■>
which is generally denoted by i'' (a, 6 ; c ; z).
If c is a negative integer, all the terms after the (1 - c)th have zero
denominators; and if either a or 6 is a negative integer the series will
terminate at the (1 — a)th or (1 — 6)th term as the case may be. We shall
suppose these cases set aside, so that a, b, and c are assumed not to be
negative integers.
In this series
(a + n-l){b-\-n-l)
I n{c + n—\) I
as ?i ^ 00 .
We see therefore, by § 2"36, that the series is absolutely convergent when
I 2: j < 1, and divergent when \z\ > 1.
When \z\ = l, we have*
nn-\-\
1 +
1 +
/>-
1
1-
n
/1\
+
0
{^i)
c-\ ^ /I
+ 0 -,
n \n^
_ a-\-b— c -1
n
Let a, b, c be complex numbers, and let them be given in terms of their real
and imaginary parts by the equations
a = a' + ia!', b = b' + ib", c = c -\- ic" .
Then we have
1 ^ ^ + 0 -
n \n-
= ^ 1 1 + «:± _*'_- «i- ly + ("""^'--"y + 0 {^ ('
a' + b' — c -1 ^ / 1
n \n-
By § 2'37, Corollary, a condition for absolute convergence is
a' +b'-c' <0.
* The symbol 0 (1/w^) does not denote the same function of n throughout. See § 2-11.
2-38-2-41] THE THEORY OF CONVERGENCE 25
Hence when \z\ = l, a sufficient condition* for the absolute convergence of
the hypergeometric series is that the real part of a + b — c shall be negative.
2*4. Effect of changing the order of the terms in a series.
In an ordinary sum the order of the terms is of no importance, for it
can be varied without affecting the result of the addition. In an infinite
series, however, this is no longer the casef, as will appear from the following
example.
Let S=l + 3-2 + 5 + 7-4 +9 + 11 -6 + ...,
J cf 1 1 1 1 1 1
and ^ = i_- + __- + -_^- + ...,
and let ^n and Sn denote the sums of their first n terms. These infinite
series are formed of the same terms, but the order of the terms is different,
and so 2^ and 8n are quite distinct functions of n.
Let cr^ = J- + 2 + • • • + .;j ' so that Sn = cr^n — CTn •
mi- V 1 1 . Ill 1
Then X3n = i + 3+...+4-ji-2-4--"-2^
2n
— ^iii o ""an o O'r,
2
Making /i -* 00 , we see that
1
— (0"4>i — 0"2n) + 2 \^2n — <^n)
2 = o + n o ;
2
this equation shews that the derangement of the terms of S has altered
its sum.
Example. If in the series
the order of the terms be altered, so that the ratio of the number of positive terms to the
number of negative terms in the first n terms is ultimately a^, shew that the sum of the
series will become log (2a). (Manning.)
2"41. The fundamental property of absolutely convergent series.
We shall shew that the sum of an absolutely convergent series is not
affected by changing the order in which the terms occur.
Let S = Ui + U2 + U3 + ...
* It may be shewn that the condition is also necessary. See Bromwich, Infinite Series,
pp. 202-204.
t We say that the series S f„ consists of the terms of 2 w,^ in a different order if a law
is given by which corresponding to each positive integer p we can find one (and only one)
integer q and vice versa, and Vq is taken equal to «„.
26 THE PROCESSES OF ANALYSIS [CHAP. ll
be an absolutely convergent series, and let S' be a series formed by the same
terms in a different order.
Let e be an arbitrary positive number, and let n be chosen so that
I I I I I I ^
I ^»i+i I + I W'»+2 I + • • • + I Ufi+p I < 2 ^
for all values of p.
Suppose that in order to obtain the first n terras of S we have to take
m terms of S' ; then if A; > m,
^k = ^n + terms of S with suffices greater than n,
so that
Sfc' — S = 8n — S + terms of S with suffices greater than n.
Now the modulus of the sum of any number of terms of 8 with suffices
greater than n does not exceed the sum of their moduli, and therefore is less
than 2 f •
Therefore \Sk' - S \< \ Sn- S \-\rle.
But \Sn-S\^ lim {I Un+i I + I Un+2 !+... + ! Un+p | }
1
<2^.
Therefore given e we can find m such that
\S,'-S\<e
when k>m; therefore Sm^8, which is the required result.
If a series of real terms converges, but not absolutely, and if >S^^ be the
sum of the first p positive terms, and if o-„ be the sum of the first n negative
terms, then Sp^ cc , an-*— oo ; and lim (Sp + an) does not exist unless we
are given some relation between p and n. It has, in fact, been shewn by
Riemann that it is possible, by choosing a suitable relation, to make
lim (Sp + an) equal to any given real number*.
2'5. Double seriesf.
Let Um^n he a number determinate for all positive integral values of 711
and n ;
consider the" array
Ul,u
Wl,2,
Wl,3, ••
U2,l,
Ur.,2,
U2,i, ■■
^3,l>
Uz,-2,
U3,3,..
* Gea. Werke, p. 221.
t A complete theory of double series, on which this account is based, is given by Pringsheim,
Miliichener Sitztrngsberichte, xxvii. pp. 101-152. See further memoirs by that writer, Math. Ann.
Liii. pp. 289-321 and by London, ibid. pp. 322-370, and also Bromwich, Infinite Series,
which, in addition to an account of Pringsheim's theory, contains many developments of the
subject. Other important theorems are given by Bromwich, Proc. London Math. Soc. i.
pp. 176-201.
2-5, 2'51] THE THEORY OF CONVERGENCE 27
Let the sum of the terms inside the rectangle, formed by the first
m rows of the first n columns of this array of terms, be denoted by <S,„^„.
If a number S exists such that, given any arbitrary positive number e, it
is possible to find integers m and n such that
\S^,.-8\<e
whenever both ix>m and v>n, we say* that the double series of which the
general element is u^^ ^ converges to the sum S, and we write
lim S^^ ^ = S.
jit -» JO, J/-* oo
If the double series, of which the general element is \u^^^\, is convergent,
we say that the given double series is absolutely convergent.
Since Wm,^ = ('S^m,i' ~ 'S^M.t— i) — ('S^Mr*- *— 'S^M-i.^-iX i* is easily seen that, if
the double series is convergent, tfien
lim w^_ „ = 0.
JUL -^ QO , V —^ 00
Stolz' necessary and sufficient j- condition for convergence. A condition for
convergence which is obviously necessary (see § 2-22) is that, given e, we can
find m and n such that j jS'^+p, ^+, — >Sf^ „ | < e whenever ix>m and v>n and
p, (T may take any of the values 0, 1, 2 The cndition is also sufficient ;
for, suppose it satisfied ; then, when /jb>ni + n, \ »S^+p,;x+p - S^^^ \ < e.
Therefore, by § 2-22, S^^^ has a limit S; and then making p and a tend to
infinity in such a way that fji. + p = v + a; we see that \ 8 - S^^^\-^e when-
ever fi>m and v>n; that is to say, the double series converges.
Corollary. An absolutely convergent double series is convergent. For if the double
series converges absolutely and if t-^n be the sum of m rows of n columns of the series of
moduli, then, given f , we can find /x such that, when p>m> jx and q> n> /x, tp^ ^ - ^,„, n < f •
But \Sp,q- Sjn,n\^tp,q-tm,n ^^d so \ Sp^q- Sm,n\<f when JO > wi > /Li, J > ?i > /x ; and this
is the condition that the double series should converge.
2"51. MethodsX of summing double ser'ies.
00 00
Let us suppose that % u^^^, converges to the sum 8^. Then S S^ is
called the sum by rows of the double series ; that is to say, the sum by rows
00/00\ 00/CO\
is S ( S u^^A. Similarly, the sum by columns is defined as S ( S u^^^j.
That these two sums are not necessarily the same is shewn by the example
S^^^= , in which the sum by rows is — 1, the sum by columns is + 1 ;
and S does not exist.
* This definition is practically due to Cauchy {Analyse Algebraique, p. 540).
t This condition, stated by Stolz (Math. Ann. xxiv.), appears to have been first proved by
Pringsheim.
X These methods are due to Cauchy.
28 THE PROCESSES OF ANALYSIS [CHAP. II
Pringsheim's theorem *: If S exists and the sums hy rows and columns
exist, then each of these sums is equal to S.
For since S exists, then we can find m such that
I ^ft, v~ S \< ^, if fJb> m., V > m.
And therefore, since lim 8^^^ exists, |( lim Sfj,^^) — 8\^e\ that is to say,
S Sp — S\i^€ when yu, > m, and so (§ 2'22) the sum by rows converges to ^S'.
In like manner the sum by columns converges to 8.
2"52. Absolutely convergent double series.
We can prove the analogue of § 2*41 for double series, namely that if the
terms of an absolutely convergent double series are taken in any order as a
simple series, their sum tends to the same limit, provided that every term occurs
in the summation.
Let a^^t, be the sum of the rectangle of jx rows and v columns of the
double series whose general element is ! u^^ „ \ ; and let the sum of this double
series be a. Then given e we can find m and n such that a — a^^^, < e
whenever both /x > 7n and v > n.
Now suppose that it is necessary to take N terms of the deranged series
(in the order in which the terms are taken) in order to include all the terms
of >S^3/_,.i j^+i, and let the sum of these terms be t]^.
Then tj^^r— 8m+i,3i+\ consists of a sum of terms of the type Up^q in which
p >m, q >n whenever M > m and M>n; and therefore
I ^A' — ^^31+1, M+1 I < O" — 0"3/+l, 31+1 < 2^'
Also, *S^ — 8m+i, 3T+1 consists of terms iip^ q in which p > m, q > n; therefore
I 8 — 8M+i,3i+-i I ^ cr — o-3[+i,3T+i < 2 ^ ' therefore \8—t2f\<e; and, corresponding
to any given number e, we can find N; and therefore tN-*'8.
Example 1. Prove that in an absolntelj convergent double series, 2 2<,„,n exists, and
n = l
thence that the sums by rows and columns respectively converge to S.
[Let the sum of ^i rows of v columns of the series of moduli be t^^v, and let t be the sum
of the series of moduli.
Then 2 | ?ija, v\<t, and so 2 Uf^, „ converges ; let its sum be h^ ; then
v=l v=\
\W + \b.2\ + ... + \h^\^ lim t^^v^t
f-*- 00
and so 2 6^ converges absolutely. Therefore the sum by rows of the double series
//. = !
exists, similarly the sum by columns exists ; and the required result then follows from
Pringsheim's theorem.]
* Loc. cit. p. 117.
2-52-2'6] THE THEORY OF CONVERGENCE 29
Example 2. Shew from first principles that if the terms of an absolutely convergent
double series be arranged in the order
Wl. 1 + («2. l + Wl, 2) + (W3.1 + ^*2.2 + «1.3) + (W4.1 + •••+«!. 4) + •..,
this series converges to S.
2"53. Cauchys theorem* on the multiplication of absolutely convergent
series.
We shall now shew that if two series
<S = Iti + Ms + ^3 + . . .
and T = Vi + V.2 + V3 + . . .
are absolutely convergent, then the series
P = UiVi + U2V1 + U1V2 + ...,
formed by the products of their tei^ms, written in any order, is absolutely con-
vergent, and has for sum ST.
Let Sn = U^ + «2 + . . . 4- Mn.
Tn = Vi+ V.+ ... +Vn.
Then ST = lim Sn Hm T,, = lim (S,, Tn)
by example 2 of § 2'2. Now
Sn Tn = WiVi + U^Vj + . . . + UnVi
+ U^V2 + U2V2 + ... + UnV2
+
+ UiVn + IhVn + . . . + UnV^.
But this double series is absolutely convergent; for if these terms are
replaced by their moduli, the result is cTnTn, where
0'n= |Wi| + 1^2!+ ••• + i^nl,
and o-„T„ is known to have a limit. Therefore, by § 2*52, if the elements of
the double series, of which the general term is u^Vn, be taken in any order,
their sum converges to ST.
Example. Shew that the series obtained by multiplying the two series
2 ^2 2^ 3* 111
1+2 + 25 + 2^ + 25+-' ^+.-+.^ + .-5 + -'
and rearranging according to powers of z, converges so long as the representative point of z
lies in the ring-shaped region bounded by the circles | 2 1 = 1 and | z | = 2.
2 '6. Power- Series f.
A series of the type
Oo + fli 2 + a-.z- + a.jZ^ + ...,
in which the coefficients a^, ai, a.^,, «y, ... are independent of ^'j is called a series
proceeding according to ascending powers of z, or briefly a poiuer-series.
* Analyse Algebraique, Note vii.
t The results of this section are due to Caucby, Analyse Algebraique, Chap. ix.
30 THE PROCESSES OF ANALYSIS [CHAP. II
We shall now shew that if a power-series converges for any value z^ of z,
it will be absolutely convergent for all values of z whose representative points
are within a circle which passes through Zq and has its centre at the origin.
00
For if z be such a point, we have 1 2^ j < | ^'o | . Now since S a„V* converges,
»=o
anZo^ must tend to zero as n ^ cc , and so we can find M (independent of n)
such that
I anZo"" I < M.
Thus I anZ'' i < 31
Therefore every term in the series 2 \anZ^ \ is less than the corresponding
71 = 0
term in the convergent geometric series
2 m\-\ ■
the series is therefore convergent ; and so the power-series is absolutely
convergent, as the series of moduli of its terms is a convergent series;
the result stated is therefore established.
00
Let lim | a„ \~^'"^ = r; then, from § 2"35, S a,i^"^ converges absolutely when
M=0
00
\z\<r; if I ^ I > r, a^^r" does not tend to zero and so S a„^" diverges (§ 2"3).
«=o
The circle \z\=r, which includes all the values of z for which the
power-series
«o + a^z + a.,z^ + agZ^ + ...
converges, is called the aVcZe of convergence of the series. The radius of
the circle is called the radius of convergence.
In practice there is usually a simpler way of finding r, derived from d'Alembert's
test (§ 2'36) ; r is lim («„/«„ + 1) if this limit exists.
A power-series may converge for all values of the variable, as happens, for
instance, in the case of the series*
z^ z^
z J
3! 5! ■"'
which represents the function sin z ; in this case the series converges over the
whole ^--plane.
On the other hand, the radius of convergence of a power-series may be
zero ; thus in the case of the series
we have , ™'±i -^ ^^^ 1 2: 1
9/ I 11'
[ "m
* The series for e*, sin z, cos z and the fundamental properties of these functions and of
log z will be assumed throughout. A brief account of the theory of the functions is given
in the Appendix.
2-61] THE THEORY OF CONVERGENCE 31
which, for all values of n after some fixed value, is greater than unity when
z has any value different from zero. The series converges therefore only at
the point z = 0, and the radius of its circle of convergence vanishes.
A power-series may or may not converge for points which are actually on
the periphery of the circle ; thus the series
z z^ z^ z*
iH 1 h — I — ^-1-...,
1* 2* 3» 4«
whose radius of convergence is unity, converges or diverges at the point z=l
according as s is greater or not greater than unity, as was seen in § 2*33.
Corollary. If («„) be a sequence of positive terms such that Hqi (a,n-i/«„) exists, this
limit is equal to lim a^}''^.
2"61. Convergence of series derived from a power-series.
Let ao + aiZ + a^z^ + asZ^ + QiZ* + ,.,
be a power-series, and consider the series
tti -I- 2a2Z + SttsZ^ + 4taiZ^ + ...,
which is obtained by differentiating the power-series term by term. We
shall now shew that the derived series has the same circle of convergence as the
original series.
For let z he a point within the circle of convergence of the power-series ;
and choose a positive number ?'i , intermediate in value between | z j and r the
radius of convergence. Then, since the series ^ ofn^i" converges absolutely, its
re = 0
terms must tend to zero as n -♦ oc ; and it must therefore be possible to find a
positive number M, independent of n, such that | a„ | < Mr-^~''^ for all values
of w.
Then the terms of the series 2 n j «„ | \z\'^~'^ are less than the corre-
M = l
spending terms of the series
M ^ n |^!"-i
ji— 1
n «=i r.
But this series converges, by § 2"36, since \z\<r-^. Therefore, by § 2-34, the
series
• • 00
n = \
oc
converges ; that is, the series S nanz''^~^ converges absolutely for all points z
situated within the circle of convergence of the original series 2 a„^". When
\ z\> r, anZ'^ does not tend to zero, and a fortiori nanZ''^ does not tend to
zero ; and so the two series have the same circle of converoence.
32 THE PROCESSES OF ANALYSIS [CHAP. II
Similarly it can be shewn that the series X — — - , which is obtained bv
»j=o w + 1 -^
integrating the original power-series term by term, has the same circle of
convergence as S <x„2^**.
2 "7. Infinite Products.
We proceed now to the consideration of a class of limits, known as
infinite products.
Let 1 + «!, 1 + a2, l+tta, ... be a sequence such that none of its members
vanish. If, as ?i -* oo , the product
(1 + aO (1 + ^2) (1 + as) ••• (1 + aj
(which we may denote by n„) tends to a definite limit other than zero, this
limit is called the value of the infinite product
n, = (1 + a,) (1 + a,) (l+tts) ...,
and the product is said to be convergent. It is almost obvious that a necessary
condition for convergence is that lim a„ = 0, since lim Hn-i — li^^ ^n =1= ^•
The limit of the product is written 11 (1 + o.^).
in Cm ^
Now n (1 + a^) = exp ] X log (1 + an)\,
and* exp { lim Um] = bni {exp Um]
if the former limit exists ; hence a sufficient condition that the product
00
should converge is that S log (1 + «») should converge when the logarithms
have their principal values. If this series of logarithms converges absolutely,
the convergence of the product is said to be absolute.
The condition for absolute convergence is given by the following theorem :
in order that the infinite product
(1 +ai)(l + a2)(l + a3)...
may he absolutely convergent, it is necessary and sufficient that the series
tti + a2 + 0.3 + . . .
should he absolutely convergent.
For, by definition, 11 is absolutely convergent or not according as the
series
log(l+ai) + log(l + fto) + log (1 + tta) + ...
is absolutely convergent or not.
* See the Appendix.
27, 271] THE THEORY OF CONVERGENCE 33
Now, since lim a„ = 0, we can find m such that when n > m, | a„ | < ^ ; and
then
dn O/fi O'n
I a„-^ log(l + a„) - 1 I = I - ^ + ^ - ^ + ...
^ 22 2» 2
And thence, when n>m, ^ ^ —^ — ^ o ' therefore, by the comparison
theorem, the absolute convergence of S log (1 + «„) entails that of 2a„ and
conversely, provided that a„ =f — 1 for any value of n.
This establishes the result*.
If, in a product, a finite number of factors vanish, and if, when these are suppressed,
the resulting product converges, the original product is said to converge to zero. But such
CO
a product as n (1 - n~^) is said to diverge to zero.
Corollary. Since, if Sn-»~l, exp {Sn)-*-exi) I, it follows from § 2'41 that the factors
of an absolutely convergent product can be deranged without afifiecting the value of the
product.
00 00
Example 1. Shew that if n (1 +a„) converges, so does 2 log (1 +«„)) if the logarithms
have their principal values.
Example 2. Shew that the infinite product
sin z sin \z sin ^z sin \z
z ' iz ' ^z ' ^z
is absolutely convergent for all values of z.
For (sin ^ ) / ( ^ ) can be written in the form 1 — ^ , where | X„ | < ^ and k is inde-
pendent of n ; and the series 2 -^ is absolutely convergent, as is seen on comparing
n=l 'i
it with 2 -^. The infinite product is therefore absolutely convergent.
2"71. Some examples of infinite products.
Consider the infinite product
which, as will be proved later (§ 7'5), represents the function z~'^ sin z.
In order to find whether it is absolutely convergent, we must consider the
series X — r-„, or— „ S -; this series is absolutely convergent, and so the
product is absolutely convergent for all values of z.
Now let the product be written in the form
* A discussion of the convergence of infinite products, in which the results are obtained
without making use of the logarithmic function, is given by Pringsheim, Math. Aim. xxxiii.
pp. 119-154, and also by Bromwich, Infinite Series, Chap. vi.
W. M. A. 3
34 THE PROCESSES OF ANALYSIS [CHAP. II
The absolute convergence of this product depends on that of the series
z z z z
IT TT 27r 27r "*
But this series is only conditionally convergent, since its series of moduli
\z\ \z\ \z\ \z\
— ' + — + ^-^ + k- +
is divergent. In this form therefore the infinite product is not absolutely
convergent, and so, if the order of the factors [ 1 + — j is deranged, there is
a risk of altering the value of the product.
Lastly, let the same product be written in the form
in which each of the expressions
I 1 + — ) e^rmt
is counted as a single factor of the infinite product. The absolute convergence
of this product depends on that of the series of which the (2m — l)th and
(2m)th terms are
\ mTTj
But it is easy to verify that
\ mirj \m^J
and so the absolute convergence of the series in question follows by comparison
with the series
- 1 1 1 1 1 1
l+l + 2^. + 25+32 + p + p+45+....
The infinite product in this last form is therefore again absolutely
± —
convergent, the adjunction of the factors e "'' having changed the con-
vergence from conditional to absolute. This result is a particular case of
the first part of the factor theorem of Weierstrass (§ 7"6).
Example 1. Prove that IT ]( 1 ] e»4 is absolutely convergent for all values of
2, if c is a constant other than a negative integer.
For the infinite product converges absolutely with the series
2 ni~--~]e^-l
n=\\\ c + n
2-71] THE THEORY OF CONVERGENCE 35
Now the general term of this series is
But 2 -^converges, and so, by §2-34, 2 \{l ■)e''-li converges absolutely,
n=l'«" n=l l\ C + 7lJ )
and therefore the product converges absolutely.
Example 2. Shew that n -|l-(l — j 2~"^ converges for all points z situated
outside a circle whose centre is the origin and radius unity.
For the infinite product is absolutely convergent provided that the series
2 1-- 2-"
n=2 \ nj
is absolutely convergent. But lim (1 — j =e, so the limit of the ratio of the (?i+l)th
term of the series to the Kth term is - ; there is therefore absolute convergence when
Example 3. Shew that
< 1, i.e. when U | > 1.
1.2.3...(7n-l)
{z + \){z+2)...{z + m-l)
tends to a finite limit as m -*- oc , unless 2 is a negative integer.
For the expression can be written as a product of which the nth factor is
This product is therefore absolutely convergent, provided the series
is absolutely convergent ; and a comparison with the convergent series 2 -^ shews that
this is the case. When 2 is a negative integer the expression does not exist because one of
the factors in the denominator vanishes.
Example 4. Prove that
.6-
1-^-111+^
1-;
1 + ;
--log2 .
: e T SHI 2.
For the given product
lim 2(1-
i-^Ui+^
1-
: lim
V 2 •.^ 4 2 2A:-1 -ilckl
1-
2ki
1 +
X2 1-
1-
2h
(i'lkn , 1 +
,> Icn
lime -'^ 2^3 "^t-i 2k)ji_^Vlfi+L\e ^ ('i _ AV,2. ^ +_l') ^"2^ ... ,
3—2
36 THE PROCESSES OF ANALYSIS [CHAP. II
since the product whose factors are
\ rr j
is absolutely convergent, and so the order of its factors can be altered.
Since log2 = l-| + ^-i + ^- •••>
this shews that the given product is equal to
--log 2 .
e T sin z.
2*8. Infinite Determinants.
Infinite series and infinite products are not by any means the only
known cases of limiting processes which can lead to intelligible results. The
researches of G. W. Hill in the Lunar Theory* brought into notice the
possibilities of infinite determinants.
The actual investigation of the convergence is due not to Hill but to Poincar^ Bull, de
la Soc. Math, de France^ xiv. (1886), p. 87. We shall follow the exposition given by
H. von Koch, Acta Math. xvi. (1892), p. 217.
Let Aik be defined for all integer values (positive and negative) of i, k,
and denote by
■L-'m^^ \_-^ik\i,k = —m,...-\-m
the determinant formed of the numbers Ai]c{i, k= — m, ...+wi); then if,
as VI ^ oc , the expression D^ tends to a determinate limit D, we shall say
that the infinite determinant
L-^ ifcJi,A; = — 00 — f-x
is convergent and has the value D. If the limit D does not exist, the deter-
minant in question will be said to be divergent.
The elements An, (where i takes all values), are said to form the principal
diagonal of the determinant D\ the elements A^k, (where i is fixed and k
takes all values), are said to form the row i; and the elements Ai]c, (where k
is fixed and i takes all values), are said to form the column k. Any element
Aijc is called a diagonal or a non-diagonal element, according as i = k ov i^k.
The element A^^^ is called the origin of the determinant.
2'81. Convergence of an infinite determinant.
Wo shall now shew that an infinite determinant converges, provided the product of the
diagonal elements converges absolutely, and the sum of the non-diagonal elements converges
absolutely.
For let the diagonal elements of an infinite determinant D be denoted by l+aji,
and let the non-diagonal elements be denoted by aj^, (^=t=^), so that the determinant is
... l + <x_i-i a-io a~n--
«() _ 1 1 -j- Cloo <^01
«!_! aio l + ai, .,
Repriuted in Acta Mathematica, viii. pp. 1-3G (1886).
2-8-2-82] THE THEORY OF CONVERGENCE _ 87
Then since the series
is convergent, the product
is convergent.
Now form the products
2 i a,-fc j
'= n (l+ 2 \aa\)
P,„= n 1+ 2 a,fc
i=~in \ k=—m
P,n= 0 1+2
i= — m \ k=—m
hk ! j ;
then if, in the expansion of /*,„, certain terms are rei^laced by zero and certain other
terms have their signs changed, we shall obtain D^ ; thus, to each term in the expansion
of 2>„ there corresponds, in the expansion of /*,„, a term of equal or greater modulus.
Now D,n + p — J)m represents the sum of those terms in the determinant i),„ + p which vanish
when the numbers an^li, k= ± (m+1) ... + {m-^p)} are replaced by zero ; and to each of
these terms there corresponds a term of equal or greater modulus in Pm + p-P-m-
Hence I D^^^ + p — D^ \ ^ P^ + p~ P.^.
Therefore, since P,„ tends to a limit as to -*• x , so also D,„ tends to a limit. This
establishes the proposition.
2'82, The rearrangement Theorem for convergent infinite determinants.
We shall now shew that a determinant, of the convergent form already considered,
remains convergent when the elements of any row are replaced by any set of ele^nents whose
'moduli are all less than some fixed positive number.
Replace, for example, the elements
■•• -^0, -wi) ••• ^0 ••• -4o,m •••
of the row through the origin by the elements
... /x_,„, ... ^0 ••• Mm •••
which satisfy the inequality
! Mr I < M>
where /x is a positive number ; and let the new values of i),„ and D be denoted by
Z)„/ and ly. Moreover, denote by /*,„' and P" the products obtained by suppressing in
/*„i and P the ftxctor corresponding to the index zero ; we see that no terms of /)„/ can
have a greater modulus than the corresponding term in the expansion of /xP,^' ; and
consequently, reasoning as in the last article, we have
which is sufficient to establish the result stated.
Example. Shew that the necessary and sufficient condition for the absolute conver-
gence of the infinite determinant
lim 1 ai 0 0 ... 0
^1 1 02
0 /32 1
0 ...
ag ...
0
0
0 ... 0 13,,
1
is that the series
shall be absolutely convergent. (von Koch.)
38
THE PROCESSES OF ANALYSIS
[chap. II
REFERENCES.
Convergent series.
A. Pringsheim, Math. Annalen, Bd. xxxv.
T. J. I'a. Bromwich, Theory of Infinite Series, Chs. ii. lii. iv.
Conditionally convergent series.
G. F. B. RiEMANN, Ges. Math. Werke, pp. 221-225.
A. Pringsheim, Math. Annalen, Bd. xxii.
Double series.
A. Pringsheim, Milnchener Sitzungsherichte, Bd. xxvii.
„ „ Math. Annalen, Bd. Liii.
G. H. Hardy, Proc. London Math. Soc, ser. 2, vol. i.
Miscellaneous Examples.
1. Evaluate lim (e~"''#), liin («~" logw) when a>0, 6>0.
2. Investigate the convergence of
3. Investigate the convergence of
ri.3,..2w-l 4?i + 3]2
°° fl-3...2?z-l 4?i + 31 2
.^i{2T4...2n ' 2n + 2) '
(Trinity, 1904.)
(Peterhouse, 1906.)
, 2n 2n
4. Find the range of values of z for which the series
2 sin2 0-4 sin^^ + Ssin^z-... + (-)» + ! 2»sin2"a+...
is convergent.
5. Shew that the series
1 _ _1_ J[ 1_
z z + l"^ z + 2 z + 3'^'"
is conditionally convergent, except for certain exceptional values of z ; hut that the series
1 1 1
+ — T + -- +
1
1
+
+ ...,
2 2 + 1 ■■" z+p-l z+p z+p + l '" z-\-2p + q-l z + 2p-\-q
in which {p + q) negative terms always follow jo positive terms, is divergent. (Simon.)
6. Shew that
i-*-i+fi-J-Hi-.-=*iog2.
7. Shew that the series
is convergent, although
8. Shew that the series
is convergent although
1111
1* 2^ 3* 4^
M2n + l/M2n-*=0-
a + l3^ + a^ + ^*+...
(Trinity, 1908.)
(l<«0)
(Ceskro.)
(0<a</3<l)
(Cesaro.)
THE THEORY OF CONVERGENCE 39
9. Shew that the series
" TOg'*-'{(l+it-^)"-l}
„!x(2»-l){0»-(l +«-')»}
converges absolutely for all values of z, except the values
\ m)
(a = 0, 1; >{;=0, 1, ... »?-l; m = l, 2, 3, ...).
10. Shew that, when «>1,
nfl »• « - 1 ^ ^ti !_%» ^ « - 1 1(% + 1 )«-! «<•-»/ J '
^ and shew that the series on the right converges when 0 < s < 1.
(de la Valine Poussin.)
11. In the series whose general term is
M„ = ^"■"^ri''(''+^>, (0<?<1<^)
where v denotes the number of digits in the expression of n in the ordinary decimal scale
of notation, shew that
J
lim u^n=q_^
and that the series is convergent, although lim m„ + i/m„ = xi .
12. Shew that the series
where
?n = r"*""% (0<J<1)
is convergent, although the ratio of the (?i+l)th term to the wth is greater than unity
when n is not a triangular number. (Cesaro.)
13. Shew that the series
«> Jlrnrix
2 ^ ,
where w is real, and where {w+7iy is understood to mean e^^°s(w+n)^ the logarithm being
taken in its arithmetic sense, is convergent for all values of s, when the imaginary part of
X is positive, and is convergent for values of s whose real part is positive, when x is real.
14. If Un>0, shew that if 2tt^ converges, then lim (nM„)=0, and that, if in addition
Un > Un + 1, then lim (nUn) — 0.
15. If
shew that
m — n (w + ?^- 1) !
"m, n gm + n
2m + n mini
2(2 a^„
»=o \»=o
— 1, 2 12 Clm,n
ji=0 \m = 0
{m, n>0)
(Trinity, 1904.)
16. By converting the series
89^ 16£^ 2V ,
^l-9^1+?2 1-93 '
(in which | g- 1 < 1), into a double series, shew that it is equal to
1 +
(1-^)2 {l+q^f (l-j3):
(Jacobi.)
40
17. Assuming that
THE PROCESSES OF ANALYSIS
sin3 = 3 n ( 1 -
[chap. II
^2^2 ] '
shew that if m -^ oo and % -*- tx) in such a way that Hm {mln)=k, where k is finite, then
lim U' U+'-\ = k^l-^^}^,
r=-n V rtrj z
the dash indicating that the factor for which r = 0 is omitted. (Math. Trip., 1904.)
18. If ■WQ = wi = tt2=0, and if, when ?i> 1,
1 111
00 00 00
then n (H-M„) converges, though 2 m„ and 2 ?«„2 ^re divergent.
«=0 n=:0 7t=0
19. Prove that
(Math. Trip., 1906.)
TT \(^ ^^"* (^^^ ^-w'^'AI
where k is any positive integer, converges absohitely for all values of z.
00 00
20. If 2 a,j be a conditionally convergent series of real terms, then 11 (l+a„) con-
«=i jt=i
verges (but not absolutely) or diverges to zero according as 2 a^ converges or diverges.
(Cauchy.)
21. Let 2 ^„ be an absolutely convergent series. Shew that the infinite determinant
A(c) =
(c- 4)2-^0
4-'-^o \ 4''i-^o 42-^0 42-^0 42-
-^1 (c-2)2-^o -e^ -fi.>
••• 22-^, 22-^ 22-^0 22-^0 22-^0 ••
-^3 -62 -Ql (C + 2)2-^0 -^1
■■■ 22-^0 22-^; 22-^0 '22-^0, 22-^0 ■■■
-^i^c + 4)2-^o
42-^0
42-^0 42-^0 42-^0 42-^0
converges ; and shew that the equation
is equivalent to the equation
A(c) = 0
sin2 -^770 = A (0) sin2 \n6()K
(Hill.)
CHAP.TER III
CONTINUOUS FUNCTIONS AND UNIFORM CONVERGENCE
3'1. The dependence of one complex number on another.
The problems with which Analysis is mainly occupied relate to the
dependence of one complex number on another. If z and f are two complex
numbers, so connected that, if z is given any one of a certain set of values,
corresponding values of ^ can be determined, e.g. if f is the square of z, or if
^=1 when z is real and ^=0 for all other values of z, then ^ is said to be a
function of z.
This dependence must not be confused with the most important case of
it, which will be explained later under the title oi analytic functionality .
If ^ is a real function of a real variable z, then the relation between f and z, which
may be written
can be visualised by a curve in a plane, namely the locus of a point whose coordinates
referred to rectangular axes in the plane are {z, (). No such simple and convenient
geometrical method can be found for visualising an equation
considered as defining the dependence of one complex number ^=^-\-ir) on another
complex number z=.v + i}/. A representation strictly analogous to the one already given
for real variables would require four-dimensional si)ace, since the number of variables
I, 77, X, y is now four.
One suggestion (made by Lie and Weierstrass) is to use a doubly-manifold system of
lines in the quadruply-manifold totality of lines in three-dimensional space.
Another suggestion is to represent ^ and x) separately by means of surfaces
^ = l(-^, y), '? = '?(•^^ .'/)•
A third siiggestion, due to HefFter* is to write
then draw the surface r = r{x, y)— which may be called the modular-surface of the
function — and on it to express the values of 6 by surface-markings. It might be
possible to modify this suggestion in various ways by representing 6 by curves drawn
on the surface r = r {x\ y).
* Zeitschriftfiir Math. u. Phys. xliv. (1899), p. 235.
42 THE PROCESSES OF ANALYSIS [CHAP. Ill
3'2. Continuity of functions of real variables.
The reader will have a general idea (derived from the graphical represen-
tation of functions of a real variable) as to what is meant by continuity.
We now have to give a precise definition which shall embody this vague
idea.
Let f{x) be a function of x defined when a ^x ^h.
Let ooi be such that a^x^^^h. If there exists a number I such that,
corresponding to an arbitrary positive number e, we can find a positive
number t] such that
\f{x)-l\<e,
whenever \x — x■^\<rj, x^^x^, and a^x ^b, then I is called the limit of f{x)
as X ^Xi.
It may happen that we can find a number l^ (even when I does not exist)
such that \f(x) — Z+ j < e when x^Kx <x^+r). We call 1+ the limit of f{x)
when X approaches x^ from the right and denote it by f(xy + 0) ; in a similar
manner we define f(xi — 0) if it exists^
If f(xi+ 0), f(xi), f(xi — 0) all exist and are equal, we say that f{x) is
continuous at x^^ ; so that ii f{x) is continuous at x-i, then, given e, we can find
7] such that
\f(x) -f{x,) i < e,
whenever \x — Xi\<7] and a^x^b.
If ^4. and L exist but are unequal, f(x) is said to have an ordinary
discontinuity* at x^; and if l+ = l^^f(x^),f{x) is said to have a removable
discontinuity at Xj.
If f(x) is a complex function of a real variable, and if f(x) = g {x) + i h {x)
where g (x) and h{x) are real, the continuity of f(x) at x^ implies the con-
tinuity of g (x) and of h (x). For when \f(x) —f{x^) \ < e, then \g{x) —g{xi) \ < e
and \h(x) — h (xi) \< e; and the result stated is obvious.
Example. From § 2-2 examples 1 and 2 deduce that if f{.v) and (f) (x) are con-
timious at .Vi, so are / (^) ± </> (.r), f{x)x4>{x) and, if (/)(.ri)4=0, f{x)l^{x).
The popular idea of continuity, so far as it relates to a real variable /(^) depending
on another real variable x, is somewhat different from that just considered, and may
perhaps best be expressed by the statement "The function /(.*■) is said to depend con-
tinuously on X if, as x passes through the series of all values intermediate between any
two adjacent values x^ and X2,f{x) passes through the series of all values intermediate
between the corresponding values /{xj) and /(*'2)."
The question thus arises, how far this popular definition is equivalent to the precise
definition given above.
Cauchy shewed that if a real function / (.r), of a real variable x, satisfies the precise
definition, then it also satisfies what we have called the popular definition ; this result
* If a function is said to have ordinary discontinuities at certain points of an interval it
is implied that it is continuous at all other points of the interval.
3-2, 3-21] CONTINUOUS FUNCTIONS AND UNIFORM CONVERGENCE 43
will be proved in § 3-63. But the converse is not true, as was shewn by Darboux. This
fact may be illustrated by the following example*.
Between x= — 1 and x= + 1 (except at ^=0), let / (a?) = sin — ; and let/(0)«»0.
It can then be proved that f(x) depends continuously on x near x = 0, in the sense of
the popular definition, but is not continuous at ^s=0 in the sense of the precise definition.
Example. If f{x) be defined and be monotonic in the range (a, b), the limits/ (a; ±0)
exist at all points in the interior of the range.
[1{ f(x) be an increasing function, a section of rational numbers can be found such
that, if a, .4 be any members of its Z-class and its iS-class, a<f{x+k) for every positive
value of h and A "^/(x + h) for some positive value of h. The number defined by this
section is/(^ + 0).]
3'21. Simple curves. Gontinua.
Let X and y be two real functions of a real variable t which are continuous
for every value of t such that a^t^b. We denote the dependence of x and y
on t by writing
x = x (t), y = y it). (a^t^b)
The functions x (t), y (t) are supposed to be such that they do not assume the
same pair of values for any two different values of t in the range a< t< b.
Then the set of points with coordinates (x, y) corresponding to these values
of t is called a simple curve. If
x(a) = x (b), y(a)==y (b),
the simple curve is said to be closed.
Example. The circle x^ + i/^=l is a simple closed curve ; for we may write t
A'=cos^, 7/ = smt. {0^t^2Tr)
A two-dimensional continuum is a set of points in a plane possessing the
following two properties :
(i) If (x, y) be the Cartesian coordinates of any point of it, a positive
number 3 (depending on x and y) can be found such that every point whose
distance from {x, y) is less than S belongs to the set.
(ii) Any two points of the set can be joined by a simple curve consisting
entirely of points of the set.
Example. The ^joints for which x'^+y'^Kl form a continuum. For if P be any
point inside the unit circle such that OP=r<l, we may take 8 = 1 -r; and any two
points inside the circle may be joined by a straight line lying wholly inside the circle.
The following two theorems^ will be assumed in this work ; simple cases
of them appear obvious from geometrical intuitions and, generally, theorems
of a similar nature will be taken for granted, as formal proofs are usually
extremely long and difficult.
* Due to Mansion, Mathens, ix. (1899).
t It may be proved that the sine and cosine are continuous functions. See the Appendix.
X Formal proofs will be found in Watson's Complex Integration and Caiiclujs Theorem.
(Cambridge Math. Tracts, No. 15.)
44 THE PROCESSES OF ANALYSIS [CHAP. Ill
(I) A simple closed curve divides the plane into two continua (the
' interior ' and the ' exterior ').
(II) If P be a point on the curve and Q be a point not on the curve,
the angle between QP and Ox increases by + 27r or by zero, as P describes
the curve, according as Q is an interior point or an exterior point. If the
increase is + 27r, P is said to describe the curve ' counterclockwise.'
A continuum formed by the interior of a simple curve is sometimes called
an open two-dimensional region, or briefly an open region, and the curve is
called its boundary ; such a continuum with its boundary is then called a
closed two-dimensional region, or briefly a closed region or domain.
A simple curve is sometimes called a closed one-dimensional region ; a
simple curve with its end-points omitted is then called an open one-dimensional
region.
3'22. Continuous functions of complex variables.
Let f{z) be a function of z defined at all points of a closed region (one- or
two-dimensional) in the Argand diagram, and let z^ be a point of the region.
Then/(^) is said to be continuous at z-i, if given any positive number e,
we can find a corresponding positive number 77 such that
\f{z)-f{z,)\<e,
whenever \z — Zy\< 7^ and ^^ is a point of the region.
3'3. Series of variable terms. Uniformity of convergence.
Consider the series
x!^ x^ x^
7.2 -I 4- -1- -I - 4-
^ \ -^ x"" {I + x-'f {1 + x^Y ""
This series converges absolutely (§ 2"33) for all real values of x.
If 8n {x) be the sum of n terms, then
and so lim 8^ {x) = 1 + a;^ ; (« ^ 0)
but >S'„(0) = 0, and therefore lim >Sf„(0)=0.
Consequently, although the series is an absolutely convergent series of
continuous functions of x, the sum is a discontinuous function of x. We
naturally enquire the reason of this rather remarkable phenomenon, which
was investigated in 1841-1848 by Stokes*, Seidelf and Weierstrass J, who
shewed that it cannot occur except in connexion with another phenomenon,
that of non-uniform convergence, which will now be explained.
* Collected Papers, i. p. 236.
+ Munchen Abhandlunpen, v. (1848), p. 381.
+ Ges. Math. Werke, 1. pp. 67, 75.
3'22-3-31] CONTINUOUS functions and uniform convergence 45
Let the functions u^ (z), u^ (z), ... be defined at all points of a closed region
of the Argand diagram. Let
Sn (z) = Wi (Z) + U^{z)+ ...+ Iln {z).
00
The condition that the series S w„ {z) should converge for any particular
M = l
value of z is that, given e, a number n should exist such that
I Sn+p {z) -8n{z)\<e
for all positive values of p, the value of n of course depending on e.
Let n have the smallest integer value for which the condition is satisfied.
This integer will in general depend on the particular value of z which has
been selected for consideration. We denote this dependence by writing
n{z) in place of n. Now it may happen that we can find a number N,
independent of z such that
n{z)< N
for all values of z in the region under consideration.
If this number N exists, the series is said to converge uniformly
throughout the region.
If no such number N exists, the convergence is said to be non-uniform*.
Uniformity of convergence is thus a property depending on a whole set ot
values of z, whereas previously we have considered the convergence of a series
for various particular values of z, the convergence for each value being con-
sidered without reference to the other values.
We define the phrase ' uniformity of convergence near a point z ' to mean
that there is a definite positive number S such that the series converges
uniformly in the domain common to the circle \z — Zi\^h and the region in
which the series converges.
3"31. On the condition for uniformity of convergence\.
If Rn,p {z) = Un+\ {z) + i*n+2 (^) + . . . + u^+p (z), WB have seen that the
necessary and sufficient condition that S Un {z) should converge uniformly
n = \
in a region is that, given any positive number e, it should be possible to
choose N independent of z (but depending on e) such that
I Rn,p (^) \ < e
for ALL positive integral values of p.
* The reader who is unacquainted with the concept of uniformity of convergence will find it
made much clearer by consulting Bromwich, Infinite Series, Chap, vii., where an illuminatiug
account of Osgood's graphical invest gation is given.
t This section shews that it is indifferent whether uniformity of convergence is defined by
means of the partial remainder Ii,^,p(z) or by R^iz). Writers differ in the definition taken
as fundamental.
46 THE PROCESSES OF ANALYSIS [CHAP. Ill
If the condition is satisfied, by § 2 '2 2, Sn {z) tends to a limit, S (z), say for
each value of z under consideration ; and then, since e is independent of p,
|{lim Rtf^p{z)]\^€,
and therefore, when n>N,
S (z) - Sn (z) = { lim Ej^,^ (^)l - Rif^ n-N (z),
and so \S{z)-Sn{z)\<^e.
Thus (writing \e for e) a necessary condition for uniformity of convergence
is that \S{z) — Sn (z) \ < e, whenever n> N and N is independent of z ; the
condition is also sufficient ; for if it is satisfied it follows as in § 2'22 (I)
that j Rn,p {z) I < 2e, which, by definition, is the condition for uniformity.
Example 1. Shew that, if x be real, the sum of the series
1 {x + \) {x+\){'-lx + \) {{n-\)x+\}{nx+\}
is discontinuous at ^=0 and the series is non-uniformly convergent near x=0.
The sum of the first n terms is easily seen to be 1 zr ; so when x — 0 the
'' nx+l
sum is 0; when ^4=0, the sum is 1.
The value of Rn {x) = S {x) - Sn{x) is -,VT ^^ •^' + 0; so when x is small, say
;» = one-hundred-millionth, the remainder after a milUon terms is — j or l-:f7rj-, so
the first million terms of the series do not contribute one per cent, of the sum. And in
general, to make r < e, it is necessary to take
° ' nx + \
X \€
Corresponding to a given e, no number N exists, independent of x, such that n<JV for
all values of x in any interval including x=0 ; for by taking x sufficiently small we can
make n greater than any number JV which is independent of x. There is therefore non-
uniform convergence near x = 0.
Examjjle 2. Discuss the series
X {n{n+l)x^ — l]
2
„Ii {l+n^x'}{l + (n + lfx^}'
in which x is real.
^, , 1 • , ^^•^' (n+l)X ry X 1
The nin term can be written ^ 5— „ - _— f-^ — tinp 9) so /b= , , -„, and
\+n^x^ \ + {n + lf X' \+x^
[Note. In this examjjle the sum of the series is not discontinuous at .r = 0.]
But (taking 6<|, and jr 4=0), \ Rn{x)\ <f iU-'^{n-{-l) \x\<l + {n + iy x^ ; i.e. if
n-t-l>^{e-i-t-v'6-^-4} 1^1-1 or n + l <i {f-i- Vf-^- 4} |^|"^
332] CONTINUOUS FUNCTIONS AND UNIFORM CONVERGENCE 47
Now it is not the case that the second inequality is satisfied for all values of n greater
than a certain value and for all values of x ; and the first inequality gives a value of
n (x) which tends to infinity as ^ -^0 ; so that, corresponding to any interval containing the
point .r = 0, there is no number N independent oi x. The series, therefore, is non-uniformly
convergent near ^=0.
The reader will observe that n{x) is discontinuous at x = Q; for n(^)-^oo as ^-^0,
but n(0) = 0.
3'32. Connexion of discontinuity with non-uniform convergence.
We shall now shew that if a series of continuous functions of z is uniformly
convergent for all values of z in a given closed domain, the sum is a continuous
function of z at all points of the domain.
For let the series be f(z) = Ui{z) + U2(z)+ ... +Un(z) + ... = Sn (z) + Rn (•s^),
where Rn (z) is the remainder after n terms.
Since the series is uniformly convergent, given any positive number e, we
can find a corresponding integer n independent of z, such that | Rniz) i < o e
for all values of z within the domain.
Now n and e being thus fixed, we can, on account of the continuity of
8n (^), find a positive number i) such that
|>Sf,,(^)->S.(/)|<^e,
whenever \z — z' \<'r].
We have then
I f{z) -f{z') \ = \[Sn (Z) - Sn (Z)} \ + \ R^ (z) - R^ {z') \
<\8n{z)-Sn{z)\+\Rn{z)\ + \ Rn{z')\
< e,
which is the condition for continuity at z.
Example 1. Shew that near x — 0 the series
Ui (x) + «2 (x) + U3{x)+ ...,
1 1
■where Ui(x) = x, M„(a7) = ^^"~^-^-"~^,
and real values of x are concerned, is discontinuous and non-iinifornily convergent.
In this example it is convenient to take a slightly different form of the test ; we shall
shew that, given an arbitrarily small number e, it is possible to choose values of x, as
small as we please, depending on n in such a way that | R^ (x) \ is not less than e for any
value of n, no matter how large. The reader will easily see that the existence of such
values of x is inconsistent with the condition for uniformity of convergence.
1
The value of *S',i(*') is x^^''~^ ; as 7i tends to infinity, S^ {x) tends to I, 0, or - 1, accord-
ing as X is positive, zero, or negative. The series is therefore absolutely convergent for all
■values oi X, and has a discontinuity at :r = 0.
48 THE PROCESSES OF ANALYSIS [CHAP. Ill
1
In this series Rji{x) = l -x^"'~^, {x>0) ; however great n maybe, by taking* ;r=e"(2"-i)
we can cause this remainder to take the value 1 — e~', which is not arbitrarily small. The
series is therefore non-uniformly convergent near x = 0.
Example 2. Shew that near z = 0 the series
-" -2g(l+2)»-i
„£i {l + (l+2)»-i}{l + (l+^)~}
is non-uniformly convergent and its sum is discontinuous.
The nth. term can be written
l-(l+2)" _ l-(l+2)»-l
1 + {1+Z)" l-t-(H-2)"-l'
1 — (1+2)"
SO the sum of the first n terms is . . . Thus, considering real values of z greater
than — 1, it is seen that the sum to infinity is 1, 0, or — 1, according as z is negative, zero,
or positive. There is thus a discontinuity at 2=0. This discontinuity is explained by the
fact that the series is non-uniformly convergent near 2=0; for the remainder after n terms
in the series when z is positive is
l+(H-2)»'
and, however great n may be, by taking 2 = 7, this can be made luimerically greater
2
than , which is not arbitrarily small. The series is therefore non-uniformly con-
vergent near 2 = 0.
3'33. The distinction between absolute and uniform convergence.
The uniform convergence of a series in a domain does not necessitate
its absolute convergence at any points of the domain, nor conversely. Thus
the series 2 p. converges absolutely, but (near z=()) not uniformly ;
(1 -|- z^Y
while in the case of the series
V V J
the series of moduli is
1 ^
which is divergent, so the series is only conditionally convergent ; but for all
real values of z, the terms of the series are alternately positive and negative
and numerically decreasing, so the sum of the series lies between the sum of
its first n terms and of its first {n+\) terms, and so the remainder after
n terms is numerically less than the nth. term. Thus we only need take a
finite number (independent of z) of terms in order to ensure that for all real
values of z the remainder is less than any assigned number e, and so the
series is uniformly convergent.
Absolutely convergent series behave like series with a finite number of
terms in that we can multiply them together and transpose their terms.
* This value of x satisfies the condition I a; j < 5 whenever 2n - 1 > log 5" 1.
i
8-33-3-341] CONTINUOUS functions and uniform convergence 49
Uniformly convergent series behave like series with a finite number of
terms in that they are continuous if each term in the series is continuous
and (as we shall see) the series can then be integrated term by term,
3'34. A condition, due to Weierstrass* , for uniform convergence.
A sufficient, though not necessary, condition for the uniform convergence
of a series may be enunciated as follows : —
If, for all values of z within a domain, the moduli of the terms of a series
8=Ui (z) + Wa (z) +U3(2)+ ...
are respectively less than the corresponding terms in a convergent series
of positive terms
where Mn is independent of z, then the series S is uniformly convergent in
this region. This follows from the fact that, the series T being convergent,
it is always possible to choose n so that the remainder after the first n terms
of T, and therefore the modulus of the remainder after the first n terms
of S, is less than an assigned positive number e; and since the value of n
thus found is independent of z, it follows (§ 3"31) that the series S is uni-
formly convergent ; by § 2"34, the series S also converges absolutely.
Example. The series
cos 2 + -, COS'' 2 4- ;Ti COS^ 2 + . . .
is uniformly convergent for all real values of 2, becavise the moduli of its terms are not
greater than the corresponding terms of the convergent series
I + 2I + 3-2+-,
whose terms are positive constants.
3*341. Uniformity of convergence of infinite products^.
A convergent product 11 {1 + Un (2)} is said to converge uniformly in a domain of values
of 2 if, given e, we can find m independent of 2 such that
I m+p m 1
n {i-i-ii„(2)}- n {i+M„(2)}|<<
I n=\ n=l 1
for all positive integral values of p.
The only condition for uniformity of convergence which will be used in this work
is that the product converges uniformly if | Un (2) j < M^ where l/„ is independent of 2 and
2 Mn converges.
n=l
* Ahhandlungen aus der Funktionenlehre, p. 70. The test given by this condition is usually
described (e.g. by Osgood, A^mals of Mathematics, Vol. in. p. 130) as the il/-test.
t The definition is, effectively, that given by Osgood, Funktionentheorie, p. 462. Tlie
condition here given for uniformity of convergence is also established in that work.
W. M. A. 4
dO THE PROCESSES OF ANALYSIS ^ [CHAP. Ill
To prove the validity of the condition we observe that n (l + Mn) converges (§ 2"7),
and so we can choose m such that
m+p m
n {l + M^}- a {l+M,,}<e;
n=l ?i=l
and then we have
m+p m
n {i+u^{z)}- n {l + u„{z)}
= n {i+u,{z)}\ n {i+«„(^)}-i
1 n=l \_n = m+l J |
^n{l + M„)\ n {l + Mn}-l\
and the choice of m is independent of z.
3*35. Hardy's tests for uniform convergence* .
The reader will see, from § 2-31, that if, in a given domain, 2 a„ (2) \^k where «„(«) is-
I n=l I
real and k is finite and independent of p and 2, and if /n (2) ^./ji + 1 (2) and /„(2)-^0
unifoTinly as n ^. oo , then 2 a„ (2)/n (■^) converges uniformly.
n=\
Also that if
^ ^ w™ (2) ^ w„ + 1 (2) ^ 0,
where k is independent of 2 and 2 «„ (2) converges uniformly, then 2 a„ (?) «„ (2) con-
verges uniformly. [To prove the latter, observe that to can be found such that
«»» + l(2), «m + l(2) + «OT + 2(4 •••, «m + l(2)+«m + 2(«) + ---+«m + p(2)
are numerically less than ilk ; and therefore (§ 2-301)
I m-Vv I
2 a„ (2) M„ (2) < eM^ + 1 (2)/^ < f ,
and the choice of t and to is independent of 2.]
' Example 1. Shew that, if S > 0, the series
* cos n^ "* sin nB
2 2
converge uniformly in the range
S < ^ ^ 27r - S.
Obtain the corresponding result for the series
°" ( — )"■ cos n6 °° ( - )" sin n6
2 , 2 ,
7i=\ n n=i n
by writing 6 + 17 for 6.
Example. 2. If, when a^x^b, |a)„(.r)|<^i and 2 | co„ + i (a;)-o)„ (^) | <^2) where
M = l
X-j, ^2 ai"6 independent of n and ^, and if 2 an is a convergent series independent of x,
n=l
then 2 a„a)„(j7) converges uniformly when a ^^^6. (Hardy.)
12=1
* Proc. London Math. Sac. Ser. 2, Vol. iv. (1907), pp. 247-265. These results, which are
generalisations of Abel's theorem (§ 3-71, below), though well known, do not appear to have been
published before 1907. From their resemblance to the tests of Dirichlet and Abel for con-
vergence, Bromwich proposes to call them Diiichlet's and Abel's tests respectively.
3-35-3"4] CONTINUOUS functions and uniform convergence 51
<i, and then, by § 2-301
I m+p
[For we can choose m, independent of x, such that 2 a„
I n=w+l
m+p
2 a„«„(j?) <{kt-i-k2)f.]
n=m+l
corollary, we have
3'4. Discussion of a particular double series.
Let tui and Wg be any constants whose ratio is not purely real; and let
a be positive.
The series S -. ^ ^i zz , in which the summation extends over
all positive and negative integral and zero values of m and n, is of great
importance in the theory of Elliptic Functions. At each of the points
z = — 2mwi — 2n.G)2 the series does not exist. It can be shewn that the series
converges absolutely for all other values of ^^ if a > 2, and the convergence is
uniform for those values of z such that | z + 2m&)i + 2nco^ j ^ S for all integral
values of m and n, where B is an arbitrary positive number.
Let S denote a summation for all integral values of m and n, the term for
which m=n = 0 being omitted.
Now, if m and n are not both zero, and if | 2r 4- 2m(Oi + 2nco2 1 ^ S > 0 for
all integral values of m and n, then we can find a positive number G, de-
pending on B but not on z, such that
1 '<c! 1
(z + 27710)1 + 2na)2)°- 1 I (2mft)i + 2weo2)°' \
Consequently, by § 3"34, the given series is absolutely and uniformly*
convergent in the domain considered if
I mcoi + 710)2 1 "
converges.
To discuss the convergence of the latter series, let
0)i = «! + i^i , 0)2 = Oa + i^2,
where a^, a^, yQj, ySg are real. Since 0)2/0)1 is not real, ai/32 — Oa/Si =f 0. Then
the series is
S 1
This converges (§ 2 "5 corollary) if the series
S = t' ^
(771- + n-f"^
converges ; for the quotient of corresponding terms is
1 + ix'
The reader will easily define uniformity of convergence of double series (see § 3-5).
4—2
62 THE PROCESSES OF ANALYSIS [CHAP. Ill
where /x = w/m. This expression, qua function of a continuous real variable fx,
can be proved to have a positive minimum * (not zero) since OjySj — aa/Si =f 0 ;
and so the quotient is always greater than a positive number K (independent
of /a).
We have therefore only to study the convergence of the series 8. Let
50 9 1 .
t
*■' r^-p rZ-q (m^ + 71^)^* '
1
<4 t t
Separating /Sp_ q into the terms for which m = n, m>n, and m<n, re-
spectively, we have
pi p m-\ 1 q n-\ 1
M-1 (2m2)i'' m=i w=o (m^ + n^)^" «=i m=o (m^ + ri^)**
But "S' ^ ^ ^
Therefore i>S^ S ^^ + 5 -^ + i — .
But these last series are known to be convergent if a — 1 > 1. So the series 8
is convergent if a > 2. The original series is therefore absolutely and uni-
formly convergent, when a > 2, for the specified range of values of z.
Example. Prove that the series
X ' ,
(»ii2 + wig^ + . . . + m^^)''
in which the summation extends over all positive and negative integral values and zero
values of mi, m2, ... m^, except the set of simultaneous zero values, is absolutely convergent
if fji>jir. (Eisenstein, Crelle, xxxv.)
3'5. The concept of uniformity.
There are processes other than that of summing a series in which the idea
of uniformity is of importance.
Let e be an arbitrary positive number; and let f(z, ^) be a function of
two variables z and ^, which, for each point ^ of a closed region, satisfies the
inequality \f{z, ^) | < e when ^ is given any one of a certain set of values
which will be denoted by (^2) ; the particular set of values of course depends
on the particular value of z under consideration. If a set (^)o can be found
such that every member of the set (^)o is a member of all the sets (Q, the
function f(z, ^) is said to satisfy the inequality uniformly for all points z of
* The reader will find no difficulty in verifying tliis statement ; the minimum value in
question is given by
K-^''^ = ^ [Cti2 + a22 + ^i2 + p,2 _ I („j _ ^^)2 + (^2 + ^i)2}i { (aj + p,)2 + (a^ - ^1)2} ij.
35, 36] CONTINUOUS FUNCTIONS AND UNIFORM CONVERGENCE 53
the region. And if a function 4> (^) possesses some property, for every positive
value of e, in virtue of the inequality \f{z, ^) | < e, ^ (z) is then said to possess
the property uniformly.
In addition to the uniformity of convergence of series and products, we shall have
to consider uniformity of convergence of integrals and also uniformity of continuity ; thus
a series is uniformly convergent when \ Rn{z)\<f, f( = n) assuming integer values in-
dependent of z only.
Further, a function f{z) is continuous in a closed region if, given t, we can find a
positive nimaber i;, such that \f{z + Ci) ~f{'^) I < f whenever
^<\Cz\<riz
and 2+f is a point of the region.
The function will be uniformly continuous if we can find a positive number rf inde-
pendent of 2, such that t] <r]g and \f{z + () -f{z) \ < e whenever
0<lC|<i7
and z + ^ is a point of the region, (in this case the set (f)o is the set of points whose
moduli are less than 17).
We shall find later (§ 3*61 ) that continuity involves uniformity of continuity ; this is
in marked contradistinction to the fact that convergence does not involve uniformity
of convergence.
3'6. The modified Heine-Borel theorem.
The following theorem is of great importance in connexion with properties
of uniformity ; we give a proof for a one-dimensional closed region *.
Given (i) a straight line CD and (ii) a law by which, corresponding to
each pointf P of CD, we can associate a closed interval I(P) of CD, P being
an interior], point of I{P).
Then the line CD can be divided into a finite number of closed intervals
Ji, J2, •■• Jk, such that each interval Jr contains at lea^t one point {not an end
point) Pr, such that no point of Jr lies outside the interval I{Pr) associated
{by means of the given law) with that point Pr^.
A closed interval of the nature just described will be called a suitable
interval, and will be said to satisfy condition {A).
If CD satisfies condition (A), what is required is prov^ed. If not, bisect CD;
if either or both of the intervals into which CD is divided is not suitable,
bisect it or themj|.
* A formal proof of the theorem for a two-dimensional region will be found in Watson's
Complex Integration and Cauchy\ Theorem (Camb. Math. Tracts, No. 15).
t Examples of such laws associating intervals with points will be found in §§ 3'61, 5"13.
X Except when P is at C or D, when it is an end point.
§ This statement of the Heine-Borel theorem is given in Hobson's Functions of a Real
Variable, p. 87, where it is pointed out that the theorem is practically given in Goursat's proof
of Cauchy's theorem {Trans. American Math. Sac. Vol. i. p. 15) ; the ordinary form of the
Heine-Borel theorem with historical references will also be found in the treatise cited.
II A suitable interval is not to be bisected ; for one of the parts into which it is divided
might not be suitable.
54 THE PROCESSES OF ANALYSIS [CHAP. Ill
This process of bisecting intervals which are not suitable either will
terminate or it will not. If it does terminate, the theorem is proved, for CD
will have been divided into suitable intervals.
Suppose that the process does not terminate ; and let an interval, which
can be divided into suitable intervals by the process of bisection just described,
be said to satisfy condition {B).
Then, by h3rpothesis, CD does not satisfy condition {B) ; therefore at least
one of the bisected portions of CD does not satisfy condition {B). Take that
one which does not (if neither satisfies condition {B) take the left-hand one) ;
bisect it and select that bisected part which does not satisfy condition (J5).
This process of bisection and selection gives an unending sequence of intervals
*oj *i> *2' ••• such that:
(i) The length of s„ is 2-"Ci).
(ii) No point of 5^+] is outside Sn.
(iii) The interval s„ does not satisfy condition {A).
Let the distances of the end points of Sn from C be Xn, yn\ then
sOn^Xn+\<yn+\'^yn- Therefore, by § 2*2, Xn and yn have limits; and, by the
condition (i) above, these limits are the same, say |^ ; let Q be the point whose
distance from C is ^. But, by hypothesis, there is a number Sq such that
every point of CD, whose distance from Q is less than Sq, is a point of the
associated interval I {Q). Choose n so large that 2~^CD < 8q; then Q is an
internal point or end point of 5„ and the distance of every point of Sn fi'om
Qis less than Sq. And therefore the interval «„ satisfies condition (A), which
is contrary to condition (iii) above. The hypothesis that the process of
bisecting intervals does not terminate therefore involves a contradiction ;
therefore the process does terminate and the theorem is proved.
In the two-dimensional form of the theorem*, the interval CD is replaced by a closed
two-dimensional region, the interval I{P) by a circlet with centre P, and the interval
J (Fn) by a square with sides parallel to the axes.
3*61. Uniformity of continuity.
From the theorem just proved, it follows without difficulty that if a
function f(x) of a real variable x is continuous when a^x^b, then f(x)
is uniformly continuous J throughout the range a^x^b.
For let e be an arbitrary positive number; then, in virtue of the con-
tinuity of f(x), corresponding to any value of x, we can find a positive
number Bx, depending on oc, such that
\f(x)-fix)\<le
for all values of x' such that \x' — x\< Bx-
* The reader will see that a proof may be constructed on similar lines by drawiug a square
circumscribing the region and carrying out a process of dividing squares into four equal
squares.
t Or the portion of the circle which lies inside the region.
X This result is due to Heine; see Crelle, lxxi. p. 361, and lxxiv. p. 188.
3'61, 362] CONTINUOUS functions and uniform convergence 55
Then by § 3"6 we can divide the range (a, b) into a, finite number of closed
intervals with the property that in each interval there is a number x^ such
that \f(x')—f{xi) I < 4 e, whenever x' lies in the interval in which a?, lies.
Let ^0 be the length of the smallest of these intervals ; and let ^, ^' be
any two numbers in the closed range (a, 6) such that | ^ — |' 1 < So- Then
^, f' lie in the same or in adjacent intervals; if they lie in adjacent intervals
let fo be the common end point. Then we can find numbers x^, x^, one in
each interval, such that
l/(n-/(^i)l<S€, l/(|o)-/WI<ie,
l/(r)-/(^2)i<i^, l/(ro)-/(^2)|<ie,
so that
i/(i)-/(r)i=ii/(^)-/(.^i)i - {/(?o)-/(^oi
-i.Ar)-/(^.)} + {/(io)-/(^.)}i
< 6.
If f, ^' lie in the same interval, we can prove similarly that
i/(i)-/(r)i<2^-
In either case we have shewn that, for any number f in the range,
we have
l/(l)-/(^+DI<6
whenever |^4- f is in the range and — So < ^<K, where So is independent of ^.
The uniformity of the continuity is therefore established.
Corollary (i). From the two-dimensional form of the theorem of § 3 "6 we can prove
that a function of a complex variable, continuous at all points of a closed region of the
Argand diagram, is uniformly continuous throughout that region.
Corollary (ii). A function f{x) which is continuous throughout the range a ^^^6 is
bounded in the range ; that is to say we can find a number k independent of x such that
\f{x) \<K for all points x in the range.
[Let n be the number of parts into which the range is divided.
Let a, £i, ^2) ••• ^n-l^ ^ be their end points ; then if x be any point of the rth interval
we can find numbers Xi, X2, ... Xn such that
LA«)-/(^i)l<if, \f(^i)-f{^i)\<h, \f{^i)-f(^2)\<b, l/(^2)-/(^2)l<K-
...!/(^r-i)-/WI<^-
Therefore \f{a)—f{x) \ < \re, and so
l/WI<l/(«)l + i^f>
which is the required result, since the right-hand side is indej^endent of x.'\
The corresponding theorem for functions of complex variables is left to the reader.
3'62. A real function, of a real variable, continuous in a closed interval,
attains its upper bound.
Let f{x) be a real continuous function of x when a^x ^b. Form a
section in which the ii-class consists of those numbers r such that r >f(x)
56 THE PROCESSES OF ANALYSIS [CHAP. Ill
for all values of x in the range (a, h), and the Z-class of all other numbers.
This section defines a number a such that f{x) ^ a, but, if Z be any positive
number, values of x in the range exist such that f{x) > a — 8. Then a is
called the upper hound oi f{x); and the theorem states that a number x
in the range can be found such that f{x') = a.
For, no matter how small 8 may be, we can find vahies of x for which
\f{x)—a\~'^>h~''^; therefore . j{/(i») - a} I ~^ is not bounded in the range;
therefore (§ 3"61 cor. (ii)) it is not continuous at some point or points of the
range; but since \f{x) — a\ is continuous at all points of the range, its re-
ciprocal is continuous at all points of the range (§ 3*2, example) except
those points at which f{x) = a ; therefore f{x) = a at some point of the
range; the theorem is therefore proved.
Corollary (i). The lower bound of a continuous function may be defined
in a similar manner ; and a continuous function attains its lower bound.
Corollary (ii). If f{z) be a function of a complex variable continuous in
a closed region, I f{z) \ attains its upper bound
3'63. A real function, of a real variable, continuous in a closed interval,
attains all values between its upper and lower bounds.
Let M, m be the upper and lower bounds off(x); then we can find numbers
^> ^. by § 3"62, such that /(^) = M,f{x) = m\ let fi be any number such that
m < fx<M. Given any positive number e, we can (by § 3"61) divide the range
{x, x) into a. finite number, r, of closed intervals such that
\f(x,^r^)-fix,^'-^)\<e,
where aJi'*"', i»2""' are any points of the 7^th interval; take *-i""', a;,*''' to be
the end points of the interval ; then there is at least one of the intervals
for which/(iCi''"') - /a, /(aJa""*) - yu. have opposite signs ; and since
\{f(x,^r^)-,M]-{f(x,ir>)-^]\<e,
it follows that \f(x,^'-^)-fi | < e.
Since we can find a number Xi^^^ to satisfy this inequality for all values
of e, no matter how small, the lower bound of the function [ f(x) — /a | is
zero ; since this is a continuous function of x, it follows from § 3"62 cor. (i)
that f{x) — [JL vanishes for some value of x.
3"64. The fluctuation of a function of a real variable*.
Let /(a;) be a real bounded function, defined when ai^x ^ b. Let
a^x-i^^x.2,1^ ... -^ Xn ^ b.
Then \f(a)-f(x,)\ + \f(x,)-f(x,)\ + ...+\f(xn)-f(b)\ is called the
fluctuation oi f(x) in the range (a, b) for the set of subdivisions x^, x.^, ... Xn.
* The terminology of this section is partly that of Hobson, Functions of a Real Variable and
partly that of Young, Sets of Points.
3-63-3-71] CONTINUOUS FUNCTIONS AND UNIFORM CONVERGENCE 57
If the fluctuation have an upper bound FJ*, independent of n, for all choices of
x^, 003, ...Xn, then f(x) is said to have limited total Jluctuation in the range
(a, b). FJ* is called the total fluctuation in the range.
Example 1. If f{x) be monotonic in the range (a, b], its total fluctuation in the range
i8|/(a)-/Wl-
Example 2. A function with limited total fluctuation can be expressed as the differ-
ence of two positive increasing monotonic functions,
[These functions may be taken to be | {/^/+/(^)}, h {Fa''-f{x)}.]
Example 3. If f{x) have limited total fluctuation in the range (a, b), then the limits
f{x±0) exist at all points in the interior of the range. [See § 3'2 example.]
Example 4. If f{x), g {x) have limited total fluctuation in the range (a, b) so has
[For \f{x')gia/)-f{x)g{x)\^\fix')\. \g{x')-g{x) \ + \g{x)\. \f{x')-f{x) |,
and so the total fluctuation of f{x) g{x) cannot exceed g . FJ'+f . GJ', where/, g are the
upper bounds of \f{x) \, \g(x) |.]
3*7. Uniformity of convergence of power series.
Let the power series
tto + a■^z 4- ... +a,i^" + ...
converge absolutely when z = z^.
Then, if | 2^ | ^ | ^o I , I o^nZ'^ | ^ | a„2o" | .
CO
But since 2 | anZ^^ \ converges, and is a series of positive terms independent
»=o
00
of z, it follows, by § 3*34, that 2 anZ'^ converges uniformly with regard to
the variable z when j ^^ | ^ | ^0 !•
Hence, by | 3"32, a power series is a continuous function of the variable
throughout the closed region formed by the interior and boundary of any
circle concentric with the circle of convergence and of smaller radius (§ 2'6).
3'71. Abel's theorem* on continuity up to the circle of convergence.
Let 2 a^z"^ be a power series, whose radius of convergence is unity, such
«=o
00
that 2 ttn converges; and let O^^a;^!; then Abel's theorem asserts that
M = 0
(00 \ 00
2 anX^ ) = 2 a^-
~ - ^ w=0 / n=0
For, with the notation of § 3'35, the function a;"- satisfies the conditions
00
laid on Un{x), when 0^«^1; consequently f(oc)= 2 a,i*'* converges loii-
n = 0
* Crelle's Journal, 1. (1826), pp. 311-339, Tlieorem iv. Abel's proof employs directly the
arguments by which the theorems of § 3-32 and § 3 "35 are proved. In the case when 2 «„ |
converges, the theorem is obvious from § 3-7.
58 THE PEOCESSES OF ANALYSIS [CHAP. Ill
formly throughout the range O^^a;^! ; it is therefore, by §3'32, a continuous
function of x throughout the range, and so lim f(x) =/(l), which is the
x^-l-O
theorem stated.
3'72. Abel's theorem* on multiplication of series.
This is a modification of the theorem of § 2*53 for absolutely convergent
series.
Let Cn = aobn-\- «! 6n-i + . . . + anbo.
00 «) 00
Then the convergence of % an, S bn and X Cn is a suficient condition that
I 2 a,J Z 6„ = 2 Cn.
\n=0 J \n=0 / M=0
For, let
A{x)= 2 a„a;", B{x)= 2 6«a;», 0(a;)= 2 c„a;".
>i = 0 « = 0 M=0
Then the series for A{x), B{x), C(x) are absolutely convergent when
I a; I < 1, (§ 2"6) ; and consequently, by § 2"53,
A{x)B(x)=C(x)
when 0 < i» < 1 ; therefore, by § 2'2 example 2,
{ lim A{x)]{ lim B(x)] = { lim C{x)^
a:-^l-0 a;-».l-0 a;^.l-0
provided that these three limits exist; but, by § 3'71, these three limits are
00 00 CO
2 o-nj X K> S c„ ; and the theorem is proved.
M=0 w=0 w=0
3'73. Power series which vanish identically.
If a convergent power series vanishes for all values of z such that \z\^r-^,
where r-^ > 0, then all the coeffi^cients in the power series vanish.
For, if not, let a^ be the first coefficient which does not vanish.
Then am-\-am+iZ + am^2z'^+ ••• vanishes for all values of z (zero excepted)
and converges absolutely when 1 2^ | ^ r < rj ; hence, if s = «,„+! -<■ am+2^ + . . . , we
have
00
I s I ^ 2 I a^-irn I ^" )
n = \
and so we can findf a positive number S ^ r such that, whenever j ^^ j ^ S,
I ^m+1'2' + am^^Z +...1^21 ^m \ \
and then | am + « I ^ |c<^jw I — | * I > 2 I ^w» |, and so \am-\-s\^0 when \z\<h.
* Crelle's Journal, i. (182G), pp. 311-339, Theorem vi. This is Abel's original proof. In
some text-books a more elaborate proof, by the use of Cesaro's sums (§ 8-43), is given.
t It is sufficient to take 5 to be the smaller of the numbers r and i | «m !^ 2 | a^+„ | r"~i.
n=l
3-72, 3-73] CONTINUOUS functions and uniform convergence 59
We have therefore arrived at a contradiction by supposing that some
coefficient does not vanish. Therefore all the coefficients vanish.
Corollary 1. We may 'equate corresponding coefficients ' in two power
series whose sums are equal throughout the region \z\<h, where S > 0.
Corollary 2. We may also equate coefficients in two power series which
are proved equal only when z is real.
REFERENCES.
T. J. Fa. Bromwich, Theory of Infinite Series, Ch. vii.
E. GouRSAT, Coxirs d' Analyse, Chs. i, xiv.
C. J. DE LA Vall^e PoussiU, Cours d^ Analyse Infinitesimale, Introduction and
Ch. vin.
G. H. Hardy, A course of Pure Mathematics, Ch. v.
W. F. Osgood, Lehrhuch der Funktionentheorie, Chs. ii, in.
G. N. Watson, Complex Integration a^ad Gauchy's Theorem (Camb. Math. Tracts,
No. 15), Chs. I, II.
Miscellaneous Examples.
1. Shew that the series
A (1-2") (1-2"^')
is equal to 7:; rx when | z I < 1 and is equal to —tt r^ when \z\>\.
{\ — zf z{\ — zY
Is this fact connected with the theory of uniform convergence ?
2. Shew that the series
28ini + 4sinl + ... + 2™sin3L + ...
converges absolutely for all values of 2 (2 = 0 excepted), but does not converge uniformly
near 2 = 0.
3. If «„(.r)=-2(?i-l)2.re-(™-i)'-'^''+2'rt,2^e-"''^,
shew that 2 Un{x) does not converge uniformly near x=0. (Math. Trip., 1907.)
4. Shew that the series —p. ^ + -r-— ... is convergent, but that its square (formed
by Abel's rule)
is divergent.
5. If the convergent series « = y7 ~ H'r + ^ ~ 2r "^•" ^'"'^^^ ^® multiplied by itself
the terms of the product being arranged as in Abel's result, shew that the resulting series
diverges if r ^ |^ but converges to the sum s^ if /• > |. (Cauchy and Cajori.)
60 THE PROCESSES OF ANALYSIS [CHAP. Ill
6. If the two conditionally convergent series
00 /'_Nn + X 00 / _'\H + 1
2 ^ — '- — and 2 ^^ — V .
where r and s lie between 0 and 1, be multiplied together, and the product arranged as in
Abel's result, shew that the necessary and sufficient condition for the convergence of the
resulting series is 7" + 5> 1. (Cajori.)
7. Shew that if the series l-|^ + i-|+...
be multiplied by itself any number of times, the terms of the product being arranged as
in Abel's result, the resulting series converges. (Cajori.)
8. Shew that the g-th power of the series
ajsin ^ + 02 sin 2^ + ... + a„sin«^+...
is convergent whenever q{\-r)<\,r being the greatest number satisfying the relation
for all values of n.
9. Shew that if 6 is not equal to 0 or a multiple of 27r, and if Uq, Mj, U2, ... be a
sequence such that Un'-*-0 steadily, then the series 2w„ cos {nd + a) is convergent.
Shew also that, if the limit of m„ is not zero, but «„ is still monotonie, the sum
6 6
of the series is oscillatory if — is rational, but that, if — is irrational, the sum may
TT TV
have any value between certaii) bounds whose difference is a cosec|^, where a= lim m„.
(Math. Trip., 1896.)
CHAPTER IV
THE THEORY OF RIEMANN INTEGRATION
4*1. The concept of integration.
The reader is doubtless familiar with the idea of integration as the
operation inverse to that of differentiation ; and he is equally well aware that
the integral (in this sense) of a given elementary function is not always
expressible in terms of elementary functions. In order therefore to give
a definition of the integral of a function which shall be always available,
even though it is not practicable to obtain a function of which the given
function is the differential coefficient, we have recourse to the result that the
integral* oi f{x) between the limits a and h is the area bounded by the
curve y=f{x), the axis of x and the ordinates x = a, x = h. We proceed to
frame a formal definition of integration with this idea as the starting-point.
4*1 1. Upper and lower integrals'^'.
Let f{x) be a bounded function of x in the range (a, h). Divide the
interval at the points x^,, x^, ... x^-i (a ^ x^ ^x^^ . . . ^x^-i^b). Let U, L be
the bounds of f{x) in the range (a, 6), and let U^, Lr be the bounds of f(x)
in the range {x^-i, x^), where Xq = a, Xn = b.
Consider the sums J
8n =Ui{x^ — a)+ U^(X2-X^) + ... + Un (b — Xn-i),
Sn = L^ (^1 — a) + X2 (^2 - ^i) + • • • + Ln (b — Xn-i).
Then U (b -a)^ Sn>Sn> L {b- a).
For a given n, Sn and Sn are bounded functions of x^, x^, ... x^-i. Let
their lower and upper bounds § respectively be S^, ««, so that Sn, Sn depend
* Defined as the (elementary) function whose differential coefficient is/(^).
t The following procedure for establishing existence theorems concerning integrals is based
on that given in Goursat's Cours (TAnalyse, Chapter iv. The concepts of upper and lower
iutegrals are due to Darboux, Annales de I' locale norm. mp. ser. 2, t. iv.
X The reader will find a figure of great assistance in following the argument of this section.
/S„ and «„ represent the sums of the areas of a number of rectangles which are respectively
greater and less than the area bounded by y~f{x), x = a, x = b and y — 0, if this area be
assumed to exist.
§ The bounds of a function of Ji variables are defined in just the same manner as the bounds
of a function of a single variable (§ 3*62).
62 THE PROCESSES OF ANALYSIS [CHAP. IV
only on n and on the form of f(x), and not on the particular way of dividing
the interval into n parts.
Let the lower and upper bounds of these functions of n be >S', s. Then
Sfi ^ S, Sn ^ S.
We proceed to shew that s is at rnost equal to 8 ; i.e. 8'^ s.
Let the intervals (a, x^, {x^, x^, ... be divided into smaller intervals by
new points of subdivision, and let
a, Vx, 2/2, ••• Vk-x, yk(=^i), Vk+i, ••• yi-i, yi(=^2), yi+i, ••• ym-i, h
be the end points of the smaller intervals; let Ur, X/ be the bounds of f{x)
in the interval (y,—!, yr)-
m m
Let Tyn^t {yr - yr-i) U,!, tm^^ {yr - yr-i) W .
r=l r=l
Since Ui , U^', ... C/*' do not exceed t/j, it follows without difficulty that
^n ^ -^ m ^ ^m ^ ^n •
Now consider the subdivision of {a, b) into intervals by the points
Xi, X.2, ... Xn-1, and also the subdivision by a different set of points
x/, X2, ... x'n'-i- Let S'n', s'n' be the sums for the second kind of sub-
division which correspond to the sums aS^„, s^ for the first kind of subdivision.
Take all the points Xi, ... ic„_i ; x/, ... a'V-i as the points yi, y^, ... ym-
Then 8,^ ^ T,„ >tm>Sn,
and 8\' >T^> Un > «'„'.
Hence every expression of the type 8,^ exceeds (or at least equals) every
expression of the type s'^ ; and therefore 8 cannot be less than s.
[For \{ 8<s and s — 8=27) we could find an 8^ and an s'„' such that
8n-8<'r], s — s'n'<'n and so Sn'>8n, which is impossible.]
The bound 8 is called the upper integral off{x), and is written / f{x) dx ;
J a
s is called the lower integral, and written 1 f{x) dx.
J a
If 8 = s, their common value is called the integral of f{x) taken between
the limits* of integration a and b.
/■*
The integral is written / f{x) dx.
J a
ra rb
We define | f{x)dx, when a<b, to mean — I f{x)dx.
J b J a
rb rb rb
Example!. \ {f{x) + ^{x)]dx=\ f(x)dx+ I (f){x)dx.
J a J a J a
Example 2. By means of example 1, define the integral of a continuous complex
function of a real variable.
* 'Extreme values' would be a more appropriate term but 'limits' has the sanction of
custom.
4-12, 413] THE THEORY OF RIEMANN INTEGRATION 08
4*12. Riemann's condition of integr ability *.
A function is said to be ' integrable in the sense of Riemann ' if (with the
notation of § 4"11) *S'„ and s„ have a common limit (called the Riemann
integral of the function) when the number of intervals {xr^i, Xj) increases
indefinitely in such a way that the length of the longest of them tends to
zero.
The necessary and sufficient condition that a bounded function should be
mtegrahle is that S^ — s^ should tend to zero vjhen the number of intervals
{Xr-iy Xr) increases indefinitely in such a way that the length of the longest tends
to zero.
The condition is obviously necessary, for if >S^^ and «„ have a common limit
S^ — s^^Osi^n-*-(X). And it is sufficient ; for, since S^'^S'^s'^s^, it follows
that if lim {S^ — s^) = 0, then
lim Sji = lim s„ = S = s.
Note. A continuous function f{x) is 'integrable.' For, given e, we can find 8 such
that \f{xf)—f{a/')\<fl{h-a) whenever \af-af'\<b. Take all the intei-vals (^g_i, Xg)
less than 8, and then Ug- Lg<f/{b — a) and so *S'„-s„<e ; therefore Sn-Sn^^-O under the
circumstances specified in the condition of integrability.
Gorollari/. If Sn and s„ have the same limit S for one mode of subdivision of (a, b)
into intervals of the specified kind, the limits of Sn and of 5„ for any other such mode of
subdivision are both ^S'.
Example 1. The product of two integrable functions is an integrable function.
Example 2. A function which is continuous except at a finite number of ordinaiy
discontinuities is integrable.
[If f{x) have an ordinary discontinuity at c, enclose c in an interval of length Si;
given f, we can find S so that \f{x')—f{x) \ <e when x' — xl<8 and x, x' are not in this
interval.
Then Sn-Sn^e{b-a — 8i)+k8i, where k is the greatest value of \f{x')—f{x)\, when
X, x' lie in the interval.
When 8i^0, ^^|/(c4-0)-/(c-0) |, and hence lim (.S'„-s„) = 0.]
Example 3. A function with limited total fluctuation and a finite number of ordinary
discontinuities is integrable. (See § 3'64 example 2.)
4*13. A general theorem on integration.
Let f{x) be integrable, and let e be any positive number. Then it is
possible to choose S so that
n rh j
t (xp - Xp.,)f(x'p_i) - I f{x) dx\< €,
p=l J a
provided that Xp — Xp^^ ^ B, Xp_-^ ^ x'p^^ ^ Xp.
* Eiemann (Ges. Math. Werke, p. 239) bases his definition of an integral on the limit of the
sum occurring in § 4*13; but it is then difficult to prove the uniqueness of the limit. A mote
general definition of integration has been given by Lebesgue, Annal'i di Mat. Ser. iii. A, t. vii.
See also his Lemons sur Vintegration.
64
THE PROCESSES OF ANALYSIS
[chap. IV
Therefore
To prove the theorem we observe that, given e, we can choose the length
of the longest interval, 8, so small that >S^„ — s„ < e.
n
Also 'Sfn^ S (a;p-Xp_,)f(x'p_,)^Sn,
J a
n 1*6 ]
t (Xp - Xj,_i)f{x'p_,) - f{x) dx\^Sr,-s^
p=\ J a
< €.
As an example* of the evaluation of a definite integral directly from the theorem
of this section consider I r, where X<\.
j 0 (1 - ^2)4
Take 8 = -arcsinX and let ^g=sinsS, (0<s8<^7r), so that
•^« + 1 ~ ^»= 2 sin ^S cos (s + 1^) 8 < 8 ;
also let jr/ = sin (« + |) S.
Then
2 " "
_ P sin«8-8in(a-l)?8
,= 1 (] _^2^_j)i 8=1 COS(5-^)S
= 2p sin \b
=-arc sin Z. {sin ^S/(^8)}.
By taking p sufficiently large we can make
arbitrarily small.
We can also make
arbitrarily small.
That is, given an arbitrary number €, we can make
Jo (l-.r2)i~ «=i (l_^'2^_j;
. „ fsiniS ]
arcsmZ.|^|--lj
f^ dx
U (1 - x^-.
•arc sin X
<(
(1 - ^2)4
by taking p sufficiently large. But the expression now under consideration does not
depend on p ; and therefore it must be zero ; for if not we could take e to be less than it,
and we should have a contradiction.
That is to say
Example 1. Shew that
'^ dx . „
, =arc sin ^i.
0 (l-a,-2)^
lim
X 2x (n — l)x
1 + cos - + cos h . . . + cos —
n n n &\nx
Example 2. If f {x) has ordinary discontinuities at the points aj, «2? •■• <^«) then
fb ( fa,-S, fa.i-S., fb ]
I f{x)dx = Y\m\ +\ +...+ f(x)dxy ,
J a [J a J ai+ei J a^^+e^ )
where the limit is taken by making Si, 82, ... 8k, (i, f^j ■•• *« tend to +0 independently.
* Netto, Zeitschriftfilr Math. xl. (1895).
4'14] THE THEORY OF RIEMANN INTEGRATION 66
Example 3. lif{x) i» integrable when ai^x^hi and if, when aj ^ a < 6< ftj, we write
/ f{x)dx'=<f){a,b),
and if/{b + 0) exists, then
lim ii^Ll±^i«'l)=/(6 + 0).
5^ + 0
Deduce that, iff{x) is continuous at a and b,
d_
da
jjix) dx= -f{a\ ^ Jy (^) dx=f{b).
Example 4. Prove by differentiation that, if </> {x) is a continuous function of x and
-J- a continuous function of t, then
J xo J t„ at
Example 5. If /' {x) and 0' (^) are continuous when a^x^b, shew from example 3
that
f /' W 0 (-y) c?^+ f 0' (*')/(-^) ^•=^=/(&) <^ (h) -f{a) <l> (a).
y a J a
Example 6. If /(:p) is integrable in the range (a, c) and a ^ 6 ^ c, shew that I /(^) ci?a'
is a continuous function of b.
4'14. Mean Value Theorems.
The two following general theorems are frequently useful.
(I) Let U and L be the upper and lower bounds of the integrable function /(.r) in the
range (a, b).
Then from the definition of an integral it is obvious that
r{U-f{x)}dx, \\f{x)-L}dx
J a J a
are not negative ; and so
U{b-a)^ l^f{x)dx^L{b-a).
J a
This is known as the First Mean Value Theorem.
lif{x) is continuoxis we can find a number ^ such that a^^^b and such that/(^) has
any given value lying between U and L (§ 3-63). Therefore we can find | such that
rf(x)dx = ib-a)f{i).
J a
If F(x) has a continuous differential coefficient F' (x) in the range (a, b), we have, on
writing F' (x) for/(^),
F{b)-F{a)^{h-a)F'{i)
for some value of ^ such that a^^^b.
- Example, li f{x) is continuous and (^ (.*;) >;0, shew that | can be found such that
-h fb
j f{x)4>{^)dx=^f{^) I 4>(x)dx.
J a J a
W. M. A.
fi6 THE PROCESSES OF ANALYSIS [CHAP, IT
(II) Let f{x) and <^ {x) be integrable in the range (a, h) and let <^ {x) be a positive
decreasing function of x. Then Bonnet's* form of the Second Mean Value Theorem is
that a number ^ exists such that a^^^h, and
I / {^) ^ (•*') dx = (l){a) j f{x) dx.
J a J a'
For, with the notation of §§ 4'1-4'13, consider the sum
p
S= 2 {Xs-X,_i)f{x,_i)(l){x,_i),
«=1
Writing (;*;g-^g_i)/(^g_i) = ag_i, (^(^g_i) = <^g_i, ao + ai + ... + a, = 6g, we have
p-i
s=l
Each term in the summation is increased by writing b for 6g_i and decreased by-
writing 6 for 6g_i, if 6, 6 be the greatest and least of 6o? ^i) ••• ^p-i > ^"d so b(f>Q ^S^b(f>Q.
m
Therefore S lies between the greatest and least of the sums (p (xq) 2 {Xg-Xg_i)f(xg^j)
s=l
where m = 1, 2, 3, ... jo. But given e we can find 8 such that when a'g — .^'g_l < S
2 (.rg-.^■g_l)/(^■g_l) 0 (^g_i) - /(.r) 4> (x) dx\<f,
I "l /"aim I
I » = 1 J Xo \
and so, writing a, b for .t'o, .v„, we find that I / (;^^) ^ (.:r) rfo; lies between the upper and
lower bounds oft <^ («) I f{x)dx±2f, where ^j may take all values between a and 6.
y o
Let fT' and Z be the upper and lower bounds of (f) (a) j f (x) dx.
J a
fb
Then U + 2f'^j f{x)(f> (x) dx'^L — 2e for all positive values of e ; therefore
J a
C^ j f{x)(f> (x) dx ^ L.
Since (f> (a) j f{x) dx qua function of ^i takes all values between its upper and lower
. . /-ft
bounds, there is some value ^, say, of |i for which it is equal to I f{x)(f) (x) dx. This
J a
proves the Second Mean Value Theorem.
Kxample. By writing I (f){x) — (fi (b) \ in place of <^ {x) in Bonnet's form of the mean
value theorem, shew that if ^ (x) is a monotonic function, then a number ^ exists
such that a^^^b and
r f{.v) (P (.r) dx.= cf, (a) I \f{x) dx + (p{b) j f{x) dx.
(Du Bois Reymond.)
* Lionville's Journal, Vol. xiv. (1849), p. 249. The proof given is a slight modification
of one due to Holder.
t By § 4-13 example 0. since /(.t) is bounded, / '^f(x) dx is a continuous function of fi.
4-2] THE THEORY OF RIEMANN INTEGRATION 67
4"2. Differentiation of integrals containing a parameter.
The equation* -j- \ f{x, a)dx= \ 4~ ^ is true if f{x, a) possesses a
bothf the variables x and a.
For -J- \ f (x, a) dx = lim •^-^— ^ 1 — '^ dx
if this limit exists. But, by the first mean value theorem, since fa is a
continuous function of a, the second integrand is fa (x, a + Oh), where
But, for any given e, a number 8 independent of x exists (since the con-
tinuity of /a is uniform J with respect to the variable x) such that
I fa (x, a') -fa (x, a) j < 6/{b - a),
whenever \a.' — a\ <B.
Taking | A | < S we see that \6h\< B, and so whenever \h\<B,
I l^f(x,a + h)-f(x,a) ^^ _ p^^ ^^^ ^^ ^^ < f l/« (^> « + Oh) -fa (x, a) \ dx
' J a "' J a J a
< €.
Therefore by the definition of a limit of a function (§ 3"2),
h^dJa h
exists and is equal to I fadx.
J a
Example 1. If a, 6 be not constants but functions of a with continuous differential
coefl&cients, shew that
Example 2. If /(^, a) is a continuous function of both variables, I f{x, a) dx is a
J «■
continuous function of a.
* This formula was given by Leibniz, without specifying the restrictions laid on/(x, o).
t <f> (x, y) is defined to be a continuous function of both variables if, given e, we can find
5 such that | <l>(x', y') -<f>(x, y) \<e whenever {(x' -x)^ + {y' -y)^}^<d. It can be shewn by § 3-6
that if </» (x, y) is a continuous function of both variables at all points of a closed region in
a Cartesian diagram, it is uniformly continuous throughout the region (the proof is almost
identical with that of § 3-61). It should be noticed that, if <p {x, y) is a continuous function
of each variable, it is not necessarily a continuous function of both ; as an example take
this is a continuous function of x and of y at (0, 0), but not of both x and y.
X It is obvious that it would have been sufficient to assume that /^ had a Eiemann integral
and was a continuous function of a. (the continuity being uniform with respect to x), instead
of assuming that / was a continuous function of both variables. This is actually done bj'
Hobsou, Functions of a Real Variable, p. 599.
5—2
68 THE PROCESSES OF ANALYSIS [CHAP. IV
4'3. Double integrals and repeated integrals.
Let f{x, y) be a function which is continuous with regard to both of the
variables x and y, when a% x^h, a^y ^^.
By § 4-2 example 2 it is clear that
a VJ^^' ^^ ^^\ ^^' J V '^^'^' ^^ ^\ ^^
both exist. These are called repeated integrals.
Also, as in § 3*62, f{x, y), being a continuous function of both variables,
attains its upper and lower bounds.
Consider the range of values of x and y to be the points inside and on a
rectangle in a Cartesian diagram ; divide it into nv rectangles by lines parallel
to the axes.
Let Um,^^, Lm,n be the upper and lower bounds oi f{x, y) in one of the
smaller rectangles whose area is, say, Am,ii', and let
n V n V
m = l ix = l m=l ;a = l
Then S^i^y >Sn,v, and, as in § 4'11, we can find numbers Sn,v, Sn,v which
are the lower and upper bounds of >S^^_„, Sn,v respectively, the values of
^n,v, Sn,v depending only on the number of the rectangles and not on their
shapes ; and S^^ „ ^ Sn, v We then find the lower and upper bounds {8 and s)
respectively of N^^^, s„^^ qua functions of n and v ; and Sn,v^ S^s'^ Sn^^, as in
§ 4-11.
Also, fi:"om the uniformity of the continuity oi f{x, y), given e, we can find
B such that
'^ 171,11. -^m, /i ■*^ f)
(for all values of m and /x.) whenever the sides of all the small rectangles are
less than the number 8 which depends only on the form of the function /(a;, y)
and on e.
And then *S„_ ^ — 5n, ^ < eih — a){j3 — a),
and so S — s<e(h — a){^— ol).
But S and s are independent of e, and so S = s.
The common value of S and s is called the double integral of f{x, y) and
is written
rb fp
f(x, y) (dxdy).
It is easy to shew that the repeated integrals and the double integral are all equal
when f{x, y) is a continuous function of both variables.
4-3, 44] THE THEOEY OF BIBMANN INTEGRATION 09
For let Y,n> Am be the upper and lower bounds of
aa a; varies between ar^.j and 0?^.
Then 2 Y„,(a7^-^^_,) > / ]/ f{x,i/)d^\dx^2AmiXm-x,n-i)'
m=l J a U "■ ) m=l
V V
But* 2 Um,y.{yy.-yy.-i)'^Yr„^A.m'^ 2 -^m.|x (^m "^M-l)-
Multiplying these last inequalities by Xm—Xm-u using the preceding inequalities and
summing, we get
2 2 C/'^^^„,,M^ I W f{x,y)dy\dx'^^ 2Z^,^^^^;
m=l (11=1 J a \J <*■ ) m=l (ut=l
and so, proceeding to the limit.
But S=s= / f{x,y){dxdy\
J a J a-
and so one of the repeated integrals is equal to the double integral. Similarly the other
repeated integral is equal to the double integral.
Corollary. lif{x, y) be a continuous function of both variables,
J dx\\ f{x,y)dyj = j dyU^ f{x,y)dx\.
4'4. Infinite integrals.
If lim [ I f{x)da;\ exists, we denote it by / f{x)dx; and the limit in
question is called an infinite integral'^.
Examples.
(3) By integrating by parts, shew that / P^e~^ dt = n !. (Euler.)
fb fb
Similarly we define / /(x) dx to mean lim / / (.r) dx, if this limit exists ; and
J ~^ a-^ — cD J a
I f{x)dx is defined as I f(x)dx+j f(x)dx. In this last definition the choice
of a is a matter of indifference.
* The upper bound of f {x, y) in the rectangle A^^^^ is not less than the upper bound
of f(x, y) on that portion of the line x=f which Hes in the rectangle.
t This phrase, due to Hardy (Proc. London Math. Soc. Vol. xxxiv. p. 16), suggests the
analogy between an infinite integral and an infinite series.
70
THE PROCESSES OF ANALYSIS
[chap. IV
4'41. Infinite integrals of continuous functions. Conditions for con-
vergence.
A necessary and sufficient condition for the convergence of 1 f{oc) dx is
J a
that, corresponding to any positive number e, a positive number X should
exist such that I f{x)dx\<e whenever
J x' \
x'^x'^X.
The condition is obviously necessary ; to prove that it is sufficient, suppose
ra+n
it is satisfied ; then, if n'^X — a and w be a positive integer and 8^ = I /(^)»
J a
we have | S^+p - ^^ | < e.
Hence, by § 2'22, S^ tends to a limit, S ; and then, if ^ > a + w,
S - f{x) dx ^ S- \ f{x) dx + I fix) dx
■'a J a J a+n
<2e;
and so lim / f(x) dx = S; so that the condition is sufficient.
f -*.oo J a
4*42. Uniformity of convergence of an infinite integral.
The integral / f{x, a) dx is said to converge uniformly with regard to a
J a
in a given domain of values of a if, corresponding to an arbitrary positive
number e, there exists a number X independent of a such that
L
< €
f(x, a) dx
for all values of a in the domain and all values of x' ^ X.
The reader will see without difficulty on comparing §§ 2*22 and 3*31 with
§ 4"41 that a necessary and sufficient condition that I f(x, a.)dx should
J a
converge uniformly in a given domain is that, corresponding to any positive
number e, there exists a number X independent of a such that
f
f {x, OL)dx\< e
for all values of a in the domain whenever x" > a;' ^ X.
4'43. Tests for the convergence of an infinite integral.
There are conditions for the convergence of an infinite integral analogous
to those given in Chapter ii for the convergence of an infinite series.
The following tests are of special importance.
4-41-4-43] THE THEORY OF RIEMANN INTEGRATION 7l
(I) Absolutely convergent integrals. It may be shewn that I f{x) dx
J a
certainly converges if 1 | f{x) \ dx does so ; and the former integral is then
J a
said to be absolutely convergent. The proof is similar to that of § 2-32.
/■«.
Example. The comparison test. If \f{x)\^g{x) and i g{x)dx converges, then
r "
/ f{x)dx converges absolutely.
[Note. It was observed by Dirichlet* that it is not necessary for the convergence of
I f{x)dx that f{x)-*-0 as x-»-od : the reader may see this by considering the function
f{x)=0 (n'^x^n+l-in + l)-^),
f(x) = {n + iy{n + l-x){x-(n + l) + {n + l)-^} (n + l-{n + \)-^^x^n + l),
where n takes all integral values.
ft fn+l
For I f(x)dx increases with ^ and / f{x)dx=^{n+l)~^; whence it follows
Jo J n
without difficulty that I / (x) dx converges. But when x = n + l —^ {n+l)~^, f{x) = ^ :
J a
and aof{x) does not tend to zero.]
(II) The Maclaurin-Cauchyf test. If f(x)>0 and f(x)^0 steadily,
I f(x) dx and S f{n) converge or diverge together.
Jl M=l ^
/"m + l
For f{m)^ f{x)dx^f{m + l),
J m
n fn+l M+l
and so 2 f{m)^ j f{x)dx^ 2 f{m).
m=l J 1 m=2 '
The first inequality shews that, if the series converges, the increasing sequence
fn+l
I f{x)dx converges (§ 2'2) when 7i-*^ao through integral values, and hence it follows
fx'
without difficulty that I f{x)dx converges when ^-'^x ; also if the integral diverges,
.' 1
so does the series.
The second shews that if the series diverges so does the integral, and if the integral
converges so does the series (§ 2"2).
(III) Bertrand's\ test. If f{x) = 0 (x^"^), 1 f{x)dx converges when
J a
\ < 0 ; and '\ff{x) = 0 (a'~* [log xY~^), I f{x) dx converges when X < 0.
J a
These results are particular cases of the comparison test given in (I).
* Crelle, Bd. xvii.
t Maclaurin {Fluxions, Vol. i. pp. 289, 290) makes a verbal statement practically equivalent
to this result. Cauchy's result is given in his Collected Works, ser. ii. t. vii. p. 269.
:J: Liouville^s Journal, t. vii. pp. 38, 39.
72 THE PROCESSES OF ANALYSIS [CHAP. IV
(IV) Ghartiers test* for integrals involving periodic functions.
is bounded as x^ <x ,
If f{x) -* 0 steadily as a; -* oo and if I </> {oc) dx
J a
then I f{x) ^ (x) dx is convergent.
J a
For if the upper bound of / (j) {x) dx he A, we can choose X such that/(;c) <e/2^
J a
when .r ^ X ; and then by the second mean vahie theorem, when x" ^x''^X, we have
r f{x)(t>(x)dx\= fix') j (}){x)dx =f{x') I r(f>{x)dx- r (t>{x)dx\ ^2Af(x')<f,
\ J X' I J X' \ j a J a I ■
which is the condition for convergence.
Example 1. / dx converges.
Jo *'
/•oc
Example 2. I x~^ sin {x^ — ax) dx converges.
J 0
4"431. Tests for uniformity of convergence of an infinite integral f .
(I) Be la Vallee Poussin's testl. The reader will easily see by using
the reasoning of § 3'34 that I f{x, a) dx converges uniformly with regard
to a in a domain of values of a if { f(w, a)\< fi {x), where yu, {x) is independent
of a and fx (x) dx converges. [For, choosing X so that I fi,{x)dx< e
when a?" ^ ic' ^ A'', we have I f(x, a)dx < e, and the choice of X is inde-
J of
pendent of a.]
Example. I o(f—'^e~'^dx converges uniformly in any interval (4, B) such
.' 0
that
\<A<B.
(II) The method of change of variable.
This may be illustrated by an example.
Consider
We have
Since
/"" sin.o
dx where a is real.
J x' X J ax' y
sm y
So
./o y
sin ax
dy converges we can find I' such that
— -- dy \<e when y^y ^ Y.
V' y \
dx '• < e whenever j a.r' | ^ F ; if | a | ^ S > 0, we therefore get
p" s
dx \ <i
* Liouvillcst Journal, t. xviri. (1853). It is remarkable that this test for conditionally
convergent integrals should have been given some years before formal definitions of absolutely
convergent integrals.
t The results of this section and of § 4-44 are due to de la Vallee Poussiu, Ann. de la Sac.
Scientifiqiie de Bru.velles, t. xvi. pp. 150-180.
X This name is due to Osgood.
4-431, 4-44]
THE THEORY OF RIEMANN INTEGRATION
78
when y ^ a/ ^ jr= Yjb ; and this choice of X is indei^endent of a. So the convergence is
uniform when a ^ 8 > 0 and when a ^ - S < 0.
Example. \ \\ sin {^'^x^) d^\ dx' is uniformly convergent in any range of real
(de la Valine Poussin.)
values of a
/:
"or*
'~hs\mdz
does not exceed a constant inde-
[ Write ff^x^=Zf and observe that
pendent of a and x since / z-i sin z dz converges.]
Jo
(III) The method of integration by parts.
If / f{x, a) dx=<f> (x, a)-]- j X (^> «) ^•^
and if <^ (:r, a) -*• 0 uniformly as .y-* qo and I ;^ (^,a) c?^ converges uniformly with regard
.' a
to a, then obviously / f(x, a) dx converges uniformly with regard to a.
J a
(IV) The method of decomposition.
Example. f °° cos^-sin a..^^^^ p sin(a + l) x-^^_^^ ^ sin (a- 1) ^^^ .
J 0 X J 0 ^ Jo X
both of the latter integrals converge uniformly in any closed domain of real values of
a from which the points a = + 1 are excluded.
4'44. Theorems concerning uniformly convergent infinite integrals.
/•«
(I) Let I f(x, a) dx converge uniformly when a lies in a domain S.
J a
Then, if f{x, a) is a continuous function of both variables when x'^a and
a lies in S, I fix, a) dx is a continuous function* of a.
J a
For, given e, we can find X independent of a, such that f{x,0L)dx\<e
whenever ^^ X.
Also we can find S independent of x and a, such that
\f(x,a)-f(x,a')\<el{X-a)
whenever j a — a' < 8.
That is to say, given e, we can find B independent of a, sucK that
I f{x, a') dx — I f(x, a) dx 1^ \ {f(x, a) —f(x, a')} dx
•la J a : ] J a
+ , I f(x, a.')dx + I f{x, a)dx
I J X • J X
<3e,
whenever | a' — a j < S ; and this is the condition for continuity.
* This result is due to Stokes. His statement is that the integral is a continuous function
of a if it does not ' converge infinitely slowly.'
74 THE PEOCESSES OF ANALYSIS [CHAP. IV
(II) If fix, a) satisfies the same conditions as in (I), and if a lies in S
when A ^a^ B, then
Therefore
I 4 I f{x, a) dxi da = i \ j f {x, a)cZa^ dx.
For, by § 4-3,
I \ I f{x, (x)dx\ da= i 11 f(x, a)dal dx.
I \j f(x, a)dxl dcL— I \j f(x,a)da[dx
— \ I \ j fi^y ci)dxl da
< eda < e(B-A),
J A
for all sufficiently large values of ^.
But, from 5S 2*1 and 4'41, this is the condition that
should exist, and be equal to
lim I \ I f (x, a) day dx
■.^aoJa [J A )
I \ I f{x, a)dx\da.
Corollary. The equation -j- \ (^{x, a)dx= \ ^— dx is true if the integral on the
da J a J a <J(^
right converges uniformly and the integrand is a continuous function of both variables,
when x'^a and a lies in a domain S, and if the integral on the left is convergent.
Let J be a point of >S^, and let ^=f{x, a), so that, by § 4-13 example 3,
va
j fix, a)da = (l> (x, a) -(f) (x, A).
JA
Then I i / (a;, a) c?a > o?^ converges, that is I {(f) (x, a) - (f) {x, A)} dx convergea,
and therefore, since / (f) (x, a) dx converges, so does i (f)(x, A) dx.
J a J a
Then T / ^ (■^j ") ^■^' r= ^ / ^^ ('^' ") ~ *^ (^' ^)} <^'^
= [ f{x,a)dx=\ ^dx,
which is the required result ; the change of the order of the integrations has been justified
above, and the differentiation of / " with regard to a is justified by § 4-44 (I) and § 4-13
example 3.
45, 4-51] THE THEORY OP RIEMANN INTEGRATION 75
4'5. Improper integrals. Principal values.
rb
If l/(^)|-* <» as a;-^a + 0, then lim f{x)dx may exist, and is
written simply / f{x) dx ; this limit is called an improper integral.
J a
If I f(x) I -* 00 as x-^c, where a <c <b, then
fc-s rb
lim / f{x)dx+ lim I f{x)dx
may exist; this is also written / f(x)dx, and is also called an improper
-' a
integral ; it might however happen that neither of these limits exists when
B, S' -^-O independently, but
lim \ I f(x) dx + I f(x) dx\
6-*.+0 Ua J c+S )
exists; this is called * Cauchy's principal value of / f(x)dx' and is written
for brevity P I f{x)dx.
J a
Results similar to those of §§ 4*4-4"44 may be obtained for improper
integrals. But all that is required in practice is (i) the idea of absolute
convergence, (ii) the analogue of Bertrand's test for convergence, (iii) the
analogue of de la Valine Poussin's test for uniformity of convergence. The
construction of these is left to the reader, as is also the consideration
of integrals in which the integrand has an infinite limit at more than one
point of the range of integration*.
Examples. (1) I ;p~"' cos ;r o?a: is an improper integral.
J 0
(2) I .t?^""-" (1 - ^)'*~^ dx is an improper integral if 0<X<1, 0<fi<l.
J 0
It does not converge for negative values of X and /x.
(3) P I dx is the principal value of an. improper integral when
J Q 1—x
0<a<l.
4'51. The inversion of the order of integration of a certain repeated integral.
General conditions for the legitimacy of inverting the order of integration when the
integrand is not continuous are difficult to obtain.
The following is a good example of the difficulties to be overcome in inverting the
order of integration in a repeated improper integral.
* For a detailed discussion of improper integrals, the reader is referred either to Hobson's or
to Pierpont's Functions of a Real Variable. The connexion between infinite integrals and
improper integrals is exhibited by Bromwich, Infinite Series, § 164.
Now
76 THE PROCESSES OF ANALYSIS [CHAP. IV
Let f{x, y) he a continuoiis functioti of both variables, and let 0<X^1, 0</i^l,
0 < j/^1 ; then
/o '^"^ ill" ^'"'y""' (1 -^-y)''"V(^, y) dy]
= \yy{\'~' x^~^y''-Hi-^-yr^f{x,y)dxY
This integral, which was first employed by Dirichlet, is of importance in the theory of
integral equations ; the investigation which we shall give is due to Hurwitz*.
Let x^~^f-~^ (1 -x-yy~'^f{x, y) = (j) (x, y) ; and let M be the upper bound of \f{x, y) \ .
Let 8 be any positive number less than J.
Draw the triangle whose sides are x = 8, y — 8, x+y=l — 8; at all points on and inside
this triangle 0 (x, y) is continuous, and hence, by § 4'3 corollary,
n-25 c n.—x—s ) fi—as ( fi—y—s 1
/■1-25 ( n-x 1 n-as c fi-x-s ■) r\-2S fi-iS
J& [Jo "f'^'^'^^^^j^ js '^^{js ^(^'^)^i'| +J5 Iidx+\ l^x,
fs n-x
where 1^= \ ^{x,y)dy, h^ \ ^{x,y)dy.
Jo J i-a-S
But I A I < f * Jifx^-^y^-"^ (1 - x-T/)"-^ dy
Jo
< i/"^-' (1 - ^ - 8)"-' r y^-' dy,
since (1 _.^_y)''-i^(l _^_ §)"-!,
Therefore, writing x={\-8)xx, we havet
/ hdx ^M8f^fi-n x^-\l~x-8Y-^dx
^ i/r ^-1 (1 - 8)^+"-' r xi"-' (l-xi)"-' dx
J 0
The reader will prove similarly that /2-*-0 as S-*>0.
Hencel \ ^'^^ A ^ (•^'' ^) M = ^^"^ /^ ' ^'^' | / ^ ^^^ 3/) '^yj
n-'iS ( fi-y-S ]
= hm / ^y \ L ^ (**'' ^) ^*j '
* Annals of Mathematics, Vol. ix (1908), p. 183.
t I xi^'l (1 - xi)"-'^ dxi = B(\, v) exists if \ > 0, >/ > 0 (§ 4-5 example 2).
J The repeated integral exists, and is, in fact, absolutely convergent ; for
j "^ \ x^-'^ y'^-'^ (1- X -y)''-\f{.r, y) dy \<M.x''-'^ (1 - xf-^"-'^ j s'^-'^ (1 ^s^'^ds,
writing ?/ = (l-a;)s ; and I Mx^~^ {l--x)'^^''~^dx . i s'*~^ (l-s)""-^ ds exists. And since the
Jo ^ Jo r
fl-e fl-2S
integral exists, its value which is lim / may be written lim |
5, e-»0 J 5 S-*0 J S
46] THE THEORY OF RIEMANN INTEGRATION T7
by what has been already proved ; but, by a precisely similar piece of work, the last
integral is
We have consequently proved the theorem in question.
Corollary. Writing ^ = a + (6-a)^, rj = h — {h-a)y,\vie see that, if 0 (|, i;) is con-
tinuous,
/„' ^^ 1 i] ^^ ~ ''^'"' ^^ " "'^'^ ^'' ~ ^^"'^ "^ ^^' ''^ "^4
This is called Dirichlet's formula.
[Note. What are now called infinite and improper integrals were defined by Cauchy,
Lecons sur le calc. inf. 1823, though the idea of infinite integrals seems to date from
Maclaurin (1742). The test for convergence was employed by Chartier (1853). Stokes
(1847) distinguished between 'essentially' (absolutely) and non-essentially convergent
integrals though he did not give a formal definition. Such a definition was given by
Dirichlet in 1854 and 1858 (see his Vorlesungen, 1904, p. 39). In the early part of the
nineteenth century improper integrals received more attention than infinite integrals,
probably because it was not fully realised that an infinite integral is really the limit
of an integral.]
4*6. Complex integration*.
Integration with regard to a real variable x may be regarded as integration
along a particular path (namely part of the real axis) in the Argand diagram.
Let f{z), (= P + iQ), be a function of a complex variable z, which is continuous
along a simple curve AB m the Argand diagram.
Let the equations of the curve be
x = X (t), y = y (t) (a^t^ b).
Let X (a) + iy (a) = z^, x (6) + iy (b) = Z.
Then iff oc (t), y(t) have continuous differential coefficients^ we define
f{z) dz taken along the simple curve AB to mean
\>-Ht--i)
-\-i-~] dt.
The 'length' of the curve ^5 will be defined as r^('^Y+ (^X dt.
doc du
It obviously exists if j7> ;^ are continuous; we have thus reduced the
discussion of a complex integral to the discussion of four real integrals, viz.
/>!-■ />!-. M-. />!-
* A treatment of complex integration based on a different set of ideas and not making
so many assumptions concerning the curve AB will be found in Watson's Complex Integration
and Cauchy^s Theorem.
t This assumption will be made throughout the subsequent work.
X Cp. § 4-13 example 4.
78 THE PROCESSES OF ANALYSIS [CHAP. IV
By § 4*13 example 4, this definition is consistent with the definition of an
integral when AB happens to be part of the real axis.
Examples, j f{z)dz= — j f{z) dz, the paths of integration being the same (but in
J Zo J Z
opposite directions) in each integral.
\jz=Z-z,. /^«^^=/^'fS-2'| + ^-(-'J+y§)}^^
= l^^-^y^+i^'y
4'61. The fundamental theorem of complex integration.
From § 4*13, the reader will easily deduce the following theorem :
Let a sequence of points be taken on a simple curve z^Z; and let the first
n of them, rearranged in order of magnitude of their parameters, be called
^■i*"', 2^2 w, . . . ^„(«> (^o^ = z^, 2'„+i<'*' = Z) ; let their parameters be ^i<"', ^.2<">, . . . t^*"',
and let the sequence be such that, given any number 8, we can find N such
that, when n>N, ^r+i"*' -^r'"' < K^ov r = 0, 1, 2, ... , ri ; let ^r**"' be any point
whose parameter lies between i;.'"', ifr+i"** ; then we can make
r = 0 -l Za
arbitrarily small by taking n sufficiently large.
4'62. An upper limit to the value of a complex integral.
Let M be the upper bound of the continuous function \f{z)\.
Then !f;/(.)rf.|«f;|/Wlj(^ + i|)|rf(
^Ml,
where I is the ' length ' of the curve z^Z.
I r^ I
That is to say, I f{z)dz\ cannot exceed Ml.
4"7. Integration of infinite series.
We shall now shew that if *S^ {z) = u-^ {z) + Wa {z) + ... is a uniformly con-
vergent series of continuous functions of z, for values of z contained within
some region, then the series
I u-^ {z) dz + I Uo {z)dz + ...,
' c J c
(where all the integrals are taken along some path G in the region) is con-
vergent, and has for sura I aS'(^) dz.
R^ {z) dz
c
4-61-4'7] THE THEORY OF RIEMANN INTEGRATION 79
For, writing
*Sf {z) = Ml {z) + M2 (^) + . . . + M„ {z) + Rn {z),
we have
1 S{z)dz=\ Ui{z)dz+ ... + Un (z) dz+ I Rn (z) dz.
Jo J c -' c J c
Now since the series is uniformly convergent, to every positive number e
there corresponds a number r independent of z, such that when n^r we have
\Rn{z)\<€, for all values of z in the region considered.
Therefore if I be the length of the path of integration, we have (§ 4*62)
Therefore the modulus of the difference between / S (z) dz and
J c
S / Urn (z) dz can be made less than any positive number, by giving n any
m=i J c
sufficiently large value. This proves both that the series 1 i u^ {z) dz is
m=l J c
convergent, and that its sum is I 8{z)dz.
J c
Corollary. As in § 4*44 corollary, it may be shewn that*
0? °° , , '^ d , .
u/Z n=0 n=o '*■'
if the series on the right converges uniformly and the series on the left is convergent.
Example 1. Consider the Series
"" 2x {n (n + 1) Hiu^ sc^ — 1} cos x^
„li {1+^2 sin2 x^}{l + (n+lf sin2 x^} '
in which x is real.
The nth term is
2a:ncosa;2 2x{n + l)Gosx'^
1 + n^sin^ x^ 1 +(rj. + l)2 sin^^^'
and the sum of n terms is therefore
2x cos x^ 2x (n + l) cos x^
l + sin^ar^ l + {n + iysm''x^'
Hence the series is absolutely convergent for all real values of x except ± sJ{miT)
where m=l, 2, ... ; but
„ 2x{n-\-\) cos x"^
" ^^' ^ H-(« + l)2sin2.r2 '
and if n be any integer, by taking x = {n -^-1)"'^ this has the limit 2 as w^- 00 . The series is
therefore non-uniformly convergent near ^ = 0.
* — ;— means lim -^^ / — where h-*-0 aloncr a definite simple curve; this definition
is modified slightly in § 5-12 in the case when f {z) is an analytic function.
80 THE PROCESSES OF ANALYSIS [CHAP. IV
2 /y* cos 7
Now the sum to infinity of the series is ^ ^ , and so the integral from 0 to a; of
the sum of the series is arc tan (sin x''). On the other hand, the sum of the integrals from
0 to X of the first n terms of the series is
arc tan (sin x"^) — arc tan (%+ 1 sin x"^),
and as « -» 00 this tends to
arc tan (sin x^) — -^.
Therefore the integral of the sum of the series differs from the sum of the integrals of
the terms by \tt.
Example 2. Discuss, in a similar manner, the series
=1 « (m + 1) (1 + e"^2) (1 + e" + 1 A-2)
for real values of x.
Example 3. Discuss the series
M1 + M2 + W3+--M
where
Ml = 26"^", Un=nze~'^"— {n - 1) ze-("-i)*^,
for real values of z.
The sum of the first n terms is nze~^^ , so the sum to infinity is 0 for all real values
of z. Since the terms «„ are real and ultimately all of the same sign, the convergence
is absolute.
In the series
I Uidz+ I U2dz+ I u^dz-'r...,
Jo Jo Jo
the sum of 7i terms is -1(1 — e"*"'), and this tends to the limit | as n tends to infinity ; this
is not equal to the integral from 0 to 2 of the sum of the series 2m„.
The explanation of this discrepancy is to be found in the non-uniformity of the
convergence near 2 = 0, for the remainder after n terms in the series U1 + U2 + ...is — nze~^ ;
and by taking z=n~^ we can make this equal to e"V», which is not arbitrarily small; the
series is therefore non-uniformly convergent near 2 = 0.
Example 4. Compare the values of
I \ "2, uA dz and 2 I ?«„ dz,
J 0 {n=\ J ft=l J 0
where
2^22 2(W + 1)22
" {l+'A:^z^)\og{n + l) {l-|-(% + l)2s2}log(;i + 2)"
(Trinity, 1903.)
REFERENCES.
G. F. B. RiEMANN, Ges. Math. Werke, pp. 239-241.
G. Lejbune-Dirichlet, Vorlesungen. (Brunswick, 1904.)
F. G. Meyer, Bestimmte Integrale. (Leipzig, 1871.)
E. GouRSAT, Cours d" Analyse, Chapters iv, xiv.
C. J. DE LA Vall^e Poussin, Cour.<i d' Analyse Infinitesimale, Chapter vi.
E. W. HoBSON, Functions of a Real Variable, Chapter v.
T. J. I'a. Bromwich, Theory of Infinite Series, Aj^pendix iii.
the theory of riemann integration 81
Miscellaneous Examples.
1. Shew that the integrals
/ sin {x^) dx, I cos (x^) dx, I x exp ( -r x*^ sin^ x) dx
Jo Jo Jo
converge. (Dirichlet and Du Bois Reymond.)
2. If a be real, the integral
cos (ax) ,
' ' dx
. 0 1+^-
is a continuous function of a. (Stokes.)
/:
3. Discuss the uniformity of the convergence of the integral
/""
/ X sin (x^ — ax) dx.
Jo
3 j X sin (x^ - ax) dx= - (- + ^-3) cos{x^-ax)
(de la Valine Poussin.)
4. Shew that / exp [ — ei<^(x^-nx)]dx converges uniformly in the range {-^n, ^ir)
J 0
of values of a. (Stokes.)
/• " x'^dx
5. Discuss the convergence or divergence of I — ■ when /x, v, p are
Jo l+^''|sin^|P
positive. (Hardy.)
6. Examine the convergence of the integrals
7o \^ 2 ^l-e^J X ' Jo of-
f^ dx
7. Shew that I „ exists.
J IT x
(Math. Trip. 1914.)
2 (sin x)^
8. Shew that I .r-^e^in^ sin 2.ro?^ converges if a>0, w>0. (Math. Trip. 1908.)
J a
00
9. If a series g{z)= 2 (cv — Cy+i) sin(2i/ + l) m, (in which Co = 0), converges uniformly
IT °° C
in an interval, shew that a (z) ■ is the derivative of the series f(z)= 2 — sin 2v7tz.
^ ^ '' sm TTZ •' ^ ' ^^i V
(Lerch.)
10. Shew that r r ... f "" d-^idx,...dx, ^^^ r r r dx,dx,...d^
J J J (Xi'^ + X^^+.-.+Xr,^)" J J J Xi'' + X.f + ...+X„''
converge when a>hn and a~^+^~^ + ...+\~^<l respectively. (Math. Trip. 1904.)
11. If/(ji;', ^) be a continuous function of both x and ^ in the ranges {a^x^b), (a^i/^b)
except that it has ordinary discontinuities at points on a finite number of curves with
continuously turning tangents, each of which meets any line parallel to the coordinate axes
only a finite number of times, then I f{x, y) dx is a continuous function of y.
[Consider I -I- I +...+ I {/(•^'j l/ + ^i)-f{^, ]/)} dx, where the nmubers
J a J ai + fi J ttn+fn
Si, 82, ... fi, f2) ••• ^^^ ^o chosen as to exclude the discontinuities of /(.r, y + /0 from the
range of integration ; aj, a2, ... being the discontinuities oi fix, y).'] (Bocher.)
W. M. A. 6
CHAPTER V
THE FUNDAMENTAL PROPERTIES OF ANALYTIC FUNCTIONS ;
TAYLOR'S, LAURENT'S AND LIOUVILLE'S THEOREMS
5'1. Property of the elementary functions.
The reader will be already familiar with the terra elementary function, as
used .(in text-books on Algebra, Trigonometry, and the Differential Calculus)
to denote certain analytical expressions* depending on a variable z, the
symbols involved therein being those of elementary algebra together with
exponentials, logarithms and the trigonometrical functions ; examples of such
expressions are
z^,, e^, log z, arc sin z"^.
Such combinations of the elementary functions of analysis have in common
a remarkable property, which will now be investigated.
Take as an example the function e^.
Write , . e'=f{z).
Then, if 2^ be a fixed point and if z' be any other point, we have ^
f{z)-f{z)_e^'-e^_^ e<^-)_-l
z — z
= e~
and since the last series in brackets is uniformly convergent for all values of
z, it follows (§ 37) that, as z ^ z, the quotient
/(^')-/(^)
z — z
tends to the limit e^, uniformly for all values of arg(ir'— z).
This shews that the limit of
fi^')-f{^)
z — z
is in this case independent of the path by which the point z tends towards
coincidence with z.
* The reader will observe that this is not the sense in which the term function is defined
(§ 3'1) in this work. Thus e.g. x-iy and | z \ ave functions of z { = x + iy) in the sense of § 3'1,
but are not elementary functions of the type under consideration.
6-1-5 12] FUNDAMENTAL PROPERTIES OF ANALYTIC FUNCTIONS 83
It will be found that this property is shared by many of the well-known
elementary functions; namely, that if f{z) be one of these functions and
h be any complex number, the limiting value of
\\f{^ + h)-f{z)]
exists and is independent of the mode in which h tends to zero.
The reader will, however, easily prove that, if/(^) = x— iy, where z=x-\- iy,
then lim*^ r — - — - is not independent of the mode in which h-*-0.
5*11. Occasional failure of the property.
For each of the elementary functions, however, there will be certain
points z at which this property will cease to hold good. Thus it does not
hold for the function at the point z = a, since
lim
..„ , J U
h^o h [z — a — h z — a]
does not exist when z = a. Similarly it does not hold for the functions log z
and z^ at the point ^= 0. • , '
These exceptional points are called singular points or singularities of the
function /(^) under consideration; at other points the function is said to be
analytic.
The property does not hold good at any point for the function \z\.
5"12. Cauchy's* definition of an analytic function of a complex variable.
The property considered in § 5*11 will be taken as the basis of our
definition of an analytic function, which may be stated as follows.
Let a two-dimensional region in the ^-plane be given; and let m be a
function of z defined uniquely at all points of the region. Let z, z-\-hz be
values of the variable z at two points, and u, u + 8u the corresponding values
ult
of u. Then, if, at any point z within the area, -^ tends to a limit when S^-*-0,
By-i-O, independently (where 8z = Sx + iBy), u is said to be a function of z,
which is analytic^ at the point. If the function is analytic and one-valued
at all points of the region, we say that the function is analytic tlirougJiout
the region\.
We shall frequently use the word ' function ' alone to denote an analytic
function, as the functions studied in this work will be almost exclusively
analytic functions.
* See the memoir cited in § 5 "2.
t The words ' regular ' and ' monogenic ' are sometimes used in place of ' analytic'
:J: See § 5-2 cor. 2, footnote.
6—2
84 THE PROCESSES OF ANALYSIS [CHAP. V
In the foregoing definition, the function u has been defined only within
a certain region in the ^-plane. As will be seen subsequently, however, the
function u can generally be defined for other values of z not included in this
region; and (as in the case of the elementary functions already discussed)
may have singularities, for which the fundamental property no longer holds,
at certain points outside the limits of the region.
We shall now state the definition of analytic functionality in a more
arithmetical form.
Let f{z) be analytic at z, and let e be an arbitrary positive number ;
then we can find numbers I and S, {h depending on e) such that
f{z')-f{^)
-I
< €
whenever \ z' — z \ < B.
l{f{z) is analytic at all points 2^ of a region, I obviously depends on ^ ; we
consequently write I =/' (2).
Hence f{z) =f(z) + {z' - z) f {z) + v (/ - z),
where w is a function of z and / such that | ?; | < e when \ z' — z \ < 8.
Example 1. Find the points at which the following functions are not analytic :
2—1
(i) z^. (ii) cosec2 (z = mtt, ?i any integer). (iii) -^ — . „ (2 = 2,3).
(iv) e^ (2 = 0). (V) {{Z-\)Z}^ (2 = 0,1).
Example 2. If z=x-iriy, f{z) = u-\-iv, where u, v, x, y are real and / is an analytic
function, shew that
d-x = d-r d-r-dx- (Riemann.)
5*13. An application of the modified Heine-Borel theorem.
Let f{z) be analytic at all points of a continuum ; and on any point z of
the boundary of the continuum let numbers f (z), 8 (8 depending on z) exist
such that
\f{z')-f{z)-iz'-z)f,{z)\<e\z'-z\
whenever \z' — z\ <8 and / is a point of the continuum or its boundary.
[We write /j (2) instead of /' (2) as the differential coefficient might not exist when
2' approaches 2 from outside the boundary so that/j (2) is not necessarily a unique derivate.]
The above inequality is obviously satisfied for all points z of the continuum
as well as boundary points.
Applying the two-dimensional form of the theorem of § 3'6, we see that
the region formed by the continuum and its boundary can be divided into
a finite number of parts (squares with sides parallel to the axes and their
5-13, 52] FUNDAMENTAL PROPERTIES OF ANALYTIC FUNCTIONS 85
interiors, or portions of such squares) such that inside or on the boundary of
any part there is one point z^ such that the inequality
I /(/) -f{z,) - {z' - z,)f, (z,) \<e\z'-z,\
is satisfied by all points z' inside or on the boundary of that part.
52. Cauchy's theorem* on the integral of a function round a contour.
A simple closed curve G in the plane of the variable z is often called
a contour; if A, B, D be points taken in order in the counter-clockwise sense
along the arc of the contour, and if f{z) be a one-valued continuous +
function of z (not necessarily analytic) at all points on the arc, then the
integral
1 / {z) dz or I f{z) dz
J ABDA J (C)
taken round the contour, starting from the point A and returning to A again,
is called the integral of f{z) taken along the contour. Clearly the value of the
integral taken along the contour is unaltered if some point in the contour
other than A is taken as the starting-point.
We shall now prove a result due to Cauchy, which may be stated as
follows. If f{z) is a function of z, analytic at all points on^ and inside a
contour G, then
\ f{z)dz = 0.
For divide up the interior of G by lines parallel to the real and imaginary
axes in the manner of § 5"13 ; then the interior of C is divided into a number
of regions whose boundaries are squares Cj, Co, ... Gm and other regions
whose boundaries Dj, D^, ... Dy are portions of sides of squares and parts
of C; consider
Mr N c
2 f{z)dz+t \ f{z)dz,
71 = 1 J iCn) « = 1 J (Dn)
each * of the paths of integration being taken counter-clockwise ; in the
complete sum each side of each square appears twice as a path of integration,
and the integrals along it are taken in opposite directions and consequently
cancel§; the only parts of the sum which survive are the integrals of/(^)
* Memoire siir les integrales defmles prises entre des limites imaginaires (1825). The proof
here given is that due to Goursat {Cours d' Analyse, t. ii.).
t It is sufficient for f(z) to be continuous when variations of z along the arc only are
considered.
:{: It is not necessary that f(z) should be analytic on C (it is sufficient that it be continuous
on and inside C), but if f{z) is not analytic on C, the theorem is much harder to prove. This
proof merely assumes that/' (z) exists at all points on and inside C. Earlier proofs made more
extended assumptions ; thus Cauchy's proof assumed the continuity of /' {z). Riemann's
proof made an equivalent assumption. Goursat's first proof assumed that f(z) was uniformly
differentiable throughout C.
§ See § 4-6, example.
86 THE PROCESSES OF ANALYSIS [CHAP. V
taken along a number of arcs which together make up G, each arc being
taken in the same sense as in I f{z)dz; these integrals therefore just make
hc)
Now consider f{z)dz. With the notation of § 5"12,
J (C„)
[ f{z) dz=\ . {f{z,) + (^ - z,) f {z,) + {z-z,)v] dz
J (Cu) J {Cn)
= {/(0-'2^i/'(^i)} I dz+f'(z,) zdz + {z-z,)vdz.
•I (Cn) UCn) J (Cn)
But [ dz = [z]., =0, f zdz= Iz^l =0,
JiCn) ^" J(Cn) L JCn
by the examples of § 4*6, since the end points of C„ coincide.
Now let l^ be the side of (7„ and A^ the area of C„,
Then, using § 4*62,
f{z)dz\ = n {z — z^)vdz <, \ ](z—Zi)vdz\ .- ■
J (Cn) I M(C„) J(Cn)'
<el^^/2. \dz\ = eln^/2.4>l^=4,eAn^/2.
In like manner
(Dn)
f{z)dz
^ I \{z — Zi) vdz I
Ur>n)
where A J is the area of the complete square of which D^ is part, Z,/ is the
side of this square and X,^ is the length of the part of G which lies inside this
square. Hence, if \ be the whole length of C, while I is the side of a square
which encloses all the squares C„ and D^,
(C)
f(z)dz
^ s
M = l I J (Cn)
f{z)dz
+ s
(Dn)
f(z)dz
<46V2 S A^+ t AJ + l % \A
(.« = ! n = l n = l }
< 46 V2 . (l' + IX).
Now e is arbitrarily small, and I, \ and I fi^) dz are independent of e.
-' iC)
It therefore follows from this inequality that the only value which 1 f(z) dz
J c
can have is zero ; and this is Cauchy's result.
5'2] FUNDAMENTAL PROPERTIES OF ANALYTIC FUNCTIONS 87
Corollary 1. If there are two paths z^AZ and z^^BZ from ^o to Z, and if f(z) is a
function of z analytic at all points on these curves and throughout the domain enclosed by
these two paths, then | f{z) dz has the same value whether the path of integration is
Zf^^AZ ox z^BZ. This follows from the fact that zqAZBzq is a contour, and so the integral
taken round it (which is the difference of the integrals along zqAZ and ZqBZ) is zero.
Thus, \i f{z) be an analytic function of z, the value of 1 f{z)dz is to a certain extent
J AB
independent of the choice of the arc AB, and depends only on the terminal points A and B.
It must be borne in mind that this is only the case when f{z) is an analytic function in the
sense of § 5'12.
Corollary 2. Suppose that two simple closed curves Co and Cy are given, such that Co
completely encloses Cj, as e.g. would be the case if Co and C\ were concentric circles or
confocal ellipses.
Suppose moreover that f{z) is a function which is analytic* at all points on Co and Cj
and throughout the ring-shaped region contained between Co and Ci . Then by di-awing a
network of intersecting lines in this ring-shaped space, we can shew, exactly as in the
theorem just proved, that the integral
I
f{z)dz
is zero, where the integration is taken round the whole boundary of the ring-shaped space;
this boundary consisting of two curves Co and Ci, the one described in the counter-clockwise
direction and the other described in the clockwise direction.
Corollary 3. In general, if any connected region be given in the 2-plane, bounded by
any number of simple closed curves Co, Ci, C2, ..., and if /(2) be any function of z which
is analytic and one- valued everywhere in this region, then
}
f{z)dz
is zero, where the integral is taken round the whole boundary of the region ; this boundary
consisting of the curves Co, Ci, ..., each described in such a sense that the region is kept
either always on the right or always on the left of a person walking in the sense in question
round the boundary.
An extension of Cauchy's theorem I f{z)dz=0, to curves lying on a cone whose vertex
is at the origin, has been made by Raout {JVouv. Annales de Math. (3) xvi. (1897),
pp. 365-7). Osgood {Bull. Amer. Math. Soc, 1896) has shewn that the property j f{z)dz=0
may be taken as the property defining an analytic function, the other properties being
deducible from it.
Example. A ring-shaped region is bounded by the two circles [ 2 1 = 1 and ; s , = 2 in the
/dz
— , where the integral is taken round the boundary
of this region, is zero.
* The phrase ' analytic throughout a region ' implies one-valuedness (§ 5-12) ; that is to say
that after z has described a closed path surrounding CQ,f{z) has returned to its initial value. A
function such as log 2 considered in the region 1^]*] ^2 will be a^iA to be 'analytic at all
points of the region.'
88 THE PROCESSES OF ANALYSIS [CHAP. V
For the boundary consists of the circumference |z| = l, described in the clockwise
direction, together with the circumference | z | = 2, described in the counter-clockwise
direction. Thus if for points on the first circumference we write z=e^'^, and for points on
the second circumference we write z = 2e'<t>, then d and <f) are real, and the integral becomes
Jo e«* Jo 2e"/>
5"21. The valube of an analytic function at a point, expressed as an integral
taken round a contour enclosing the point.
Let C be a contour within and on which f{z) is an analytic function of z.
Then, if a be any point within the contour,
.fJA
z — a
is a function of z, which is analytic at all points within the contour C except
the point z = a.
Now, given e, we can find S such that
I /(^) -/(") -(z-a)f(a)\^e\z-a\
whenever \z — a\< B; with the point a as centre describe a circle 7 of radius
r < B, r being so small that 7 lies wholly inside C.
Then in the space between 7 and G f(z)/(z — a) is analytic, and so, by '
S 5'2 corollary 2, we have
j f(z)dz^ff(z)dz
J c z — a j y z — a
where I and 1 denote integrals taken counter-clockwise along the curves
C and 7 respectively.
But, since | 2: — a | < 8 on 7, we have
r f(z)dz ^ f f(a)+(z-a)f(a) + v(z-a) ^^
J y z — a J y z — a '
where j ^' j < e ; and so
Now, if z be on 7, we may write
z — a = re^^,
where r is the radius of the circle 7, and consequently
jyZ-a Jo re'^ Jo
and I dz = j ire'^dd = 0 ;
also, by § 4'62,
//
dz \ ■C € . 27rr.
521, 522] FUNDAMENTAL PROPERTIES OF ANALYTIC FUNCTIONS 89
Thus I [ f^^ - 27nf{a) = f ,dz
\Jc z-a -^ ^ ' jy
^ 27rre.
I
But the left-hand side is independent of e, and so it must be zero, since e
is arbitrary ; that is to say
•^ ^ ' Ittx J c z - a
This remarkable result expresses the value of a function 'f{z) (which is
analytic on and inside G) at any point a within a contour G, in terms of an
integral which depends only on the value of f{z) at points on the contour
itself
Corollary. If f{z) is an analytic one-valued function of « in a ring-shaped region
bounded by two curves C and C", and a is a point in the region, then
2nt J c Z — a ZttI J c' z — ct
where C is the outer of the curves and the integrals are taken counter-clockwise.
5"22. The derivates of an analytic function f(z).
The function /' (z), which is the limit of
f(z + h)-f(z)
h
as h tends to zero, is called the derivate oi f(z). We shall now shew that
/' (z) is itself an analytic function of z, and consequently itself possesses a
derivate.
For if C be a contour surrounding the point a, and situated entirely
within the region in which f{z) is analytic, we have
/(a)=lim-^(" + ^> ->'<">
^-♦0 h
= Um ,1^ I ( f^^, - [ /i^l
h-^n ZTTih [J c z — a — h J c z — a
j,^Q 'Zwi ] c{z- a) {z-a- h)
27ri J c (^ — «)^ h^o 27rt J c (^ — (^y (z— a — h)'
Now, on C, f(z) is continuous and therefore bounded, and so is (z — a)'
while we can take j h \ less than the upper bound of ^\z — a\.
90 - ' THE PROCESSES OF ANALYSIS [CHAP. V
Therefore
/(^)
is bounded ; let its upper bound be K.
{z — af{z- a — h)
Then, if I be the length ' of C,
lim ^. f , {y^^ ,, k lim I A I (2^)-^Kl = 0,
and consequently / (a) = ^. j^ ^^", ,
a formula which expresses the value of the derivate of a function at a point
as an integral taken along a contour enclosing the point.
From this formula we have, if the points a and a + h are inside C,
f(a + h)-f(a)_ 1 rf(z)dz\ 1 1^
h ^TTiJc h \(z-a-hy {z - of
2iz-a--^h
' f{z)dz
2'7riJ c ' {z-a- hf (z - af
_ 2 f mdz
and it is easily seen that A^ is a bounded function of z when \h\< ^l^ — ci\'
Therefore, as h tends to zero, h~~^ {/' (a + h) — f {a}] tends to a limit,
namely
2 /• f(z)dz
i
27ri J c {z — af
Since /' (a) has a unique differential coefficient, it is an analytic function
of a; its derivate, which is represented by the expression just given, is
denoted by f" {a), and is called the second derivate of /(a).
Similarly it can be shewn t\ia.tf"{a) is an analytic function of a, possessing
a derivate equal to
27ri Jclz- ay '
this is denoted by /"' (a), and is called the thi7-d derivate of /(a). And in
general an nth derivate /'"^' (a) of f{a) exists, expressible by the integral
7?! [ f(z)dz
27ri J c{z- af^^
and having itself a derivate of the form
{n + 1) ! r f{z)dz .
2Tri Jc(z-ay+''
the reader will see that this can be proved by induction without difficulty.
5-23, 5*3] FUNDAMENTAL PROPERTIES OF ANALYTIC FUNCTIONS 91
A function which possesses a first derivate with respect to the complex
variable z at all points of a closed two-dimensional region in the 2^-plane
therefore possesses derivates of all orders at all points inside the region.
5'23. Cauchys inequality for /'"' (a).
Let f{z) be analytic on and inside a circle C with centre a and radius r.
Let M be the upper bound of f{z) on the circle. Then, by § 4-62,
' -^ ^ ^ ' 'Ztr J c ^"^' ■
M.nl
Example. If/ (2) is canalytic, z = x-\-iy and V'' = ^-^ + ^-2 5 shew that
V21ogi/(2)| = 0; and V-i 1/(3) |>0
iinless/(2)=0 or/'(2) = 0. (Trinity, 1910.)
5*3. Analytic functions represented by uniformly convergent series.
Let 2 /w(2') be a series such that (i) it converges uniformly along a
n=0
contour C, (ii) f^ (z) is analytic throughout C and its interior.
00
Then % fn (z) converges, and the sum of the series is an analytic
n=0
function throughout C and its interior.
CO
For let a be any point inside C; on C, let S /„ (z) = $ (z).
«=o
dz
Then ![*(£)<;, l.[ J i/. (4
z — a
„=o r^TTi J c^-a )
00
by* §4*7. But this last series, by § 5"21, is S fn{a); the series under
w=0
consideration therefore converges at all points inside C; let its sum inside
G (as well as on C) be called ^(z). Then the function is analytic if it
has a unique differential coefficient at all points inside C.
But if a and a + h be inside C,
^{a + h)-^ (a) 1 f ^(z)dz
h 2iTi j c{z — a){z — a — h)'
and hence, as in § 5-22, lim [{^ (a + h) — ^ {a)] h~^'] exists and is equal to
* Since j z - a |~i is bounded when a is fixed and 2 is on C the uniformity of the convergence
of 2 /„(z)/(z-a) follows from that of S f^,{z).
n=0 w=0
92 THE PROCESSES OF ANALYSIS [CHAP. Y
1 r <E>(^) . .
^ — . I 7 ^dz; and therefore ^(z) is analytic inside G. Further, by
27ri J c{z — o) '' ''
transforming the last integral in the same way as we transformed the first
00 oo
one, we see that <!>' {a) = 2 // (a), so that S /„ (a) may be ' differentiated
M=0 w=0
term by term.'
If a series of analytic functions converges only at points of a curve which is not closed
nothing can be inferred as to the convergence of the derived series*.
"^ COS 7X3C
Thus 2 ( — )" n— converges uniformly for real values of x {% 3"34). But the derived
n=\ n^ ''
00 sin Thz
series 2 ( — )"~^ converges non-uniformly near 2 = (2m+l)7r, {m any integer) ; and
n=l ^*
the derived series of this, viz. 2 ( — )"~^ cos nz, does not converge at all.
Corollary. By § 3*7, the sum of a power series is analytic inside its circle of con-
vergence.
5'31. Analytic functions represented hy integrals.
Let f{t, z) satisfy the following conditions when t lies on a certain path
of integration (a, h) and z is any point of a region 8 :
?)f
(i) / and ~- are continuous functions of t.
(ii) / is an analytic function of z.
(iii) The continuity of ^- qua function of z is uniform with respect to
the variable t.
Then f{t, z) dt is an analytic function of z. For, by § 4*2, it has the
unique derivate \' — dt.
J a OZ
5'32. Analytic functions represented by infinite integrals.
From § 4-44 (II) corollary, it follows that f{t, z) dt is an analytic
J a
function of z at all points of a region ^S^ if (i) the integral converges, (ii) f{t, z)
is an analytic function of z when t is on the path of integration and z is on aS^,
(iii) V is a continuous function of both variables, (iv) / --^' — dt
^ dz Ja dz
converges uniformly throughout S.
For if these conditions are satisfied 1 f{t, z) dt has the unique derivate
J a
f
J a
OZ
* This might have been anticipated as the main theorem of this section deals with uniformity
of convergence over a two-dimensional region.
5-31-5-4] Taylor's, Laurent's and liouville's theorems 93
A case of very great importance is afforded by the integral / e~'^/(<) dt,
where f{t) is continuous and |/(0I < ^^'^^ where K, r are independent of t;
it is obvious from the conditions stated that the integral is an analytic
function of z when R (z) ^ r, > r. [Condition (iv) is satisfied, by § 4431 (I),
since I te^^~^^^^dt converges.]
5*4. Taylors Theorem*.
Consider a function f{z), which is analytic in the neighbourhood of a
point z = a. Let C be a circle with a as centre in the ^-plane, which does
not have any singular point of the function f{z) on or inside it ; so that f{z)
is analytic at all points on and inside G. Let z = a+h be any point inside
the circle C. Then, by § 5*21, we have
•^ 27rr J c z — a — h
_J_r ( 1 h A" ^1"+^ I
''liriic^^''^ \z-a^{z~af^ ---'^ {z-aY+^'^{z-aY^'{z-a-h)]
= f{o) + hf (a) + %, /' (a) + . . . + ^ /"»' {a) + ^. 1 fif>dz-h''^'
y V /^ y V y-Tgi^ V / ^t-^ V / ^iri } c {z - aY+^ {z - a - h)
f(z)
But when z is on C, the modulus of — p is continuous, and so,
z— a — h
by § 3"61 cor. (ii), will not exceed some finite number M.
Therefore, by §4-62,
•I'm J c(z -
f(z) dz . k'^^
^ M.27rR A h ly+i
^ 27r \R) '
a)'»+^ {z-a- h)
where R is the radius of the circle C, so that 'i.irR is the length of the path
of integration in the last integral, and R = \z — a\ for points z on the cir-
cumference of G.
The right-hand side of the last inequality tends to zero as n^ cc . We
have therefore
/(a + /0=/(a)-h;./'(a) + |',/"(a) + ...+^/<-)(rO+...,
which we can write
This result is known as Taylor s Theorem ; and the proof given is due to
Cauchy. It follows that the radius of convergence of a power series is always
* The formal expansion was first published by Dr Brook Taylor (1715) in his Methodus
Incrementorum.
94 THE PROCESSES OF ANALYSIS [CHAP. V
at least so large as only just to exclude from the interior of the circle of con-
vergence the nearest singularity of the function represented hy the series. And
by § 5*3 corollary, it follows that the radius of convergence is not larger
than the number just specified. Hence the radius of convergence is just such
as to exclude from the interior of the circle that singularity of the function
which is nearest to a.
At this stage we may introduce some terms which will be frequently
used.
If f(a) = 0, the function f{z) is said to have a zero at the point z = a.
If at such a point /' (a) is different from zero, .the zero of /(a) is said to be
simple; if, however, /'(a), /"(a), .../<"~^> (a) are all zero, so that the Taylor's
expansion of f(z) aX z = a begins with a term in {z — a)", then the function
f{z) is said to have a zero of the nth 07^der at the point z = a.
Example 1. Find the function /(s), which is analytic throughout the circle C and its
interior, whose centre is at the origin and whose radius is unity, and has the value
a — cos 6 . sin 6
+ 2
a^ - 2a cos ^ + 1 a^ — 2a cos ^ + 1
(where a>l and 6 is the vectorial angle) at points on the circumference of O.
We have
f{z)dz . ., , , ',..
•' ^ ' 2m J c 2"^
27n J 0
"277 jo a-e"»9 ~"27rt jc'S"(a-2)^L^«- J
«,7> -7/1 a-cos(94-isui^ , ,,. ...
e - "'« . idQ . — r. — jT n — r ■■< (puttuig z = e^^)
a^-2acos^+l ' ^^ e. i
Therefore by Maclaurin's Theorem*,
or f{z) = {a-z)~'^ for all points within the circle.
This example raises the interesting question, Will it still be convenient to define f{z)
as (a-z)~i. at points outside the circle ? This will be discussed in i^ 5-51.
Example 2. Prove that the arithmetic mean of all values of s"" 2 (XyS", for points z on
the circumference of the circle |2| = 1, is a„, if "SavZ" is analytic throughout the circle and
its interior.
/■(") (0)
Let 2 at,z''=f{z), so that a„=- — -— . Then, writing 2 = e'^, and calling C the circle
1 r^-f{z)d6 1 [ f{z)dz /(")(0) ^,
1_ f2nf(z)dB^ 1
27r j 0 2" 271
* The result / (2) =/ (0) + 2/' {0)+-/"(0) + ..., obtained by putting a = 0 in Taylor's Theorem,
is usually called Maclaurin's Theorem ; it was discovered by Stirling (1717) and published by
Maclaurin (1742) in his Fluxions.
5'41] Taylor's, Laurent's and liouville's theorems 95
Example 3. Let /(«) = «'■ ; then f{z-\-h) is an analytic function of h when | A | < | a |,
T (t — 1 )
for all values of r ; and so (« + A)*' = «'" + r«'"~iA+ --- — -z'^~^h'^ + ..., this series converging
when I A I < \ z\. This is the binomial theorem.
Example 4. Prove that if A is a positive constant, and (1 - 'izh-^h^)~^ is expanded in
the form
, , 1+AP,(2)+A2P2(2)+A3P3(2) + (A),
(where Pn{z) is easily seen to be a polynomial of degree n in z), then this series converges
so long as z is in the interior of an ellipse whose foci are the points 2 = 1 and z= — 1, and
whose semi-major axis is \{h + h~'^).
Let the series be first regarded as a function of h. It is a power series in A, and
therefore converges so long as the point h lies within a circle in the A-plane. The centre
of this circle is the point A = 0, and its circumference will be such as to pass through that
singularity of {\ — 2zh-\-h?')~^ which is nearest to h = 0.
But I -<izh + h'^ = {h-z + {z^ -\)^} [h- z-{z^ -l)^,
So the singularities of {l-'izh + h?)'^ are the points h = z-{z'^-\)^ and h = z + {z'^-\)^.
[These singularities are branch points (see § 5'7).]
Thus the series (A) converges so long as \h\ is less than both
|2-(22-i)i! and |2 + (22-i)i|.
Draw an ellipse in the z-plane passing through the point z and having its foci at ±1.
Let a be its semi-major axis, and 6 the eccentric angle of z on it.
Then' ' ' 2 = acos ^-t-t"(a2— 1)2 sin^,
which gives 2±(22-l)^= {a + (a2- l)i} (cos^ + isiu^),
so .,: .,• |z±(s2-l)*| = a±(a2_i)4.
Thus the series (A) converges so long as h is less than the smaller of the numbers
a + {a^-\)^ and a — {a?-\)^, i.e. so long as h is less than a— (a^ — 1)2. But A = a — (a^- l)i
when a=\ {h + h~^).
Therefore the series (A) converges so long as z is within an ellipse whose foci are 1 and
— 1, and whose semi-major axis is ^{h + h~^).
5'41. Forms of the remainder in Taylor s series.
Let f{x) be a real function of a real variable ; and let it have continuous
differential coefficients of the first n orders when a % x % a-{-h.
If 0^t%l, we have
(1 (n-l hm \ hn {^ ^ f\n—i
Integrating this between the limits 0 and 1, we have
f{a + h)=f(a)+:^ ^, /*"'(«)+ A 1 f-Ha + th)dL
Let R^ = ^-^^, £ (1 - 0"-' /'"' (a + th) dt ;
and let jj be a positive integer such that p $ n.
96 THE PROCESSES OF ANALYSIS [CHAP. V
hn n
Then Rn = 7 ^, (1 - ty-' ■ (1 - 0""^ /'"» (a + th) dt.
{n— L)]j 0
Let U, L be the upper and lower bounds of (1 - ty-Pf''^ (a + th).
Then
r L{1 - t)P-' dt< f (1 - t)P-^ . (1 - ty-Pf''^ (a + th) dt<[ U{1 - t)P-' dt.
Jo Jo Jo
Since (1 — i)"~^y<"' (a + th) is a continuous function it passes through all
values between U and L, and hence we can find 6 such that 0 ^ ^ ^ 1, and
/:
(1 _ ty-ifin) (a 4- th) dt=p-' (1 - ^)«-i'/(") {a + Oh).
0
h^
Therefore Rn = , ttt" (1 " QY'P f^'^^ (a + Qh).
{n — 1)1 p ^ ' *'
h'^
Writing p = n,we get Rn = —;/*"' («• + 6h), which is Lagrange's form for
h^
the remaiTider ; and writing p = l, we get Rn = , _ ^x, (1 — ^y'~^/''*' {a + Qh),
which is Cauohy s form, for the remainder.
Taking tj = 1 in this result, we get
f{a + h)-f{a) = hf{a + eh)
if f {x) is continuous when a^x^a+h ; this result is usually known as the First
Mean Value Theorem (see also § 4"14).
Darboux gave in 1876 {Journal de Math. (3) 11. p. 291) a form for the remainder in
Taylor's Series, which is applicable to complex variables and resembles the above form
given by Lagrange for the case of real variables.
5'5. The Process of Continuation.
Near every point P,Zq, in the neighbourhood of which a function f{z) is
analytic, we have seen that an expansion exists for the function as a series
of ascending positive integral powers of {z — z^, the coefficients in which
involve the successive derivates of the function at Zq.
Now let A be the singularity of fiz) which is nearest to P. Then the
circle within which this expansion is valid has P for centre and PA for
radius.
Suppose that we are merely given the values of a function at all points of
the circumference of a circle slightly smaller than the circle of convergence
and concentric with it together with the condition that the function is to be
analytic throughout the interior of the larger circle. Then the preceding
theorems enable us to find its value at all points within the smaller circle
and to determine the coefficients in the Taylor series proceeding in powers
of z — Z(^. The question arises, Is it possible to define the function at points
outside the circle in such a way that the function is analytic throughout
a larger domain than the interior of the circle ?
6'6] Taylor's, Laurent's and liouville's theorems 97
In other words, given a power series which converges and represents a
function only at points within a circle, to define hy means of it the values
of the function at points outside the circle.
For this purpose choose any point Pj within the circle, not on the line
PA. We know the value of the function and all its derivates at Pj, from
the series, and so we can form the Taylor series (for the same function)
with Pi as origin, which will define a function analytic throughout some
circle of centre Pj. Now this circle will extend as far as the singularity*
which is nearest to P,, which may or not be A; but in either case, this new
circle will usuallyf lie partly outside the old circle of convergence, and for
points in the region which is included in the new circle hut not in the old circle,.
the new series may he used to define the values of the function, although the
old series failed to do so.
Similarly we can take any other point P^, in the region for which the
values of the function are now known, and form the Taylor series with Pg
as origin, which will in general enable us to define the function at other
points, at which its values were not previously known ; and so on.
This process is called continuation^. By means of it, starting from a
representation of a function by any one power series we can find any number
of other power series, which between them define the value of the function
at all points of a domain, any point of which can be reached from P without
passing through a singularity of the function ; and the aggregate § of all
the power series thus obtained constitutes the analytical expression of the
function.
It is impoi'tant to know whether continuation by two different PBQ, PB'Q paths will
give the same final power series ; it will be seen that this is the case, if the function
have no singularity inside the closed curve PBQB'P, in the following way : Let Pj be any
point on PBQ, inside the circle C with centre P ; obtain the continuation of the function
with Pi as origin, and let it converge inside a circle Oj ; let Pj' be any point inside both
circles and also inside the curve PBQB'P ; let *S', Si, Si be the poAver series with P, Pi,
Pi' as origins ; then 1 1 aSj = Si over a certain domain which will contain Pi , if P/ be taken
sufficiently near Pj ; and hence »S'i will be the continuation of Si ; for if Ti were the
continuation of Si, we have 7\ = Si over a domain containing Pj, and so (§ 3-73)
corresponding coefficients in *S'i and 7\ are the same. By carrying out such a process a
sufficient number of times, we deform the path PBQ into the path PB'Q if no singular
point is inside PBQB'P. The reader will convince himself by drawing a figure that
the process can be carried out in a finite luimber of steps.
* Of the function defined by the new series.
i The word 'usually' must be taken as referring to the cases which are likely to come
under the reader's notice while studying the less advanced parts of the subject.
X French, prolongement ; German, Fortsetznng .
§ Such an aggregate of power series has been obtained for various functions by M. J. M. Hill,
by purely algebraical processes, Proc. Londo)i Math. Soc. Vol. xxxv. (1903), pp. 388 et seq.
II Since each is equal to S.
W. M. A. 7
98 THE PROCESSES OF ANALYSIS [CHAP. V
Example. The series
1 Z ^2 23
represents the function
f^'^ = a-.z
only for points z within the circle |z| = ja|.
But any number of other power series exist, of the type
- 1,^-6 {z-hf {z-hf .
a _ 6 + (a _ 6)2 -^ (a - 6)3 "^ (a - 6)4 "^ • ■ ■ '
if hja is not real and positive these converge at points inside a circle which is partly
inside and partly outside |2| = |a|; these series represent this same function at points
outside this circle.
5 "SOI. On functions to xohich the continuation-process cannot he applied.
It is not always possible to carry out the process of continuation. Take as an example
the function/ (2) defined by the power series
/(3) = l+2-'^ + 24 + 28 + 2l6+..,+22»+...^
which clearly converges in the interior of a circle whose radius is unity and whose centre
is at the origin.
Now it is obvious that, as 2^1-0, /(2)-^-f x ; the point +1 is therefore a
singularity of/ (2).
But /(2) = 22+/(22),
and if 22-»-l -0, /(22)-»-qo and so f (z) ->• cc , and hence the points for which 2^=1 are
singularities of f{z) ; the point 2= - 1 is therefore also a singularity of f(z).
Similarly since
/(2) = 22 + 24+/(2*),
we see that if 2 is such that 2*= 1, then 2 is a singularity of f(z) ; and, in general, any root
of any of the equations
22 = 1, 24 = 1, 28=1, 2i6=l, ...,
is a singularity of /(z). But these points all lie on the circle |2| = 1 ; and in any arc
of this circle, however small, there are an unlimited immber of them. The attempt to
carry out the process of continuation will therefore be frustrated by the existence of this
unbroken front of singularities, beyond which it is impossible to pass.
In such a case the function f{z) cannot be continued at all to points 2 situated outside
the circle |2| = 1 ; such a function is called a lacunary function, and the circle is said to be
a limiting circle for the function.
5'51. The identity of two functions.
The two series
1+Z + z'' + z'^-h ...
and -1 4- (^-2) -(5 -2)- + (2 -2)^ -(^-2)4+...
do not both converge for any value of z, and are distinct expansions.
Nevertheless, we generally say that they represent the same function, on the
strength of the fact that they can both be represented by the same rational
1
expression — — .
5-501 -5-6] Taylor's, Laurent's and liouville's theorems 99
This raises the question of the identity of two functions. Under what
circumstances shall we say that two different expansions represent the same
function ?
We might define a function (after Weierstrass), by means of the last
article, as consisting of one power series together with all the other power
series which can be derived from it by the process of continuation. Two
different analytical expressions will therefore be regarded as defining the
same function, if they represent power series which can be derived from each
other by continuation.
Since if a function is analytic (in the sense of Cauchy, § 5*12) at and near
a point it can be expanded into a Taylor's series, and since a convergent
power series has a unique differential coefficient (§ 5"3), it follows that the
definition of Weierstrass is really equivalent to that of Cauchy.
It is important to observe that the limit of a combination of analytic
functions can represent different analytic functions in different parts of the
plane. This can be seen from the following example.
Consider the series
1 / 1\ V / 1\/ 1 1
The sum of the first n terms of this series is
1 / 1\ 1
z \ zj l+z""
The series therefore converges for all values of z (zero excepted) not on the
circle | 2; j = 1. But, as ?i -»- 00 , | ^i*^ | ^ 0 or | ^" | -* 00 according as | ^ j is less
or greater than unity ; hence we see that the sum to infinity of the series is
z when J2;|<1, and - when J2'|>1. This series therefore represents one
function at points in the interior of the circle \z\ = l, and an entirely different
function at points outside the same circle. The reader will see from § 5 '3
that this result is connected with the non-uniformity of the convergence of
the series near \z\ — l.
Z Z'' z'
Example. Shew that the series
represent the same function in the common part of their domain of convei-gence.
2 4 1 / 2^
5"6. Laurent's Theorem.
A very important theorem was published in 1843 by Laurent* ; it relates
to expansions of functions to which Taylor's Theorem cannot be applied.
* Comptes Rendus, t. xvii.
100 THE PROCESSES OF ANALYSIS [CHAP. V
Let C and C be two concentric circles of centre a, of which C" is the inner ;
and let /(^) be a function which is analytic* at all points on C and C and
throughout the annulus between C and C Let a + h be any point in this
ring-shaped space. Then we have (§ 5-21 corollary)
where the integrals are supposed taken in the positive or counter-clockwise
direction round the circles.
This can be written
1 r ( 1 h h"' A**"*"^ )
/(« + ^) = 2:;^- j ^ /(^) I JT^ + (^3^. + • • • + (T^oT 1 + (T-^^
We find, as in the proof of Taylor's Theorem, that
f(z)dz.h''+' [ f(z)dz(z-aT+'
c{2- ay+' {z-a- K) J c {z - a - h) /i"+i
tend to zero as w ^ oc ; and thus we have
• / 7 \ 7 7 o bi bo
y ((t + h) = a-o 4- ttj/i 4- a^h^ + ... + j + j^„ + ...,
where t «,, = ---. j-^-Vn+i and 6,, = -—. (z - ay-' f(z)dz.
This result is Laurent's Theorem ; changing the notation, it can be
expressed in the following form: If f(z) be analytic on the concentric circles
C and C of centre a, and throughout the annulus betiveen them, then at any
point z of the annulus f{z) can be expanded in the form
f{z) = Oo + a^ (z - a) + a, (z - af +...+ -^jira) ^ G-af ^'"'
where «. = ^.j^^, ^^, and b^. = ^,j^^, (t - ar~' f (t) dt.
An important case of Laurent's Theorem arises when there is only one
singularity within the inner circle C, namely at the centre a. In this case
the circle C can be taken as small as we please, and so Laurent's expansion
is valid for all points in the interior of the circle C, except the centre a.
Example 1. Prove that
;(-")=x
(x) + zJ, (x) + z'-J, (.r) + . . . + s»/„ (,r) + . . .
- „ -A (,r) + -, J., (x) -... + ^ J J, (X) + . .
1 f"^"'
where J^^ (.(;) = -- coh {nd - x >ihi 6) dd .
■In J 0
* See § 5'2 corollary 2, footnote.
t We cannot write (;„=/(«) («)/« ! as in Taylor's Theorem since /'(£) is not necessarily analytic
inside C".
56] Taylor's, Laurent's and liouville's theorems 101
For tho function of z under consideration is analytic in any domain which does not
include the point 2 = 0 ; and so by Laurent's Theorem,
jH) = ao + ai2 + a222 + ... + ^ + ^' + ...,
and where G and C are any circles with the origin as centre. Taking C to be the circle of
radius unity, and writing 2 = e'*, we have
1 r^"-
1 r^"'
= ;r— I cos {n6 - X sin 6) d6,
ZTT J 0
since I sin hid — r sin 6) dd vanishes, as may be seen by writing 27r -(f) for B. Thus
Now &n = (-)"«n» since the function expanded is unaltered if — 2"^ be written for z.
Thus
which completes the proof.
Example 2. Shew that, in the annulus defined by |a| <[2| < |ft|, the function
h
\{z-a)(b-z)\
can be expanded in the form
, -, " 1.3. ..(2^-1). 1.3.. .(2? + 2n-l)/a\2
where 8.,= 2 ^ojx — 7 .^7 — Vi ij}'
The function is one-valued and analytic in the annulus (see § 5'7), for the branch-points
0, a neutralise each other, and so, by Laurent's Theorem, if C denote the circle |2| = ?*,
where | a | < /• < 1 6 1 , the coefficient of s" in the required expansion is
1 r _dz_ ( bz 1 i
27ri j cz'^*'^ \{z^){b - 2) j *
Putting z=re^, this becomes
- I ft-nitfr-nr/H\ I t>w \
27
i^J%_»«.-.rf«(l-;-e»)-*(l -%-»)-*,
1 /'S'
-_ I e-ni0r-nde 2
2n- yo
fc=o 2*./{:! b" iZo 2Kl\ r^ '
the series being absolutely convergent and uniformly convergent with regard to B.
The only terms which give integrals different from zero are those for which k = l-\-n.
So the coefficient of 2" is
1 r^n- 1 r
^ dB2 —
2ir J Q ,
3. ..(21-1) 1 . 3 ■..(2^-|-2?t-l) g'
6"*
102 THE PROCESSES OF ANALYSIS [CHAP. V
Similarly it Ccan be shewn that the coefficient of - is .S^a".
Example 3. Shew that
e« + ''/« = ao + a,2 + a222 + ... + j + ^ + ...,
where «„= — I " e("*'"'^'^''^ GOii{{u-v)s,m 6-ne]dd,
•Itt J 0
and bn= — | '^e(" + '')'=°'*cos {(t;- w) sin ^-7i^}o?^.
2rr jo
5"61. The nature of the singularities of one-valued functions.
Consider first a function f{z) which is analytic throughout a closed
region S, except at a single point a inside the region.
Let it be possible to define a function ^ (z) such that
(i) (ji (z) is analytic throughout S,
(ii) when z i= a, f{z) = (f){z)-\ ^~+/ ^ \2+-" +
z — a ' (z — af {z — ay^
Then f(z) is said to have a 'pole of order n at a' ; and the terms
' + ^ + ... + , ^i are called the principal part of f(z) near a.
z — a {z — af '" (z — a)"
By the definition of a singularity (§ 5'12) a pole is a singularity. If w = 1,
the singularity is called a simple pole.
Any singularity of a one-valued function other than a pole is called an
essential singularity.
If the essential singularity, a, is isolated (i.e. if a region, of which a is an
interior point, can be found containing no singularities other than a), then a
Laurent expansion can be found, in ascending and descending powers of a
valid when A >\z — a\> B, where A depends on the other singularities of the
function, and B is arbitrarily small. Hence the ' principal part ' of a function
near an isolated essential singularity consists of an infinite series.
It should be noted that a pole is, by definition, an isolated singularity, so
that all singularities which are not isolated (e.g. the limiting point of a
sequence of poles) are essential singularities.
There does not exist, in general, an expansion of a function valid near a non-isolated
singularity in the way that Laurent's expansion is valid near an isolated singularity.
CoroUaty. If /{z) has a pole of order n at a, and i|/-(2) = (2-«)"/(2) {z^a\
y^{a)=\\xn{z — aYf{z), then y\r{z) is analytic at a.
Example 1. A function is not bounded near an isolated essential singularity.
[Trove that if the function were bounded near z = a, the coefficients of negative powers
of : — « would all vanish.]
561, 5-62] Taylor's, Laurent's and liouville's theorems 103
c z
Example 2. Find the singularities of the function 6*~*/{e'* — 1}.
At z = 0, the numerator is analytic, and the denominator has a simple zero. Hence
the function has a simple pole at 2 = 0.
Similarly there is a simple pole at each of the points 2niria (n= ±1, ±2, ±3, ...); the
denominator is analytic and does not vanish for other values of z.
At z = a, the numerator has an isolated singularity, so Laurent's Theorem is applicable,
and the coefl&cients in the Laurent expansion may he obtained from the quotient
z — a
,3.
IMz-af'^'
/ -. z — a
e 1 + +...
\ «
which gives an expansion involving all positive and negative powers of {z — a). So there is
an essential singularity at z = a.
Example 3. Shew that the function defined by the series
I ?t2»-l{(l+?i-l)"-l}
„=l(0«-l){2»-(H-7i-i)»}
has simple poles at the points 2 = (1 + 71-1)6^*^"/", {k=Q, 1, 2,... n- 1 ; n=\, 2, 3, ...).
(Math. Trip. 1899.)
5*62. The 'point at infinity.'
The behaviour of a function f{z) as \z\-* cc can be treated in a similar
way to its behaviour as z tends to a finite limit.
If we write z = -, so that large values of z are represented by small
values of z' in the ^^'-plane, there is a one-one correspondence between
z and z', provided that neither is zero ; and to make the correspondence
complete it is sometimes convenient to say that when z' is the origin, z is
the ' point at infinity.' But the reader must be careful to observe that this
is not a definite point, and any proposition about it is really a proposition
concerning the point z = 0.
Let f{z) = ^ {z'). Then <^ (/) is not defined at z = 0, but its behaviour
near z =0 is determined by its Taylor (or Laurent) expansion in powers
of z ; and we define <^ (0) as lim ^ {z') if that limit exists. For instance
the function ^{z') may have a zero of order w at the point / = 0; in this
case the Taylor expansion of ^ {z') will be of the form
and so the expansion of/(^) valid for sufficiently large values of | ^ j will be
of the form
A JS C
/ \ ) ^m "^ ^»n+i 2'""^"
In this case, f{z) is said to have a zero of order m at ' infinity.'
104 THE PROCESSES OF ANALYSIS [CHAP. V
Again, the function <f) (z) may have a pole of order m at the point / = 0 ;
in this case
^ u') ^ A + _^ + ^, + ... + ^ + il/+ iV^/ + Pz' + ...;
and so, for sufficiently large values of j z\,f{z) can be expanded in the form
N P
f{z) = Az'" + Bz'»-' + Cz"'-^ + ...+Lz + M + - + - + ....
In this case, f(z) is said to have a pole of order m at ' infinity.'
Similarly f(z) is said to have an essential singularity at infinity, if <f) (z)
has an essential singularity at the point z' = 0. Thus the function &" has an
1
essential singularity at infinity, since the function e^' or
1 J^ 1
^'^ z''^2\z~''^Slz'■''^'■^
hsiS an essential singularity at z' = 0.
Example. Discuss the function represented by the series
2 — . :, ^r~o5 (a>l).
The function represented by this series has singularities at z = — and z= ,
^ a^ a^
(m = l, 2, 3, ...), since at each of these points the denominator of one of the terms in the
series is zero. These singularities are on the imaginary axis, and have 2 = 0 as a limiting
point ; so no Taylor or Laurent expansion can be formed for the function valid throughout
any region of which the origin is an interior point.
For values of z, other than these singularities, the series converges absolutely, since the
limit of the ratio of the (?i + l)th term to the ?ith is Vmi {n + 1)- ^ a-'^ = 0. The function is
an even function of z (i.e. is unchanged if the sign of z be changed), tends to zero as
1 2 1 -s-Qo , and is analytic on and outside a circle C of radius greater than unity and centre
at the origin. So, for points outside this circle, it can be expanded in the form
22+24+2C + "-'
where, by Laurent's Theorem,
Now 2 -p7 ^- ^, = 2 2 , (_)mo,-2mn2-2m_
This double series converges absolutely when j 2 | > 1, and if it be rearranged in powers
of 2 it converges uniformly.
00 / \k — \fj,— 'ikn
Since the coefficient of z~^ is 2 and the only term which furnishes a non-
zero integral is the term in z~^, we liave
1 r CO (_)Jr-l„-2t,t ^2
^77?. ; (- „=o n ! 2
5'63, 5 64] Taylor's, Laurent's and liouville's theorems 105
Therefore, when \z\ > 1, the function can be expanded in the form
ai a* a*
e e e
The function has a zero of the second order at infinity, since the expansion begins with
a term in «~2,
563. Liouville's Theorem*.
Let f{z) he analytic for all values of z and let \f{z) \ < K for all values
of z, where K is a constant (so that \f(z)\ is bounded as \z\^oo). Then
f(z) is a constant.
Let z, z' be any two points and let C be a contour such that z, z' are
inside it. Then, by § 521,
/(^')-/(.) = 5^./jf^,-fi-j/(f)<^f;
take C to be a circle whose centre is z and whose radius is /a ^ 2 | ^' — 0 j ; on
0 write ^=z + pe'^ ; since \^- z'l'^^P when ^ is on C it follows from § 4-62
that
^2n \z' -z\ . K
27rJo y
<2\z-z\Kp-\
Make /o ^ oo , keeping z and / fixed; then it is obvious that f{z') —f{z) = 0 ;
that is to say, f(z) is constant.
As will be seen in the next article, and again frequently in the latter half of this
volume, Liouville's theorem furnishes short and convenient proofs for some of the most
important results in Analysis.
5'64. Functions with no essential singularities.
We shall now shew that the only one-valued functions which have no
singularities, except poles, at any point (including x ) are rational functions.
For let f(z) be such a function ; let its singularities in the finite part
of the plane be at the points Ci, Ca, ... Ck'. and let the principal part (§ 5"61)
of its expansion at the pole Cr be
Z — Cr (Z — CrY '" (z — C^)"'' '
Let the principal part of its expansion at the pole at infinity be
ttiZ + a^z^-\- ... + a,i2";
if there is not a pole at infinity, then all the coefficients in this expansion
will be zero.
• This theorem (first published in 1844 by Cauchy, Comptes Rendus, xix) was given this name
by Borchardt (Crelle, Lxxxviii), who heard it in Liouville's lectures in 1847.
106 THE PROCESSES OF ANALYSIS [CHAP. V
Now the function
f{z)- S |-^^ + ^S2 + -- + r-^'vrl-«i^-«^^'----«-^"
•^ ^ ^ r=l V - Cr {Z - CrY {z - Crf^ j
has clearly no singularities at the points Ci, Cg, ... Cu, or at infinity; it is
therefore analytic everywhere and is bounded as \z\-* qc , and so, by
Liouville's Theorem, is a constant; that is,
/(.)= C+a.. + „,.'+ ... +a...+ I j-«:; + ^-j^,+ ... + (-^"^ j ,
where C is constant; f{z) is therefore a rational function, and the theorem is
established.
It is evident from Liouville's theorem (combined with § 3'61 corollary (ii))
that a function which is analytic everywhere (including oo ) is merely a
constant. Functions which are analytic everywhere except at x are of
considerable importance; they are known as integral functions* . Examples
of such functions are e^, sin z, e*". From § 5*4 it is apparent that there is no
finite radius of convergence of a Taylor's series which represents an integral
function ; and from the result of this section it is evident that all integral
functions (except mere polynomials) have essential singularities at oo .
5T. Many-valued functions.
In all the previous work, the functions under consideration have had a
unique value (or limit) corresponding to each value (other than singularities)
of ^.
But functions may be defined which have more than one value for each
value of z ; thus \i z = r (cos 6 -^ i sin 6), the function z^ has the two values
ri fcos \d-\-iQm\d\, r^ -jcos \{d + 27r) + i sin I (6 + 27r)|- ;
and the function arc tan x (x real) has an unlimited number of values, viz.
Arc tan a; + WTT, where - 2 7r< Arctana;< 2 tt and n is any integer; further
examples of many-valued functions are log 2^, z~^, sin(^^).
Either of the two functions which z^ represents is, however, analytic
except at z = 0, and we can apply to them the theorems of this chapter ; and
the two functions are called 'branches of the many-valued function zk'
There will be certain points in general at which two or more branches
coincide or at which one branch has an infinite limit; these points are called
'branch-points.' Thus ^i has a branch-point at 0; and, if we consider the
change in z^ as z describes a circle counter-clockwise round 0, we see that 6
French, foiict ion flntiere ; German, (janze Funktion.
57] TAYLOR'S, Laurent's and liouville's theorems 107
increases by 27r, r remains unchanged, and either branch of the function passes
over into the other branch. This will be found to be a general characteristic
of branch-points. It is not the purpose of this book to give a full discussion
of the properties of many-valued functions, as we shall always have to
consider particular branches of functions in regions not containing branch-
points, so that there will be comparatively little difficulty in seeing whether
or not Cauchy's Theorem may be applied.
Thus we cannot apply Cauchy's Theorem to such a function as zi when the path of
integration is a circle surrounding the origin ; but it is permissible to apply it to one of
the branches of z^ when the path of integration is like that shewn in § 6'24, for through-
out the contour and its interior the function has a single definite value.
Example. Prove that if the different values of a*, corresponding to a given value of z,
are represented on an Argand diagram, the representative points will be the vertices of an
equiangular polygon inscribed in an equiangular spiral, the angle of the spiral being
independent of a.
(Math. Trip. 1899.)
The idea of the different branches of a function helps us to understand such a paradox
as the following.
Consider the function y = xr^i
for which ' -^ = .r^(l -f-log.r).
When X is negative and real, ~- is not real. But if x is negative and of the form
Q (where p and q are positive or negative integers), y is real.
Lq +1
If therefore we draw the real curve
y=xF,
we have for negative values of x a set of conjugate points, one point corresponding to each
rational value of x with an odd denominator ; and thus we might think of proceeding to
form the tangent as the limit of the chord, just as if the curve were continuous; and
thus -—- , when derived from the inclination of the tangent to the axis of x, would appear
to be real. The question thus arises, Why does the ordinary process of differentiation
give a non-real value for -^ 1 The explanation is, that these conjugate points do not all
arise from the same branch of the function y = x^. We have in fact
y = g.r log X + Ikvix ^
where k is any integer. To each value of k corresponds one branch of the function y.
Now in order to get a real value of y when x is negative, we have to choose a suitable
value for k: and this vahie of k varies as we go from one conjugate point to an adjacent one.
So the conjugate points do not represent values of y arising from the same branch of the
function y — x^, and consequently we cannot expect the value of -f^ when evaluated
for a definite branch to be given by the tangent of the inclination to the axis of x of the
line joining two arbitrarily close members of the series of conjugate points.
108 THE PROCESSES OF ANALYSIS [CHAP. V
REFERENCES.
E. GoURSAT, Cours d' Analyse, Chs. xiv and xvi.
J. Hadamard, La Serie de Taylor et son prolongement analytique (Scientia).
E. LiNDELOF, Le Calcul des Residus (Borel Tracts).
C. J. DE LA Vall^e Poussin, Cours d' Analyse Infinitesimale, Ch. x.
E. Borel, Lecons sur les Fonctions Entieres.
G. N. Watson, Complex Integration and Camhy's Theorem (Camb. Math. Tracts).
Miscellaneous Examples.
1. Obtain the expansion
/(.)=/(a) + 2|--/ (^-2--;+ 23-731-^ V'2~;+2^.X!-^V~2~; + •••]'
and determine the circumstances and range of its validity.
2. Obtain, under suitable circumstances, the expansion
z-aV .J z-ci\ r, { 3(0-a)l ,, f (2m-l) (z-a)!"]
+ .... • (Corey.)
3. Shew that for the series
1
the region of convergence consists of two distinct areas, namely outside and inside a circle
of radius unity, and that in each of these the series represents one function and represents
it completely.
(Weierstrass.)
4. Shew that the function
00
11 = 0
tends to infinity as .r-»-exp {2nip/m !) along the radius through the point ; where m is any
integer and p takes the values 0, 1, 2, ... {m\ - 1).
Deduce that the function cannot be continued beyond the unit circle. (Lerch.)
5. Shew that, if 2- - 1 is not a positive real number, then
1.3...(2»-1)
2 2.4 2.4...27i
(Jacobi and Scheibner.)
Taylor's, Laurent's and liouville's theorems 109
6. Shew that, if z—1 is not a positive real number, then
m(m+l) ...(m + n) ,, , f^ ,-. . , ,
n\ Jo
(Jacobi and Scheibner.)
7. Shew that, if z and 1 — z are not negative real numbers, then
^ Jo m + l\ ^m+3 ^ {m + 3)...(m + 2n-l) J
+ ^^ '^ (m + l)(m + 3)...(«i + 2»-l) Jo ^ ^
(Jacobi and Scheibner.)
8. If, in the expansion of (a + a^z + a2Z^)'"^ by the multinomial theorem, the remainder
after n terms be denoted by Rn (z), so that
(a + ai2+a2s2)'»=Jo + JiS + J22^+...+^»-i2"~^ + /?»(2),
shew that
R^iz) = ia + a,z+a,z^rj^ (,^1^,^,^,.)../ ^^-
(Jacobi and Scheibner.)
9. If {aQ + aiZ + a2z'^)-"^-'^ j {aQ + a^t+aot^)"'dt
J 0
be expanded in ascending powers of z in the form
^12 + ^2^^...,
shew that the remainder after (n — l) terms is
{aQ + aiZ + a^z^)-"^-^ j {aQ + ait + a^f^y^ {naQA^-{27n + 7i + l) a2A^_it} (''-^ di.
J 0
(Jacobi and Scheibner.)
10. Shew that the series
2 {H-X„(2)e'}--J^^
Z~ 3'' 2™
where ^n (2)= -1 +2-2I + 3I -•••+(-)"— ,,
and where ^(2) is analytic near 2 = 0, is convergent near the point 2=0 ; and shew that if
the sum of the series be denoted by/(2), then/ (2) satisfies the differential equation
/'(-')=/ (2) -0(2)- (Pincherle.)
11. Shew that the arithmetic mean of the squares of the moduli of all the values of
the series 2 a^z'' on a circle |2| = r, situated within its circle of convergence, is equal
0
to the sum of the squares of the moduli of the separate terms.
(Gutzmer.)
12. Shew that the series
i c--('"»)i2'»-i
111= I
110 THE PROCESSES OF ANALYSIS [CHAP. V
converges when \z\ < 1; and that, when a > 0, the function which it represents can also
be represented when |2| < 1 by the integral
(S)-
~ 3 f
0 e" -z xii
and that it has no singularities except at the point 2=1.
(Lerch, Monatshefte fur Math, und Phys. vili.)
13. Shew that the series
2 _ 2 f z_ z^ 1
-{z+z )+^ 2 |^i_2„_2,/'2i)(2i/ + 2i/'2i')2'*'(l-2i/-2./'2-ii)(2i' + 2i/'2-ii)2| '
in which the summation extends over all integral values of v, v\ except the combination
(j/ = 0, ^' = 0), converges absolutely for all values of z except purely imaginary values; and
that its sum is +1 or - 1, according as the real part of z is positive or negative.
(Weierstrass.)
14. Shew that sin | m[ 2 + - 1 [ can be expanded in a series of the type
o ^1 ^2
ao + ai2 + «22 +... + - + -2 + .
in which the coefficients, both of 2" and of z ", are
1 r2T
;r— I sin (2 m cos 6) cos tiddd.
2Tr J 0
shew that f{z) is finite and continuous for all real values of z, but cannot be expanded as
a Maclaurin's series in ascending powers of z ; and explain this a^jparent anomaly.
CHAPTER VI
THE THEOEY OF EESIDUES ; APPLICATION TO THE EVALUATION OF
DEFINITE INTEGRALS
6*1. Residues.
If the function f{z) has a pole of order m aX z=-a, then, by the definition
of a pole, an equation of the form
where <f> (z) is analytic near and at a, is true near a.
The coefficient a_i in this expansion is called the residue of the function
f(z) relative to the pole a.
Consider now the value of the integral I f(z)dz, where the path of
integration is a circle* a, whose centre is the point a and whose radius p is so
small that (/> (z) is analytic inside and on the circle.
f . ''^^ f dz f
We have I f{z) dz = ^ a^r I ; — ' — w +1 4> (■^) dz.
J a r = m J a \^ (^) J a
Now 1 (f) (z) dz = 0 hy ^ 5 2 ; and (putting z — a = pe^) we have, if r ^jfe 1,
J a
L{z-aY Jo p'-e'^' ^ Jo ^ LI -'•Jo
But, when ?' = 1, we have
JaZ-a Jo
Hence finally / f (z) dz = 27ria_i.
J a
Now let C be any contour, containing in the region interior to it a number
of poles a,h, c, ... of a function f{z), with residues a_i, 6_i, c_i, ... respec-
tively : and suppose that the function f{z) is analytic throughout G and its
interior, except at these poles.
* The existence of such a circle is implied in the definition of a pole as an isolated
singularity.
112 THE PROCESSES OF ANALYSIS [CHAP. VI
Surround the points a,b, c, ... by circles a, /3, 7, ... so small that their
respective centres are the only singularities inside or on each circle ; then the
function /(2') is analytic in the closed region bounded by C, a, /3, 7, ....
Hence, by § 5"2 corollary 3,
f f(z)dz=( f{z)dz+! f{z)dz+...
= 27rm_i + 27ri6_i + —
Thus we have the theorem of residues, namely that if f{z) he analytic
throughout a contour C and its interior except at a number of poles inside the
contour, then
\ f{z)dz=2'7ri'ER,
J c
where %R denotes the sum of the residues of the function f(z) at those of its
poles which are situated within the contour C.
This is an extension of the theorem of § 5'21.
Note. If a is a simple pole oi f{z) the residue of f{z) at that pole is lim {{z — a)f{z)].
6"2. The evaluation of definite integrals.
We shall now apply the result of § Ql to evaluating various classes
of definite integrals ; the methods to be employed in any particular case may
usually be seen from the following typical examples.
6'21. The evaluation of the integrals of certain periodic functions taken
between the limits 0 and tir.
An integral of the type
2Tr
R (cos d, sin 6) dd,
where the integrand is a rational function of cos 6 and sin 6 finite on the
range of integration, can be evaluated by writing e^^ = 2^ ; since
cos d = -^{z + z~^), sin 6 = ~.{z — z~'^),
the integral takes the form S (z) dz, where S (z) is a rational function of z
■ c
finite on the path of integration C, the circle of radius unity whose centre is
the origin.
Therefore, by § G'l, the integral is equal to ^-ni times the sum of the residues
if S (z) at those of its poles ivhich are inside that circle.
Example 1. If 0 < p < 1,
'2'^ cie f dz
0 l-2pcOs6+p'^ J (;l{l -pz) {Z—p)'
6-2-6"22] THE THEORY OF RESIDUES 118
The only pole of the integrand inside the circle is a simple pole at /? ; and the residue
there is
lim "^ ^
Hence
d6 27r
tin
2p cos 6 +p'^ \—p^
Example 2. If 0 < jo < 1,
jo 1 - 229 cos 26 +p^ "^ J c iz Vi ^ 2 ^ ; (1 -pz^) (1 -pz-^)
— 27r2R,
where 2R denotes the sum of the residues of , . ,/ :;—-, r at its poles inside C : these
42° ( 1 —pz^) {z^-p) •
poles are 0, -pa pi ■ and the residues at them are , , , jt^-t^ — -s^, ^r^, — -:k 5
'^ 4p3 8p^{l-p^) 8p^{l—p^)
and hence the integral is equal to
ir(l-p+p^)
1 —p
k Example 3. If 71 be a positive integer,
[ ^^ e""*' " cos (m^ - sin ^) rf^ = ^ , P" 6*^°' " sin {nO -8md)de= 0.
Example 4. U a > b > 0,
p'^ dd _ 27ra p'T d6 _7r(2a + b)
Jo (a + bcosef (a2-62)t' h (a + bcos'^ef a^{a+b)^'
6*22. The evaluation of certain types of integrals taken between the
limits — 00 and + 00 .
We shall now evaluate I Q (x) dx, where Q {z) is a function such that
(i) it is analytic when the imaginary part of z is positive or zero (except at a
finite number of poles), (ii) it has no poles on the real axis and (iii) as 1 2^ | ^ 00 ,
zQ {z) -* 0 uniformly for all values of arg z such that 0 ^ arg z-^ir; provided
that (iv) when x is real, a;Q(«)-*0, as x^±<X), in such a way* that
ro
Q (x) dx and I Q (x) dx both converge.
/,
0
Given e, we can choose p^ (independent of arg 2^) such that \zQ{z)\ < ejir
whenever \z\> p^ and 0 ■> arg z ^ it.
Consider Q {z) dz taken round a contour C consisting of the part of the
J c
real axis joining the points ± p (where p > po) and a semicircle F, of radius p,
having its centre at the origin, above the real axis.
* The condition xQ (x)-^O is not in itself sufficient to secure the convergence of I Q{x) dx;
consider Q (x) = (x log x)~^.
W. M. A. 8
114 THE PROCESSES OF ANALYSIS [CHAP. VI
Then, by §6-1, [ Q{z)dz= 2iril.R, where 2E denotes the sum of the
residues oi Q{z) at its poles above the real axis*.
Therefore If Q{z)dz-2TritR\ = \\ Q(z)dz .
\J -p \ \JT
In the last integral write z = pe^^, and then
f Q(z)dz = r Q{pe^)pe^Hde
Jt J a
< r(€/7r)dd
Jo
= €,
by § 4-62.
Hence lim / Q (z) dz = 2^"^ R.
P~^odJ — p
But the meaning of / Q (x) dx is lim I Q {x) dx ; and since
lim I Q (x) dx and lim | Q (x) dx both exist, this double limit is the
O-^CdJO p-»-OD j -p
same as lim I Q{x)dx.
p-^-oo J — p
Hence we have proved that
f" Q{x)dx = 2'!ritR.
J —to
This theorem is particularly useful in the special case when Q{a;) is a
rational function.
[Note. Even if condition (iv) is not satisfied, we still have
f {Q(x) + Q{-x)}dx=lim I Q (x) dx=27ri2R.]
J 0 p-*=o J - p
Example 1. The only pole of {z^ + l)-^ in the upper half plane is a pole at 2:=t with
dx 3
3
residue there - ^^ i. Therefore
lb
/:
(x^ + lf 8
Example 2. If a > 0, 6 > 0, shew that
r°° x'^dx
• Example 3. By integrating \e~^^^dz round a parallelogram whose corners are
- R, R, R-\- ai, - R + ai a,nd making /? -* co , shew that, if X > 0, then
I e->^x''coH{2\nx)dx = e-'"^'- j e->^x- dx = 2X~i e->^<^'' j e-^^dx.
* Q (z) lias no poles above the real axis outside the contour.
6-221, 6-222]
THE THEORY. OF RESIDUES
115
6*221. Certain infinite integrals involving sines and cosines.
If Q (z) satisfies the conditions (i), (ii) and (iii) of § 6*22, and m > 0, then
Q (z) e"*" also satisfies those conditions.
Hence f {Q (x) e'^*' + Q (- x) e-^^] dx is equal to 27ril>R', where IR
Jo
means the sum of the residues of Q (z) e™*^ at its poles in the upper half plane;
and so
(i) If Q (x) is an even function, i.e. if Q(— x) = Q (x),
I Q (x) cos (thx) dx = TriXR'.
Jo
(ii) If Q (x) is an odd function,
Q (x) sin (mx) dx = tt^R'.
f
Jo
6*222. Jordan's lemma*.
The results of §6*221 are true if Q{z) be subject to the less stringent
condition Q{z)^0 uniformly when 0 ^ arg z^ tt as | ^^ | -» oo in place of the
condition zQ{z)--0 uniformly.
To prove this we require a theorem known as Jordan's lemma, viz.
If Q(z)-*-0 uniformly with regard to arg z as \z\^oo when 0 ^ arg z^ir,
and if Q (z) is analytic when both \ z\ > c {a constant) and 0 ^ arg z "^tt, then
lim ( e'^'^Q(z)dz)=0,
where T is a semicircle of radius p above the real axis with centre at the origin.
Given e, choose po so that | Q (^) | < e/ir when \z\ > p^ and 0 ^ arg z ■^tt]
then, if p > poj
/.
e^^''Q{z)dz
gmi(pcosfl+tpsin9) Q {pe'">) pe^« idd
But le^itpcosoi = 1^ and so
e'^i''Q{z)dz
L
< I {e/'Tr)pe-'^p^''^^d6
Jo
= (2e/7r) pe-'^'o^'^Hd.
Jo
Now sin 6 ^ lOjir, whenf 0 ^ 6 ^^ir, and so
/.
e'^^'^Q {z) dz
< (2e/7r) f pe-^^p^l^
Jo
dd
< {2e/'rr) . (7r/2m) - g-^'^pe/'r
0
< ejm.
* Jordan, Cours d' Analyse, t. ii. § 270.
t This inequality appears obvious when we draw the graphs y^sinx, y = 2xlw\ it may be
proved by shewing that (sin 6)16 decreases as 6 increases from 0 to ^tt.
« 0
116 THE PROCESSES OF ANALYSIS [CHAP. VI
Hence lim f e'^^Q(z)dz = 0.
p-
This result is Jordan's lemma.
Now
f " {e'»^Q (x) + e-'^^Q (- a;)] da; = 2'jriXR' - f e'^^Q (e) dz,
Jo •' r
and, making p-*' ao , we see at once that
r {e'^^^'Q (x) + e-'^^'^Q (- x)} dx = liritR',
Jo
which is the result corresponding to the result of § 6-221.
Example 1. Shew that, if a > 0, then
/:
cos a; , TT
ax= — -e"".
Example 2. Shew that, if a ^ 0, 6^0, then
/:
cos 2a^ - cos 26a; , ,, ,
2 ax= TT (o - a).
(Take a contour consisting of a large semicircle of radius p, a small semicircle of
radius 8, both having their centres at the origin, and the parts of the real axis joining their
ends ; then make p -♦- oo , 8 -»- 0.)
Example 3. Shew that, if 6 > 0, m ^ 0, then
j 0 {xU^^f ^^^ ^^'^^ ^ "^^W *^^^ -a^-mh (362 + «2)},
Example 4. Shew that, if ^ > 0, a > 0, then
' X sin ax
/,
q ,„ dx=hive'
0 A'-' + Z:^
Example 5. Shew that, if m ^ 0, a > 0, then
/
(Take the contour of example 2.)
Example 6. Shew that, if the real part of z be positive,
/ {e-i-e-t^)—=\ogz.
J i) t
[We have
-dt
lim /["^'^..["^-''^
-O.p^oc lyS ^ J Sz U J
{/rT*-/:?*}.
since i!-i e-< is analytic inside the quadrilateral whose corners are 8, 8z, pz, p.
«-*-0,p-
— lim
6*23, 6*24] THE THEORY OF RESIDUES 117
f pz
Now / t-^e-^dt-^O&s p-t-ao when B(t)>0; and
fsz , . r««
/ t-U-tdt=\ogz- j t-^{l-e-i)dt-^logz,
since <-i(l-e-')-*l as «-*0.]
6 '23. Principal voUues of integrals.
It was assumed in §§ 6'22, 6-221, 6-222 that the function Q{x) had no poles on the real
axis ; if the function has a finite number of simple poles on the real axis, we can obtain
theorems corresponding to those already obtained, except that the integrals are all principal
values (§ 4-5) and 2^ has to be replaced by 2R + ^2Rq, where 2^ means the sum of
the residues at the poles on the real axis. To obtain this result we see that, instead of
the former contour, we have to take as contour a circle of radius p and the portions of the
real axis joining the points
-p, a-8i; a+8i, b-82; b + 82, c-83, ...
and small semicircles above the real axis of radii 81, 82, ... with* centres a,b,c,..., where
a,b,c,... are the poles of Q{z) on the real axis; and then we have to make Si, 82 > ...-^0;
call these semicircles yi, y2> •••• Then instead of the equation
j ^ Q(z) dz+ / Q (z) dz = 2iri2R,
we get P I Q{z)dz + ^ lira [ Q{z)dz+ Q{z) dz=2iri2B.
Let a' be the residue of Q (z) at a ; then writing z=a + 8ieio on y^ we get
f Q{z)dz= rQ{a+8ieiO)8ieiHdd.
J yi J IT
But $(a+8ie»"^)Sie»*-^a' uniformly as Si-*-0; and therefore lim 1 Q{z)dz= —nia' ;
we thus get
P r Q(z)dz+j Q{z) dz=2Tri2R + niSRo,
and hence, using the arguments of § 6-22, we get
pf Q{x)dx=2viC2R+^2Ro).
J -QO
The reader will see at once that the theorems of §§ 6-221, 6-222 have precisely similar
generalisations. ^
The process employed above of inserting arcs of small circles so as to diminish the area
of the contour is called indenting the contour.
6"24. Evaluation of integrals of the form x^ ^Q {x) dx.
Jo
Let Q(x) be a rational function of x such that it has no poles on the
positive part of the real axis and af'Q{x)-»-0 both when x--0 and when
x-*- 00 .
118 THE PROCESSES OF ANALYSIS [CHAP. VI
ure
Consider 1 (— zf-^ Q (z) dz taken round the contour C shewn in the fig
consisting of the arcs of circles of radii
p, 8 and the straight lines joining their
end points ; (— z)"'~^ is to be interpreted
as
exp{(a-l)log(-^)}
and
log (- ^) = log j ^ i + * arg (- z),
where — tt ^ arg (— z) ^ tt ;
with these conventions the integrand is
one-valued and analytic on and within
the contour save at the poles of Q (z).
Hence if 'Zr denote the sum of the
residues of (— zY'^ Q (z) at all its poles,
[ (_ zf-i Q (z) dz = liritr.
J c
On the small circle write —z = 8e^*, and the integral along it becomes
— I {—zYQ{z)idd, which tends to zero as h-*-0.
■J n
On the large semicircle write — z = pe^^, and the integral along it becomes
— I (— zY Q (z) idd, which tends to zero as p -»• oo .
On one of the lines we write — z = x&'^, on the other — z-= xe~^^ and
(_^)a-i becomes a;«-^e* <«-!'''*.
Hence
lim \ {fl:;«-i6i-(«-i)'^Q {x) - a;«-ie<«-^"^*Q {x)\ dx = ^iritr ;
(S-*-0, p-*-oo) J 8
/•OO
and therefore 1 af-^Q{x)dx = '7r coBec{a'ir)%r.
Jo
Corollary. If Q (x) have a number of simple poles on the positive part
of the real axis, it may be shewn by indenting the contour that
P 1 x^-'^ Q (x) dx = 7r cosec (air) Xr — ir cot (a7r) Xr',
where tr' is the sum of the residues o{z^-^Q(z) at these poles.
Example 1. If 0 < a < 1,
I f—:^a.T=Tr cosec an, PI o?*- = tt cot ajr.
63, 6*31] THE THEORY OF RESIDUES 119
Example 2. If 0 < « < 1 aud — tt < a < jt,
/:
« f + e*" sin TTZ
Example 3. Shew that, if - 1 < a < 3, then
^»-l „gi(»-l)a
cfe = — ^. . (M inding. )
/:
^ ,<&="('-''
0 (1 +0^2)2 Acos^itz'
Example 4. Shew that, if - 1 < jo < 1 and - tt < X < tt, then
x~^dx _ TT sinjoX
y 0
(Elder.)
0 \+2x cos \-\-x^ sin /jtt sin X
6*3. Cauckifs integral.
We shall next discuss a class of contour-integrals which are sometimes found useful
in analytical investigations.
Let Cbe a contour in the 2-plane, and let /(«) be a function analytic inside and on C
Let 0 (2) be another function which is analytic inside and on C except at a finite number
of poles; let the zeros of ^(2) in the interior* of Cbe ai, a^, ..., and let their degrees of
multiplicity be rj, /•2, ... ; and let its poles in the interior of Cbe 61, h^, ..., and let their
degrees of multiplicity be «i, S2) ••••
Then, by the fundamental theorem of residues, - — . / fiz) \ // dz is equal to the sum
27nJ c-^ <t>{z)
of the residues of !;, - at its poles inside C.
(t>{z)
Now 1T\ ^^" have singularities only at the poles and zeros of </> (2). Near one
of the zeros, say aj , we have
(f){z)==A {z-ai)r, + B(z-ai)rL+l + ....
Therefore 0' (2) = Jrj (2-ai)»-i-i + 5(ri + l)(2-ai)n + ...,
and /(2)=/(«i)+(^-«i)/'(«i) + ....
Therefore \^^^ - '-^^^^l is analytic at a,.
\ (f>{z) z-ai j ^
Thus the residue of jjs , at the point z = ai, is ri/(ai).
Similarly the residue at 2 = 61 is —Sif{bi) ; for near z = bi, we have
0(2) = (7(2-6i)-».-|-Z)(2-5i)-»i + l-i-...,
and f{z)=f{W) + (z-b,)f'{br)+...,
so ^^'l f}'^ + '^ is analytic at b,.
(^(2) ^ 2-61 ^ ^ ^
Hence -^. f /(2) ^^ c^2 = 2/-i/(a,)-2«i/(fei),
2ni J c 9(2)
the summations being extended over all the zeros and poles of <^ (2).
6 "SI. The number of roots of an equation contained within a contour.
The result of the preceding paragraph can be at once applied to find how many roots of
an equation cf){z) = 0 lie within a contour C.
For, on putting f{z)=l in the preceding result, we obtain the result that
1 ( *-(^)^,
2TriJ C(f)iz)
is equal to the excess of the number of zeros over the number of poles of (f) (2) contained in
the interior of C, each pole and zero being reckoned according to its degree of multiplicity.
* <p(z) must not have any zeros or poles on C.
120 THE PROCESSES OF ANALYSIS [CHAP. VI
Example 1. Shew that a polynomial <^ (2) of degree m has m roots.
Let <^(2) = ao2'"+«i2'""^ + ••• + «>«, («o=f=0).
^) ^ maog"'-H...+«m-i
•^^^ </)(25 ao2'»+. .. + «,„ '
Consequently, for large values of [s],
(^ (2) 2 \z^J
Thus, if C be a circle of radius p whose centre is at the origin, we have
_L,f ■ma.^i * + J-.f of^)*,™^!^ ofi)*.
But, as in § 6-22, f 0 (^^ dz^O
as p-t-cc ; and hence as (f) (2) has no poles in the interior of C, the total number of
zeros of (f>{z) is
lim — — . / , , , az=m.
p^^ 2iri J c (j) (2)
Example 2. If at all points of a contour C the inequality
is satisfied, then the contour contains k roots of the equation
a,„2'" + a„j_i2"''-i + ..,+ai2 + ao = 0.
For write /(2) = a,„2'" + a„_i2'"-i + ...+ai2 + ao.
Then /(2) = a,2^ ^^^a^2»^ + ...-f %.,2^^^.t.^^_^,.-i + .„+^^
= a,2''(l + C^),
where | fj ^a < 1 on the contour, a being independent* of 2.
Therefore the number of i-oots of / (2) contained in C
_ J_ /" /lii) ^ _ _1_ /■ /^' , ^_ ^\ ^
~ 27ri J a fiz) 27riJcVl+UdzJ "*'■
f dz
But I — = 27ri; and, since | f/"! < 1, we can expand (1 + £/■)-! in the uniformly con-
vergent series
Therefore the number of roots contained in C is equal to k.
Example 3. Find how many roots of the equation
2« + 62+10 = 0
lie in each quadrant of the Argand diagram. (Clare, 1900.)
• I 17 1 is a coutinuous function of 2 on C, and so attains its upper bound (§ 3-62). Hence its
upper bound a must be less than 1.
6'4] THE THEORY OF RESIDUES 121
6*4. Connexion between the zeros of a function and the zeros of its derivate.
Macdonald* has shewn that if f{z) he a function of z analytic throughout the interior of
a single closed contour G, defined hy the equation \f{z)\ = M, where M is a constant, then the
number of zeros of f{z) in this region exceeds the number of zeros of the derived function
f'{z) in the same region hy unity.
On C let f{z) = JKei^; then at points on C
Hence, by § 6*31, the excess of the number of zeros of f{z) over the number of zeros
of f (z) inside t C is
_L [ IMrf. J- [ f-^d.- J- [ f^/'^\^.
2»ri J c f(z) " 27r^• j cf {z) 2W j c Wl dz) '*'•
Let s be the arc of C measured from a fixed point and let \/r be the angle the tangent to
C makes with Ox ; then
1 /• /rf2^ Id6\ _, 1 r d6^
-^ijc Kd^ldz) ^^= - 2^-rS dz\c
1 r, de , dz~\
d6
Now log -r- is purely real and its initial value is the same as its final value ; and
dz
log-j- = i-^; hence the excess of the number of zeros oi f{z) over the number of zeros of
/' (2) is the change in -^12^ in describing the curve C; and it is obvious J that if G is any
ordinary curve, yjr increases by 27r as the point of contact of the tangent describes the
curve G; this gives the required result.
Example 1. Deduce from Macdonald's result the theorem that a polynomial of degree
n has n zeros.
Example 2. Deduce from Macdonald's result that if a function f{z), analytic for real
values of 2, has all its coefficients real, and all its zeros real and different, then between
two consecutive zeros of f(z) there is one zero and one only of/' (2).
REFERENCES.
M. C. Jordan, Cours d'Analyse, t. 11. Chap. vi.
E. GrOURSAT, Gours d^ Analyse, Chap. xiv.
E. LiNDELOF, Le Calcid des Residus, Chap. il.
* Proc. London Math. Sac. xxix. (1898).
t /' (2) does not vanish on C unless G has a node or other singular point ; for, if f=(j> + i\p,
where </> and \p are real, since i J- = ^, it follows that if f'{z) = 0 at any point, then
^ ^ ox dy
^ -^ , J^ -J^ all vanish ; and these are sufficient conditions for a singular point on
ox dy ax ay
X A formal proof could be given, but it would be long and difficult.
122 THE PROCESSES OF ANALYSIS [CHAP. VI
Miscellaneous Examples.
1. A function </> {z) is zero when z=0, and is real when z is real, and is analytic when
\z\^\; if ■f{x,y)\s, the coefficient of i in 0 {x + iy), prove that if - 1< ^ < 1,
'2t ^sin^
/,
; / (cos 6, sin 6)dd = TT(f) (x).
0 l-2^cos^ + .r2
(Trinity, 1898.)
*-aia
' 2. By integrating -| round a contour formed by the rectangle whose corners are
0, R, R+i, i (the rectangle being indented at 0 and i) and making R-*-cc, shew that
/,
sin a^ , 16"+ 1 1 /T J \
dx= =- — ;;—• (Legendre.)
0 e^'^*-! 4e»-l 2a ^ ^ '
3. By integrating log {-2)Q (2) round the contour of § 6*24, where Q (z) is a rational
function such that zQ{z)-f~0 as |2|^-0 and as \z\-^ao, shew that if Q{z) has no poles
on the positive part of the real axis, I Q (x) dx is equal to the sum of the residues of
log {-z)Q (z) at the poles of Q (2) ; where the imaginary part of log ( - 2) lies between + it.
4, Shew that, if a > 0, 6 > 0,
/
gacosJia: gin (^^ gin 6^) — =|7r (e" - 1).
0 ^
5. Shew that
/
a sin 2.r , 1 , /, , \ / -, ^ ^ i\
,xdx = -T\og{\ + a), (-l<a<l)
0 1 -2a cos 207+ a^ 4
= l,rlog(l+a-i), (a2>l)
(Cauchy.)
6. Shew that
/sin d)ix sin (b„x s\\id>„x sm a^ , t , , ,
^-^— ... — cos aiX ... cos a„,x — dx= - q)\(p2 ... (pnt
Q X X X X 2^"^"^
if ^1, (^.2, ... <pn, oi, 02, ... a,rt be real and a be positive and
a> !0i| + |02l + --. + l<^n| + |ail + ... + |aml- (Stormer.)
7. If a point z describes a circle C of centre a, and if f{z) be analytic throughout
C and its interior except at a number of poles inside C, then the point u=f{z) will
describe a closed curve y in the w-plane. Shew that if to each element of y be attributed
a mass proportional to the corresponding element of C, the centre of gravity of y is the
fiz)
pomt r, where r is the sum of the residues of -^-^ at its poles in the interior of C.
z — a
(Amigues.)
8. Shew that
dx IT (2a + h)
i:
9. Shew that
dx TT 1 .3...(2?i-3) 1
/
0 {a + hx'^y^ 2"h^ l.2...(n-\) a'*"*'
THE THEORY OF RESIDUES 128
10. If Fn (^)= n n (1 - x^), shew that the series
/(^)=- 2
F^{xn-^)
is an analytic function when x is not a root of any of the equations ^=n" ; and that the
sum of the residues of f{x) contained in the ring-shaped space included between two
circles whose centres are at the origin, one having a small radius and the other having
a radius between n and »+ 1, is equal to the number of prime numbers less than w-f 1.
(Laurent.)
11. li A and B represent on the Argand diagram two given roots (real or imaginary)
of the equation/ (2;) = 0 of degree n, with real or imaginary coefficients, shew that there is
at least one root of the equation /' (^) = 0 within a circle whose centre is the middle point
o{ AB and whose radius is ^AB cot—. (Grace, Proc. Camb. Phil. Soc. xi.)
12. Shew that, if 0 < ./ < 1,
= - — ; lim 2
[Consider /
1 - e2«^ 2»ri „^« fc=_„ ^ - ^ *
■round a circle of radius n+j^ ; and make w-^oo .]
sm irz z — x
(Kronecker.)
13. Shew that, if m > 0, then
sin" mt
/.
' dt
„m^~i ( „ , n, „s„ , n(n — l), ,. , n(n-l)(n—2) , ., , "1
= 2M^^=^!r l^"""^^ + 2! ("-^) ^^3! ^(^-6)"-^ + ...}
Discuss the discontinuity of the integral at wi = 0.
14. If A + B + C+...=0 and a, b, c, ... are positive, shew that
A cos ax + B cos bx+ ... + Kcoskx
I
dx= —A loga — 51og 6— ... — Klogk.
0 *^
(Wolstenholme.)
15. By considering I . . dt taken round a rectangle indented at the origin, shew
that, if y?; > 0,
/p gz (*; + M) r p gxti
~f — --rdt = 7ri+ lim Pi — dt,
— p IC-\-tl a-*-oo J -p ^
and thence deduce, by using the contour of § 6-222 example 2, or its reflexion in the real
axis (according SiS x^O or x <0), that
1 fp e^(* + «) ',
hm - , , dt = 2, 1 orO,
a^oo «• J -p K + tl
according as a; > 0, .r=0 or .r < 0.
[This integral is known as Cauchy's discontinuous factor.']
16. Shew that, if 0 < a < 2, 6 > 0, r > 0, then
Jo ' x^-Vr^ ~
124 THE PROCESSES OF ANALYSIS [CHAP. VI
- 17. Let <>0 and let 2 e-n^t=y\t{t).
n=—co
By considering / -r— ; dz round a rectangle whose comers are ± {N-\-\)±i, where
N is an integer, and making iV-»-oo , shew that
By expanding these integrands in powers of e'^''^^ e^'"*'^ respectively and integrating
term-by-term, deduce from § 6*22 example 3 that
(irtr J -=0
Hence, by putting t = l shew that
V.(0=«-*^(i/0.
[This result is due to Jacobi, Ges. Werke, ii. p. 188.]
18. Shew that, if «>0,
jl + 2 2 e " "'''/^ cos 2w7ral .
(Jacobi ; and Landsberg, Crelle, cxi.)
2 e-«'T<-2»T««^^-ig,ra2<^j_j.2 2 g-
n=-(x>
CHAPTER VII
THE EXPANSION OF FUNCTIONS IN INFINITE SERIES
7"1. A formula due to Darhoux*.
Let f{z) be analytic at all points of the straight line joining a to z, and
let <^ {t) be any polynomial of degree n in t.
Then if 0 $ < ^ 1, we have by differentiation
I- 2 (-)"» {z - a)'" c/)'*^-"** (0/*"^' {a + t{z- a))
= -{z-a) <^<«» (t)f {a-\-t(z- a)) + (-f {z - ay+' </> (0/<"+^* {a + t(z- a)).
Noting that 0'''' (t) is constant = ^<™' (0), and integrating between the
limits 0 and 1 of t, we get
</>"^'(0){/(^)-/(a)}
= i (-)'""' (^ - «)"* {<!)<''""" (^)/'"'* (^) - </>*""'"* (0)/""' (a)]
m=l
+ (-)»» (z - ay^^' I <b (t)p'+'^ (a + t(z- a)) dt,
Jo
which is the formula in question.
Taylor's series may be obtained as a special case of this by writing
(^ (ty = {t — 1)" and making n^ oo .
Example. By substituting 2n for n in the fonnula of Darboux, and taking 0 (t) = t" {t — l )",
obtain the expansion (supposed convergent)
f{z)-f(a)= i ^ ~ ^"." ',f r "^" {/"" (^) + ( - )" - v<"> (^)},
and find the expression for the remainder after « terms in this series.
* Liouville's Journal (3), ii. (1876), p. 271.
126 THE PROCESSES OF ANALYSIS [CHAP. VII
7'2. The Bemoullian numbers and the Bernoullian polynomials.
The function \zcot^z is analytic when \z\<2'rr, and since it is an even
function of z it can be expanded into a Taylor's series thus
2^ cot 2 ^ = 1 - 5i 2j - -Sa ^j - 5s gj - ... ;
then Bn is called the nth Bernoullian number*. It is found thatf
p 1 T> L 7? L R 7? A
-Di — 6 ' -"2 — 30 ' -03 — 42 ' -"4 30 ' ^6 66 '
These numbers can be expressed as definite integrals as follows:
We have, by example 2 (p. 122) of Chapter vi,
sin pxdx 1 i , ■
- 2p^ 2p r^^' 2! ^-^ 4! +•••[•
1 \
niTj
dx
f
Jo
Since
x'^ sin [px + -E nir j
e^oc_i
converges uniformly (by de la Vallee Poussin's test) near p = 0 we may, by
§ 4*44 corollary, differentiate both sides of this equation any number of
times and then put p = 0 ; doing so and writing 2t for x, we obtain
Bn = ^n -—. — -
Jo e^'^'-l
A proof of this result, depending on contour integration, is given by Carda, Monatshefte
fur Math, und Phys. v. (1894), PP- 321-4.
Example. Shew that
" 7r2»(22»-l) jo sinh^ -^^■
gZt _ 1
Now consider the function t i>— jy, which may be expanded into a
Maclaurin series in powers of t valid when | ^ | < 27r.
The Bernoullian polynomialX of order n is defined to be the coefficient of
— in this expansion. It is denoted by <^„ {z), so that
n = l
* These numbers were introduced by Jakob Bernoulli in his Ars Conjectandi.
t Tables of the first sixty-two Bernoullian numbers have been given by Adams, Brit. Ass.
Rep. 1877.
X The name was given by llaabe, Journal filr Math. Bd. xlii. (1851), p. 348.
'<f-^=^r^r w-
7-2, 7*21] THE EXPANSION OF FUNCTIONS IN INFINITE SERIES 127
This polynomial possesses several important properties. Writing (^ + 1)
for z in the preceding equation and subtracting the two results, we have
te^= i {<^„(^ + i)-<^„(^)j^,.
n=l '''•
On equating coefficients of V^ on both sides of this equation we obtain
which is a difference-equation satisfied by the function <^„ {£).
An explicit expression for the Bemoullian poljoiomials can be obtained
as follows. We have
^e_i = ,^ + _ + _ + ...^
J t t t t
~ 2 "*" 2! 4! "*"
Hence
From this, by equating coefficients of P^ (§ 3"73), we have
the last term being that in z or z^ and JJ^, nPi, ... being the binomial
coefficients ; this is the Maclaurin series for the /ith Bernoullian polynomial.
Example. Shew that, when «. > 1,
7'21. The Euler- Maclaurin expansion.
In the formula of Darboux (§ 7"1) write ^n(0 for ^ (0' where ^n{t) is the
nth Bernoullian polynomial.
Differentiating the equation
n — k times, we have
<^n'"-*> {t + l)- <^„'"-*> {t) = n{n-\)... kt^-\
Putting ^ = 0 in this, we have </)^<"-*' (1) = (/)n <"'"*' (0).
Now, from the Maclaurin series for ^„ (z), we have if A: > 0
^^<n-.fc-i) (0) ^ 0, (/)J--^) (0) = ^-^ {~f-'Bk,
128 THE PROCESSES OF ANALYSIS [CHAP. VII
Substituting the values of <^n'""*' (1) and <^„<»-*) (0) thus obtained in
Darboux's result, we find what is usually* known as the Euler-Maclaurin
sum formula,
(, _ a)f'{a) = f{z)-f{a) - ^ {f'{z) -/'(«)}
m=l (2w) !
-^^-J^jy^^(^^f'^'^'' {c^ + (^-<^)t]dt.
In certain cases the last term tends to zero as n-*oc , and we can thus
obtain an infinite series for f(z) — f{a).
Writing tu for ^ — a and (p {x) for /' {cc), the last formula becomes
^{x)dx = 2(^[4> («) + <^ (a + «)}
J a
Writing a + co, a + 2&), ... a + (r- — 1) o) for a in this result and adding up,
we get
ra+ru) [j 1 , .1
(f) (x) dx = a}U<f> (a) + (j) {a + co) + (f) (a+ 2o}) + . . . + ~2(f> (» + rcon
where 7?,^ - - r^^--. ^,n (t) 2 (^<^*^> (ci + moj + o)^)^ dt.
{Zn)iJo („j^o J
This last formula is of the utmost importance in connexion with the
numerical evaluation of definite integrals and also in the higher theory of the
Gamma function. It is valid if (f) (x) is analytic at all points of the straight
line joining a to a + rw.
Example 1. If f{z) be an odd function of z, shew that
where (f)^ (t) in the Bernoullian polynomial of order n.
* A history of the formula is given by Barnes, Froc. London Math. Soc. Ser. 2, Vol. iii.
p. 253. It was discovered by Euler (1732), and rediscovered by Maclaurin (1742).
7-3] THE EXPANSION OF FUNCTIONS IN INFINITE SERIES 129
Example 2. Shew, by integrating by parts, that the remainder after n terms of the
expansion of ^ 2 cot ^ z may be written in the form
(2»)!8m« /o^
(Math. Trip. 1904.)
73. Burmann's theorem'*.
We shall next consider several theorems which have for their object the
expansion of one function in powers of another function.
Let <^ {z) be a function of z which is analytic in a closed region *Sf of which
a is an interior point ; and let
<^{a)=h.
Suppose also that <^' (a) =|= 0. Then Taylor's theorem furnishes the
expansion
<f>(z)-b = <(>' (a)(z -a) + ^^^^^ (z-ay+ ...,
and if it is legitimate to revert this series we obtain
which expresses z as an analytic function of the variable {(f)(z) — b}, for
sufficiently small values of j^; — a|. If then /(2^) be analytic near z = a, it
follows that y* (2') is an analytic function of {(^ (z) — b} when | z — a | is sufficiently
small, and so there will be an expansion of the form
f{z)=f(a)+a,{<i>(z)-b]+^^{cp(z)-bY + ^^{<f>(z)-bY+....
The actual coefficients in the expansion are given by the following
theorem, which is generally known as Burmann's theorem.
Let i/r (z) be a function of z defined by the equation
z -a
then an analytic function f(z) can, in a certain domain of values of z, be
expanded in the form
»*"! [ck(z\ —bV"' d"'^~^
f(z) =f(a) + S ^^^{^ '^ '-^_, [f (a) [f (a.)}-] + R^,
m = l
where Rn = 7: — . I
m\ do''
cf>{z)-b'
:^(t)-b_
-' f (t) (f>' (z) dtdz
c}>{t)-4>(z) '
and 7 is a contour in the t-jdane, enclosing the points a and z and such that, if
^ be any point inside it, the equation (f) {t) = (f> (^) has no roots on or inside the
contour except j" a simple 7'oot t = ^.
* Memoires de Vhistitut, ii. p. 13.
t It is assumed that such a contour can be chosen if j 2 - a | be sufficiently small ; see § 7'iil.
W. M. A. 9
130
THE PROCESSES OF ANALYSIS
[chap. VII
To prove this, we have
2'7ri]aJ
-2 (^(Q-h
+
But, by § 4-3,
mt)-h]--^[^{t)-4>{t;)]\
^-rrl JaJy L^
^/'(Of (0^^t^r_{<^(^)-6}-+^ [ f'(t)dt
<f) {t) -h\ 4>{t)-h
m+l r
1) i.
27ri(m + l) J^ [</>(0-&}
m+l
~ ^Triini + 1) J, (^ - cir+^ {m + 1) ! cZa^ K ^''^ ^^ ^ ^^ -■•
Therefore, writing m — 1 for m,
m=n-\ \fk(2:\ — h]^^^ (Im-i
+
m!
1 '
2^'
l4>{t)-b]
-^f(t)cf>'(Odtd^
</>(^)-«/'(0 '
If the last integral tends to zero as n -*- oo , we may write the right-hand
side of this equation as an infinite series.
Example 1. Prove that
where
^a+ 2
(-)""^(?«(2-a)"e"(^'-«^)
C,i = (2na)" 1 ^ ^^,^ {2na)"^-\ ^ — — g,-- — ~ (2«a)"'^ —
1!
2!
To obtain this expansion, write
in the above expression of Burmann's theorem ; we thus have
But
z=a+ 2 — 1(2- «)» e" ("' - «') ^ ~, 7 e" C' " ^')
\d;2"-i
g« (a2 — 32
g-)i(2a( + t3)
(putting 2 = a + 0
= (7i- 1) I X the coefficient of <"~^ in the expansion of e~"*(^"'^')
= (« - 1) ! X tlie coefficient of ^""^ in 2
^ ^- ,=oCH-l-r)I(2r-7i + l)!-
(-)'-?i'V(2a + 0''
The highest value of r which gives a term in the summation is r = 7i-l. Arranging
therefore the summation in descending indices /•, beginning with r = ?i — 1, we have
■(-)"-'£■.,
which gives the required resuU.
7*31] THE EXPANSION OF FUNCTIONS IN INFINITE SERIES 131
Example 2. Obtain the expansion
„ . o 2 1.. 2.4 1 . „
Example 3. Let a line p be drawn through the origin in the 2-plane, perpendicular to
the line which joins the origin to any point a. If z be any point on the 2-plane which is
on the same side of the line p as the point a is, shew that
log2=loga + 2 2 ^ — -[ )
* ^ ™=i 2»i + l \z + aj
7"31. Teixeira's extended form of Burmann's theorem.
In the last section we have not investigated closely the conditions of
convergence of Burmann's series, for the reason that a much more general
form of the theorem will next be stated ; this generalisation bears the same
relation to the theorem just given that Laurent's theorem bears to Taylor's
theorem : viz., in the last paragraph we were concerned only with the
expansion of a function in positive powers of another function, whereas we
shall now discuss the expansion of a function in positive and negative powers
of the second function.
The general statement of the theorem is due to Teixeira*, whose exposi-
tion we shall follow in this section.
Suppose (i) that f{z) is a function of z analytic in a ring-shaped region A ,
bounded by an outer curve G and an inner curve c ; (ii) that 6 {z) is a function
analytic on and inside C\ and has only one zero a within this contour, the zero
being a simple one ; (iii) that ^ is a given point within A ; (iv) that for all
points z of G we have
and for all points ^ of c we have
\d(x)\>\d(z)\.
The equation 6 (z) - 6 (x) = 0
has, in this case, a single root z = x in the interior of G, as is seen from the
equation f
1 C d'{z)dz _ 1 \l ^'(^) ^ a, ^ f 0'(^) ^
27rilc0(z)-d(x)-27ri J c ^ (^) "^ ^"''M c I^MF "^ ' '
c0{z)-d(x) 27rilJceiz) "'ici^W
0' (z) dz
2inir
of which the left-hand and right-hand members represent respectively the
number of roots of the equation considered (§ 6'31) and the number of the
roots of the equation 0{z)={) contained within G.
* CreUe's Journal, cxxii. (1900), pp. 97-123.
°° id Ix)] "
t The expansion is justified by § 4*7, since S L ,Tf converges uniformly when z is on C.
n=i I'' \^} )
9—2
132 THE PROCESSES OF ANALYSIS [CHAP. VII
Cauchy's theorem therefore gives
1
27ri
/(^) = 9i:;-
c f{z)e'{z)dz rf(z)e'(z)dz
i.d{z)-d(x) Je
\.d{z)-e{x) },e{z)-e{x)
The integrals in this formula can, as in Laurent's theorem, be expanded
in powers of 6 {x), by the formulae
/• f{z)e'{z)dz_ - .. ( f{z)d'{z)dz
We thus have the formula
where
Integrating by parts we get, if ?i =i= 0,
This gives a development of f{x) in positive and negative powers of
6 {x), valid for all points x within the ring-shaped space A.
If the zeros and poles of /(2^) and 6 {z) inside G are known, An and Bn can
be evaluated by § 5"22 or by § 6"1.
Example 1. Shew that, if |.t'| < 1, then
_1 ( ^x \ 1 / '2x Y 1.3 / 2^ Y
^~2 Vi+^V """2.4 vT+^v ■*■ 27476 Vi+^^V "^ ■■■■
Shew that, when \x\> 1, the second member represents x~'^.
Example 2. If ^v* denote the sum of all combinations of the numbers
22, 42, 62,...(2n-2)2,
taken m together, shew that
i=j_+iizL):i:ij_i ^(Lh)+ +tZ::^)i(3in.)^»-
2 sinz „=o(27i + 2)! (271 + 3 2?i+l 3 j^ ^ '
the expansion being valid for all values of z represented by points within the oval whose
equation is |sin 2! = 1 and which contains the point z = Q. (Teixeira.)
7"32. Lagrange s tJieorem.
Suppose now that the function f{z) of §7'31 is analytic at all points in
the interior of G, and let 6 (x) = (x — a) 6^ {x). Then 6^ {x) is analytic and
not zero on or inside G and the contour c can be dispensed with ; therefore
the formulae which give A^ and B^ now become, by § 5*22 and § 6"1,
"■ 277^^ Jciz- af [6, (^)P n ! da^-' V," (a)| ^ ^ ^'
_ 1 i f{z)d'{z) dz
B, = 0.
7 -32] THE EXPANSION OP FUNCTIONS IN INFINITE SERIES 133
The theorem of the last section accordingly takes the following form, if
we write 0^ {z)= 1/^ {z)'-
Let f(z) and <j> (z) be functions of z analytic on and inside a contour C
surrounding a point a, and if the such that the inequality
\t<f>{z)\<\z-a\
is satisfied at all points z on the perimeter of C ; then the equation
e=a+<</,(r),
regarded as an equation in ^, has one root in the interior of G ; and further
any function of ^ analytic on and inside G can be expanded as a power series
in t by the formula
This result was published by Lagrange* in 1770.
Example 1. Within the contour surrounding a defined by the inequality 1 2 (z - a) | > \a\,
where |a| < ^|a|, the equation
z—a = 0
z
has one root {", the expansion of which is given by Lagrange's theorem in the form
Now, from the elementary theory of quadratic equations, we know that the equation
z
has two roots, namely 9ll + \/l + "^r ^'^^ 9 l^~\/^'''~^l ' ^"^^ ^^'^ expansion re-
presents the former ■^ of these only — an example of the need for care in the discussion of
these series.
Example 2. If y be that one of the roots of the equation
3/= 1+2/
which tends to 1 when 2-»-0, shew that
, n(ii + b){n + %){n + 1) ^ , n{n + 6){7i + 7)(n + 8)(n + 9)
+ 4! z +~ gj - 2' +...
so long as \z\ < |-.
Example 3. If x be that one of the roots of the equation
x=l +yx^
which tends to 1 when y^-0, shew that
, 2a
logx=y+—^
the expansion being valid so long as
, 2a- 1 , (3a -1) (3a -2) ,
log:r=y+-^-/ + ^ 2 73 ^y'+-,
\y\ < |(a-l)«-ia-«|. (McClintock.)
* Mem. de VAcad. de Berlin, t. xxiv. ; Oeuvres, t. 11. p. 25.
t The latter is outside the given contour.
134 THE PROCESSES OF ANALYSIS [CHAP. VII
7"4. The expansion of a class of functions in rational fractions*.
Consider a function f{z), whose only singularities in the finite part
of the plane are simple poles a^, a^, as, ..., where [ Oi | ^ | aj | ^ | ag | ^ ... : let
61, 62. h, ••• he the residues at these poles, and let it be possible to choose a
sequence of circles C,^ (the radius of C^ being Rm) with centre at 0, not
passing through any poles, such that |/(^) | is bounded on C^. (The function
cosec z may be cited as an example of the class of functions considered, and
we take R^ = (m + ^) tt.) Suppose further that R^^ -* 00 as m ^ qo and that
the upper bound f of f(z) on C„^ is itself bounded as:|: m-* 00 ; so that for all
points on the circle G^, | f(z) | < M, where M is independent of m.
Then, if x be not a pole of f(z), since the only poles of the integrand are
the poles of f{z) and the point 2 = x, we have, by § 6'1,
^irl J C^Z — X -"^ r dr — X
where the summation extends over all poles in the interior of C^.
But J-.f /(i^rf.= If /(i^f^ + JLf /«,d.
'iTTlJC^Z — X ^TTl J C^ Z 2-771 J C^Z (Z - X)
= /•(0)+ S ^ + — [ fi^i
7 «>• ^"rri J c^ z(z — x)'
if we suppose the function f(z) to be analytic at the origin.
Now as m ^ c» , -^ , is 0 (R~^), and so tends to zero as m tends
JC„,2{Z-X)
to infinity.
Therefore, making m^ ao , we have
0 =f(x) -/(O) +Xbn( - ~- - - ) - lim ^. f IMA^ ,
„ = 1 \an-^ aJ m^oo^TTl JC^Z{Z-X)'
which is an expansion of/(^) in rational fractions of x; and the summation
extends over all the poles of f(x).
If |%|<|a„+i| this series converges uniformly throughout the region given by
|.r|^a, where a is any constant (except near the points a^). For if R,^ be the radius
of the circle which encloses the points \ai\, ... |«,i|, the modulus of the remainder of the
terms of the series after the fii-st n is
f{z)dz
Ma
IL, - a '
I 2nt J (,',„ z{z — x)
by § 4-62 ; and, given f, we can choose ?i independent of x such that Maj(R„,-a) < e.
* Mittag-Leffler, Acta Soc. Fennicae, Vol. xi. p. 273.
t Which is a function of m.
t Of course R„ need not (and frequently must not) tend to infinity continuously; e.g. in the
example taken R„^={m + ^) ir, where m assumes only integer values.
ex
j c z-x
1 1 . ^ . ^p+'
z
7-4] . THE EXPANSION OF FUNCTIONS IN INFINITE SERIES 185
The convergence is obviously still uniform even if I«,ii <|an + i! provided the terms of
the series are grouped so as to combine the terms corresponding to poles of equal moduli.
If instead of the condition \f{z) | < J/" we have the condition | i~^f(z) I < J/", where M is
independent of m when z is on (7^, and jo is a positive integer, then we should have to
pand I —^—r fjy writnig
and should obtain a similar but somewhat more complicated expansion.
Example 1. Prove that
1 / s / 1- 1 \
cosec2 = - + 2(-)" - + — ,
Z \2 - TiTT nir)
the summation extending to all positive and negative values of n.
To obtain this result, let cosec * — =fi?)- The singularities of this function are at the
points z=niT, where n is any positive or negative integer.
The residue of f(z) at the singularity nn is therefore (- )", and the reader will easily
see that \f{z)\ is bounded on the circle \z\ = {7i+^)Tr as n-^x> .
Applying now the general theorem
f{z)=f{0) + 2cJ:r^ + y\,
|_« - a^ «„ J
where c„ is the residue at the singularity «„, we have
/(.)=/(0) + .(-).^^ + J-},
But ■' ■■ /(0)= lim— !"— ^ = 0.
■^ ' ' ^^0 ^ sin s
Therefore cosec s = - + 2 ( - )'M -\ ,
z \_z — 7i7r nirj
which is the required result.
Example 2. If 0 < a < I, shew that
go? ^i ^22 cos 2na7r - Aim sin 2nair
Example 3. Prove that
1 1112 1
+
27r^^(cosh;i?-cos A') 27r.r^ t"^ — c-'^ tt^ + ^a'* e^n-- e-2n- (27r)4-f^.r*
_ _^ 1
The general term of the series on the right is
{ev^-e-rn){(^r7rf + \x*y
which is the residue at each of the four singularities r, — ;•, ri, —ri of the function
(7r*2* + |.r-*) {eirz — f- irz) sin ttZ '
136 THE PROCESSES OF ANALYSIS [CHAP. VII
The singularities of this latter function which are not of the type r, - r, ri, - ri are
at the five points
0 -^^^ — ~ ' .
2
At 2=0 the residue is — -i ;
at each of the four points ^= ^Z ' *^^ residue is
{inx^ (cos X — cosh x)] ~ ^ .
Therefore
oc (-!)'•/• 1 2 2
^^^ grTT _ e - j-TT (,.^)4 + 1 X* 77^* 77^2 (cosh X - cos *■)
1 . C TTzdz
^ 2^1 }!^i J c (77*02 _ia,-*)(e-^-e-T^) sin ,72 '
where C is the circle whose radius is n+-, {n an integer), and whose centre is the origin.
But, at points on C, this integrand is 0 (j^l-^) ; the limit of the integral round Cis there-
fore zero.
From the last equation the required result is now obvious.
Example 4. Prove that sec ^-=477 (^^^34^., - ^^2 '^4^2 + ^r^J_ 4^2 -•••)•
Example 5. Prove that cosech x = - - 2,r (^^^ + ^72 " 4^2 ^.^.2 + 9,^2+^2 -•••)•
Example 6. Prove that sech ^.=477 {~_-^~i - g^^^^^ + 25772 + 4^.2 " -) '
Example 7. Prove that coth -'^= - +2.*' ( 2 , ,2 + . 2_i. 2 + q 2T .2 + ••• ) •
X \ 77" -J- tt' ^TT ~x' X tjTT 'J' X J
cc X "I 2
Example 8. Prove that 2 2 7--^ ... , .. — y-k = ^ coth 77a coth 776.
(Math. Trip. 1899.)
7'5. TA-e expansion of a class of functions as infinite products.
The theorem of the hist article can be applied to the expansion of
a certain class of functions as infinite products.
For let f{z) be a function which has simple zeros at the points*
tt), a.,, a-i, ..., where lim a,^ j is infinite ; and let f(z) be analytic for all values
of Z.
Then/'(2') is analytic for all values of z (§ 5"22), and so • . can have
singularities only at the points ctj, a.,, a-^, ....
Consequently, by Taylor's theorem,
f{z) = {z- a,)f' (a,) + t-.«zl'/" (,,^.) + . . .
and f'{z)= f {a,) + (^ - a,)/" {a,) + . . . .
* These being the only zeros of/ (2); and«,, \:0.
7*5, 7*6] THE EXPANSION OF FUNCTIONS IN INFINITE SERIES 137
f (z)
It follows immediately that at each of the points a^, the function "^ , .
has a simple pole, with residue + 1.
If then we can find a sequence of circles C,„ of the nature described in
%. 7 '4, such that -^r-!. is bounded on C,„ as m -- cc , it follows, from the
expansion given in § 7 '4, that
f(^) /(O) ,1=1 [2 -an a„]
Since this series converges uniformly when the terms are suitably grouped
(§ 7*4), we may integrate term-by-term (§ 4*7 ). Doing so, and taking the
exponential of each side, we get
/W = o./.». n{(i-£).».|,
where c is independent of z.
Putting z =0, we see that /(O) = c, and thus the general result becomes
/(^)=/(0)./(") n (l-^je
This furnishes the expansion, in the form of an infinite product, of any
function f(z) which fulfils the conditions stated.
Example 1. Consider the function f{z) = , which has simple zeros at the points
z
Tit, where r is any positive or negative integer.
In this case we have /(O) = 1, /' (0) =0,
and so the theorem gives immediately
sin 2 =" (A z\ -] r/ z\ -—1
z n=i {\ rnvj ) W nnj J
f (z)
for it is easily seen that the condition concerning the behaviour of •-~r- as |^]-*-X! is
fulfilled.
Example 2. Prove that
{-m hG.-^.)} {-G.U} l-GM \^<J^\
_ cosh k — cos .r
1 — cos X
(Trinity, 1899.)
7*6. The factor theorem of Weierstrass*.
The theorem of § 7*5 is very similar to a more general theorem in which
the character of the function /(^•), as | ^^ | -* oo , is not so nan-owl}- restricted.
* Math. Werke, Bd. ii. pp. 77-124.
138
THE PROCESSES OF ANALYSIS
[chap. VII
Let f{z) be a function of z with no essential singularities (except at ' the
point infinity'); and let the zeros and poles oi f{z) be at (h, a^, a^, ..., where
0 < I tti I ^ I tta ] ^ I as I . . . . Let the zero* at an be of (integer) order m„.
If the number of zeros and poles is unlimited, it is necessary that
! a„ I ^ 00 , as n^ oo; for if not, the points a„ would have a limit pointf,
which would be an essential singularity of f(z).
We proceed to shew first of all that it is possible to find polynomials
(jf,i (z) such that
n
1 _ Jl ) efl-nC^)
a
converges for rWI finite values of z.
Let K be any constant, and let \z\< K; then, since \an\^^ , we can
find N such that, when n > iV, | a„ | > 2K.
The first iV factors of the product do not affect its convergence:]: ; consider
any value of n> N, and let
^"(") = a: + 5t'+- +
k„ — l\a
kn-l
Then
m = k
1 /zy
Z
smce |2^rt,r':<2
Hence
< 2 I {Ka-'f- I ,
(1 )e3,J'n =e"-<^>
V aj 1
u,,{z)\^'2\m„{Ka-^fn\,
Now ?/i„ and a,^ are given, but k,^ is at our disposal ;"[since Ka^"'^ < 1, we
choose k„ to be the smallest number such that 2 | m„ (^a,,~^)*=» | < 6,^ where
S h,^ is any convergent series§ of positive terms.
M=l
Hence
n
g<7„(2)
n=.Y+l
where [ w,^ {z) \<h„; and therefore, since 6,^ is independent of z, the product
converges absolutely and uniformly when \z\< K, except near the points a„.
* We here regard a pole as being a zero of negative order.
t From the two-dimensional analogue of § 2-21.
:;: Provided that z is not at one of the points «„ for which in^^ is negative.
§ E.g. we might take ?>„=2-".
7-7] THE EXPANSION OF FUNCTIONS IN INFINITE SERIES 139
Now let F{z)= n
n = l
Then \i f{z)^ F{z) = Oi{z), Gi{z) is an integral function (§ 5-64) of z
and has no zeros.
It follows that ri / \ j^^it-^) is analytic for all finite values of z; and
(jt^yZj CiZ
so, by Taylor's theorem, this function can be expressed as a series 2 n6„2"~^
converging everywhere ; integrating, it follows that
00
where G{z)= 2 bnZ"- and c is a constant ; this series converges everywhere,
and so G (z) is an integral function.
Therefore, finally,
f(z)=:fiO)eOi^) U^ r|(i _-i)e^„(^)|"''
where G (z) is some integral function such that G (0) = 0.
[Note. The presence of the arbitrary element G(z) which occurs in this formula for
f{z) is due to the lack of conditions as to the behaviour of /(z) as | z | -»- oc .]
Corollary. If m,i==l, it is sufficient to take kn=n, by § 2'36.
7'7. The expansion of a class of periodic functions in a, series of
cotangents.
Let f{z) be a periodic function of z, analytic except at a certain number
of simple poles ; for convenience, let tt be the period of f{z) so that
f{z)=f{z^'rr).
Let z = x-\-iy and let f{z) -* I uniformly with respect to x as y ^ + oo ,
when O^x^ir; similarly let f{z) -* I' uniformly as y ^ — x .
Let the poles oi f{z) in the strip 0 < ^ ^ tt be at ttj, a.2, ... a„ ; and let the
residues at them be Cj, c^, ... c,^.
Further, let ABGD be a rectangle whose corners are* —ip, ir — ip,
TT+ip' and ip' in order.
Consider -— . I f (t) cot (t - z) dt
taken round this rectangle ; the residue of the integrand at a^ is c,. cot (a^. — z),
and the residue at z is f(z).
Also the integrals along DA and CB cancel on account of the periodicity
of the integrand ; and as p^cc , the integi-and on AB tends uniformly to I'i,
while as p' ^ GO the integrand on CD tends uniformly to — li ; therefore
2 (l' — I) =f{z) + 2 Cy cot («,. — z).
)• = !
* If any of the poles are on x = ir, shift the rectangle slightly to the right ; p, p' are to be
taken so large that aj, ao, ...a^ are inside the rectangle.
140 THE PROCESSES OF ANALYSIS [CHAP. VII
That is to say, we have the expansion
1 "
f{z) = 2 {V - ^) + S Cr cot {Z — ttr).
r=l
Example 1.
n ,
cot {x — ai) cot (.r - a2) • • • cot {x — a„) = 2 ' cot (a,. — a^) ...*.. . cot (a^ — a„) cot (;r — a,.) + ( - )3",
r=l
or = 2' cot (a,. — CTj) ...*... cot (a^- <^n) cot {x — a^,
according as n is even or odd ; the * means that the factor cot (a,. - a^) is omitted from the
product.
Example 2. Prove that
sin {x-hi) sin (^ — 62) • • • sin (,-r — hn) _ sin («! -&i) ... sin (at - 6,^)
sin (.r — tti) sin {x-a^j ... sin (^ - «„) sin (aj — 02) • • • sin (aj — a„)
sin (a.2 — 6,) . . . sin (a2 — b„)
sin («2 — «i) ••• sin (a2 — an)
cot (.r — ai)
cot (.r - a2)
+ .
+ cos{ai + a2 + ...+an-bi-b2-...-bn).
7'8. Borel's integral^.
Let/(0) = S a^s^** be analytic when \z\^r, so that, by § 5"23, | a„?'" j < if,
re=0
where M is independent of n.
00 yn, ^71
Hence if <p{z) = X -^^ , </> (2^) is an integral function, and
and similarly | <^<"' (z) \ < ilfel^l/'Y?'".
/•CO
Now consider /i(^)= I e~^ (f) {zt) dt ; this integral is an analytic function
of z when \z\ <r, by § 5%32.
Also, on integrating by parts.
M^) =
-e-f(p(zt)
+ z e-*<^'{zt)dt
= t z"
m = 0
e-t^m (-^^)
+ ^'^+1 I e-*(li^''+'^(zt)dt.
0 J 0
But liin e-«<^('«) (^i) = a,,^ ; and, when \z\<r, lim e-«<^<'«) (t) = 0.
Therefore
/i(2)= 2 «,«2'" + i2„,
t LeQons sur les series divcrgcntes, Ch. iii. See also the memoirs cited on p. 94 of that
work.
7-8, 7*81] THE EXPANSION OF FUNCTIONS IN INFINITE SERIES
where I i2„ k | ^«+M f e"' . Me l^HIrr-^-^ dt
Jo
< I zr-^ i^+^ilf {1 - I ^ ! r-^}-^ -► 0, as w -«- (30 .
Consequently when \z\< r, '
Mz)= 2 arr^z-^^fiz);
141
>»=0
and so
f{z)=re-'<f>{zt)dt,
Jo
00 fji vU oo
where <f>(z)= 2 ^-^ ; <f> (z) is called BoreVs function associated with % ttnZ^.
M=0 U I n=0
If >S'= 2 a„ and (}>{z)= 2 -^ and if we can establish the relation S= I e"'d> (t) dt,
the series *S' is said (§ 8"41) to be '• summahle (BY; so that the theorem just proved
shews that a Taylor's series representing an analytic function is summable (B).
7'81. BoreVs integral* and analytic continuation.
It can now be shewn that Borel's integral represents an analytic function in a more
extended region than the interior of the circle |2|=r.
This extended region is obtained as follows : take the singularities a, h, c, ... oi f{z) and
through each of them draw a line perpendicular to the line joining that singularity to the
origin. The lines so drawn will divide the plane into regions of which one is a polygon
with the origin inside it.
Then BoreVs integral represents an analytic function (which, by § 5 "5 and § 7 "8, is
obviously that defined by f{z) and its continuations) throughout the interior of this
polygon. The reader will observe that this is the first actual formula obtained for the
analytic continuation of a function, exce^jt the trivial one of § 5"5, exam})le.
* Borel, Le<jons sur les series divergentes, pp. 122-129.
142 THE PROCESSES OF ANALYSIS [CHAP. VII
For, take any point P with affix ( inside the polygon ; then the circle on OP as
diameter has no singularity on or inside it* ; and consequently we can draw a slightly
larger concentric circlet G with no singularity on or inside it. Then, by § 5*4,
and SO ^{(t) = -^. 2 ^ '^[dz;
but 2 ^ — ■ =^-^ converges uniformly (§ 3*34) on C since f{z) is bounded and | 2 1 ^ 8 > 0
7i=o n\ 2" + i
where b is independent of z.
Therefore, by § 4"7,
0aO = A. f 2-V(2)exp(C«0-i)c?0,
and so, when t is real, i 0 {^t) \<F{^) e^^, where F (^) is bounded in any closed region lying
wholly inside the polygon and is independent of t ; and X is the greatest value of the
real part of ^jz on C.
If we draw the circle traced out by the point zj^, we see that the real part of ^jz is
greatest wlien z is at the extremity of the diameter through ^, and so the value of X is
lf|.{|fl + S}-i<l.
We can get a similar inequality for 0' (CO and hence, by § 5'32, / e~^cf) (^t) dt is
./ 0
analytic at f and is obviously a one-valued function of ^.
This is the result stated above.
7'82. Expansions in series of inverse factorials.
A mode of development of functions, which, although first investigated
by SchlomilchiJ: as long ago as 1863, has only recently been recognised as
being of considerable importance § is that of expansion in a series of inverse
factorials.
To obtain such an expansion of a function analytic when l^l > r, we let
the function he f{z)= "t anZ~'\ and use the formula /(^) = I ze~^^^{t)dt,
11 = 0 Jo
00
where (}>(t)= X anf^/(n !) ; this result may be obtained in the same way as
that of § 7-8. Modify this by writing e-« = 1 - ^, <^ (t) = F{^) ; then
Jo
Now if t = u + iv and if t be confined to the strip — 'n-<v<7r, tisa one-
valued function of ^ and F (^) is an analytic function of ^ ; and | is restricted
* The reader will "easily verify this by considering the figure ; for if there were such a
singularity the side of the polygon corresponding to it would pass between 0 and P ; i.e. P would
be outside the polygon.
t The difference of the radii of the circles being, say, 5.
J Compendium der Iwheren Analysis.
§ By Kluyver, Nielsen and Pincberle. See Comptes Rendus, Vols. 133, 134, Annales de VEcole
norm, sup., Ser. 3, Vols. 19, 21, 23, and Palermo Eendiconti, Vol. 34.
7-82]
THE EXPANSION OF FUNCTIONS IN INFINITE SERIES
143
SO that — TT < arg (1 — ^) < tt. Also the interior of the circle | ^ = 1 corresponds
to the interior of the curve traced out by the point t = — log (2 cos 5^) + h*^,
(writing ^ = exp [i{d + tt)}) ; and inside this curve
\t\-R{t)^WR{t)Y-\-ir^-\^-R{t)-^0,
as J2 (<) •♦ 00 .
It follows that, when |||^1, |F(^)| < ife''l<l < il/i| e*"'!, where My is in-
dependent of t ; and so F (|) < M, \ (1 - |)-^ | .
Now suppose that 0 < | < 1 ; then, by § 5-23, j J?**"' (|) \< M^.n lp-», where
M2 is the upper bound of \F(z)\ on a circle with centre ^ and radius
/,<l-e
Taking p = r- (1 — |) and observing that* (1 + ??"*)" < e we find that
|i^(">(^)|< Jfi
lHl + :r^^^
;r:ri^|J -^'j-VTl
< Mye (n + ly .nl(l- ^)-'-«.
Remembering that, by § 4-5, means lim / , we have, by repeated
Jo f^+0^0
integrations by parts.
f{z) = lim
e-*- + 0
= lim
0 Jo
1-e 1^
0 ^^ + 1
1 -1-e
1-e
0
= bo +
h
z + 1'^ {z + I) {z + 2)'^ "- '^ (z +l){z + 2) ... {z + 11)
+
+ ...+
h
where
bn = lim
e-»-0
-(1 - ^)^+«i^<«) (^)
1-e
= F"" (0),
if the real part of z + n — r — n>0, i.e. if R{z)> r; further
\Rn\ <
lim |(l-|^)"+"i^<"+^*(^)'^^
\(z+l){z + 2)...(z + n)\ ,^oJo
Mie(n + 2y. n I
^ \(z + l)(z + 2) ... (z + n)\ . R{z -r)
M,e(n + 2y.n\
^ (r + 1 + B){r + 2 + B) ... {r + n + 8) .8'
where 8 = R (z — r).
* (l + x^^" increases with x; for > e", when w<l, and so log ( ) > n. That is to
1-2/ ^ ' Vl-2// -^
say, putting y-i = 1 + x, --- x log (1 + x-i) = log (1 + x-i) - :; > 0.
ax }. + X
144 THE PROCESSES OF ANALYSIS [CHAP. VII
A{i'-'i.y"]
tends to a limit (§ 2-71) as n-*oo , and so j -KJ— 0 if {n + ^fe-^'^'^^'"^ tends
to zero ; but
n f n+1 fl^
1 l/m> - = log(n + l),
by § 4-43 (II), and {n + 2y (n + l)-**-^ -^0 when 8 >0; therefore Rn^O as
n^oc , and so when R(z)>r we have the convergent expansion
J{^) = (^o + ^-:^rl+(^+l)(z + 2y^•''^(z + l)(z+2)...iz + ny""
Example 1. Obtain the same expansion by using the results
1 1 r^
(3+1) (2 + 2)... (2 + 71 + 1) ?l!./o ^ ^ '
f fjfydt^^l ^^ (' f(t)(i-^iY-t--^du.
Example 2. Obtain the expansion
/ 1\ 1 or-i a^
where «„= / t(l-t){2-t)...{n-l-t)dt,
J 0
and discuss the region in which it converges. (Schlomilch.)
REFERENCES.
E. GouRSAT, Cours d^ Analyse, Chs. xv, xvi.
E. BoREL, Lecons sur les series divergentes.
T. J. Pa. Bromwich* Theory of Infinite Series, Chs. viii, x, xi.
0. Schlomilch, Compendium der hiiheren Analysis, ii (Dresden, 1874).
Miscellaneous Examples.
1. If y -.v-(j} [y) = 0, whei'e (/> is a given function of its argument, obtain the
expansion
/(„)=/(,.)+ J_ A (^ (,)j,. (^ _^,^^^ ^)'V(,,),
where / denotes any analytic function of its argument, and discuss the range of its
validity. (Levi-Civitk..)
2. Obtain (from Darboux's formula or otherwise) the expansion
/(2)-/(«)= i ^"j"n^'7v5^{/<"K^)-^-"/'"H«)};
n=l '<' • \i- ~ ' )
find the remainder after n terms, and discuss the convergence of the series.
* The expansions considered by Bromwich are obtained by elementary methods, i.e. without
the use of Cauchy's theorem.
THE EXPANSION OF FUNCTIONS IN INFINITE SERIES 145
3. Shew that
+
where
J 0
and shew that •/„ (a;) is the coefficient of n ! «" in the expansion of {(1 -tx) (1 + < - tx)} ~ ^ in
ascending powers of t.
4. By taking
in the formula of Darboux, shew that
f{x+h)-f{x)= - J^ am ^ {/(™) {x + h) - l/C") (^)|
+ (-)«A™ + i \(f){t)fC''*'^){x->rht)dt,
J 0
1 - r ^ M tfi u^
1 — re " 1 -* 2 ! ^ 3 !
where . ^ = 1 - a, - + a„ -^ - a„ — +
5. Shew that
2% !
,- al^t + i /"i
2nl
where ^^(t) = -^[P^ (^^Y]
6. Prove that
- Ci {Z2 - ^i) V^ (^i) + ••• + (-)" (^2 - hY ^' I' \^, (e'" sech H)l /(» + 1) (3i + ^^2 - ^) (^^ ;
/ 0 l«W. j „^g
in the series plus signs and minus signs occur in pairs, and the last term before the
integral is that involving {z^ - Sj)" ; also C„ is the coefficient of &"• in the expansion of
cot f --- j in ascending powers of 2. (Trinity, 1899.)
* 7. If x^ and x^ are integers, and <^ {z) is a function which is analytic and bounded for
all values of z such that x^^ R {z)^X2, shew (bv integrating
/ (^ {z) dz
J e±27i-i5r _ 1
round indented rectangles whose corners are x^, x<,, X2± oc i, Xi ± oo {) that
I (^ (.ri) + (^ (:?;i + 1 ) + (/) (.Ti + 2) + . . . + (^ (.i-2 - 1 ) + 1 <|) (o-'g)
= ('''cb (z)dz+- f " </> (^-2 + ^^y) - <^ (.^i + ty) - 0 (a-2 - ^>/) + (f>(x^- >>/) ,
J x> i J 0 e2Tj/ - 1 •^'
W, M. A. 10
146 THE PROCESSES OF ANALYSIS [CHAP. VII
Hence by applying the theorem
where B^, B.^, ... are Bernoulli's numbers, shew that
<l>{l) + cl>{2) + ... + (f>{n) = C+^_cp{n)+ j''(f> (z) dz+l^^-^^^^' (l>i'^-'){n),
(where C is a constant not involving n) provided that the last series converges.
. (Plana.)
8. Obtain the expansion
X -^ , , ,1.3... (2n-S) x"
for one root of the equation x = 2u + u% and shew that it converges so long as \x\ < 1.
9. If aSI"^', , denote the sum of all combinations of the numbers
V, 32, 52, ... (27i-l)2,
taken m together, shew that
z smz^nto{2n + 2)\\2n + 3 ^^n+i) 271 + I ^ •••^^ ^'^2(»+i)3j'
fTeixeira.)
10. If the function f{z) is analytic in the interior of that one of the ovals whose
equation is |sins| = C (where C^ 1), which includes the origin, shew that f{z) can, for all
points z within this oval, be expanded in the form
/(.)=/(0)+ i /"-U0)+.n^"-"(0)+...+<r"/»(0)^,^^..„^
n=i 2n !
- /'^»-^)(0) + <V,/(^"-MO) + - + <',,.r(0) . ^,^^^
«=o (2n + l)! '
where S , is the sum of all combinations of the numbers
in
2% 42, 62, ... {2n-2)\
taken m together, and 'S'^™* denotes the sum of all combinations of the numbers
12, 32, 52, ...(2W-1)2,
taken m together. (Teixeira.)
11. Shew that the two series
2z^ 2z^
2z + ^ + ^ + ...,
, 2z 2 / 2z y 2.4 / 2z
and +.
1-22 1.32 Vl-32; ^3.52 Vl-£2
represent the same function in a certain region of the z plane, and can be transformed
into each other by Burmann's theorem. (Kapteyn.)
12. If a function /(z) is periodic, of period 27r, and is analytic at all points in the
infinite strip of the plane, included between the two branches of the curve |sin2| = C
(where C > 1), shew that at all points in the strip it can be expanded in an infinite series
of the form
/(2) = .lo + ^iSin3 + ...+^l„sin"2+
+ cos2(Z?, + 52sins + ...+i?„sin»-i2+...);
and find the coefficients A„ and B^.
THE EXPANSION OF FUNCTIONS IN INFINITE SERIES
147
13. If (f) and/ be connected by the equation
of which one root is a, shew that
X 1
! 2
the general term being ( - )
X3
^^^)-^ 1(^'-^^''^1!2!(^'3|<^" (/2i^') I 1!2!3!>'8
multiplied by a determinant in which
1! 2!...m!((^')^"*('» + i)
the elements of the first row are 0', (0^)', (0')', ..., {<i>^~^)',{p F') and each row is the
diflferential coefficient of the preceding one ; and F, /, F\ ... denote
dF{a)
J'W, /(«),
da '
(Wronski, Philosophie de la Technie, Section ii. p. 381.)
14. If the function W {a, b, x) be defined by the series
Tir / X V a — h„ (a-b) (a — 26) ,
W{a, b, ^)=^+-Yr^+ 3! ^^ + .-,
which converges so long as
shew that
I 77 I ^
^^^ W{a, b, x) = l + (a-b) W{a-b, b, x) ;
and shew that if ?/= W(a, b, x),
then x= Tf (6, a, y).
Examples of this function are
ir(i, 0, x)= 6=^-1,
W{0, 1, .r) = log(l +.»;),
{l + xY-l
W{a, 1, x)--
(Jezek.)
15. Prove that
_1 . % (-)"^" p
2 a„a;"
re=0
where
6?„=
2ai
4a2
«0
5a,
ao 1 ?i ! ao"
0
4ai
0
0
3an
(2»-2)a„_i (ri.-l)ao
7ia„ {n- l)a„_i «!
and obtain a similar expression for
(-- \h
\ 2 a„.^•'7 .
l»=o J
(Mangeot, Jti?*. de VEcole norm. sup. (3) xiv.)
,•=0 ?* + l oai
16. Shew that
2 a^x'^
0
10—2
148 THE PROCESSES OF ANALYSIS [CHAP. VII
where Sj. is the sura of the rth powers of the reciprocals of the roots of the equation
n
r=0
(Gambioli, Bologna Memdrie, 1892.)
17. If /n(z) denote the nth derivate of /(z), and if /_„(z) denote that one of the nth.
integrals of /(z) which has an 7i-ple zero at 2 = 0, shew that
f{z+a;)g{2+x)= 2 fn{z)g-n{x) ;
and obtain Taylor's series from this result, by putting g (z) = l. (Guichard.)
18. Shew that, if x be not an integer,
" " ^x+m + n
J'-.nLA^'rmf^x + nf*
as 1/ -*-Q0 , provided that all terms for which m — n ai-e omitted from the summation.
(Math. Trip. 1895.)
19. Sum the series
n = -q\{-Yx-a-n nj''
where the value w = 0 is omitted, and jo, q are positive integers to be increased without
limit.
(Math. Trip. 1896.)
20. If i^(.r) = J>"^"'^"''^''^ shew that
V [il--\ e^+i'-^
F{x) = e--
"i 1
n ^(l+:
-x+y-
7
and that the function thus defined satisfies the relations
Further, if
shew that
when
21. Shew that
[-©"]
1 +
V'(2) = 2 + 2-2 + 32+-=-j^log(l-0y,
F{x)-=e 2771^^
I l_e-27rM; I < 1.
(Trinity, 1898.)
i-.)l-G-^.)"][-(-r/.:.)"][-(a-J]
n {l-2e-a^cos (;r+/3p)+e-^'4a {1 - 2e-a^ cos (.r-/33) + e-2«i,}i
7 • 2(7-1 ^ ,
Ug = k sui -^— — TT, Py — K cos
2^(1 -cos xY^ e
2g-l
A:cos-
where
and 0<x<2tt.
22. If j a; I < 1 and a is not a })ositive integer, shew that
Z ^'" iirix'^ X
2 = f-
ji=i n — a 1— e^airj i^gZani
fa-l__^,a-l
t — X
dt,
where C is a contour in the ^plane enclosing the points 0, x.
(Mildner.)
(Lerch.)
THE EXPANSION OF FUNCTIONS IN INFINITE SERIES 149
23. If 01 (e), <f)2(2), ... are any polynomials in z, and if F(z) be any integrable
function, and if ^i(z), ^(^2(2), ... be polynomials defined by the equations
J a z — a;
J a ? — X
/ ' Fia;) 4>, (.;) <t>2 (^) ... <t>.n-i (^) '^"^ ^^ ~ t" ^^'^ '^•^=^'» (^)'
/a Z — CO
J a S-X 01 (2) 01 (2) 02 (2) 01
^^3(2)
(2) 02 (j) 03 (Z)
+
^/'mC^)
+ 0.(.)0.(.)...0.(.) j ;^^") *^^") 0.(-)-«^.n(-),— ,
01 (2) 02(2). •.0m (2)
24. A system of functions p^ (z), p^ {z), p^ (z), ... is defined by the equations
P(,(z)=l, p^ + i (z) = (z2 + UnZ + bn) fn (z),
where a„ and 6„ are given functions of n, which tend respectively to the limits 0 and - 1
as n-*- QC .
Shew that the region of convergence of a series of the form
2e„p„ (2),
where ej, e^, ... are independent of z, is a Cassini's oval with the foci +1, — 1.
Shew that every function f{z), which is analytic on and inside the oval, can, for points
inside the oval, be expanded in a series
f{z) = 2(Cn + ZCn')pn{z),
where
2« =27nj ^^" "^ ^^ ^" ^^^-^^^^ ^^' ^"' "" 2^" / ^" ^^^-^^^^ ^^'
the integrals being taken round the boundary of the region, and the functions qn (z) being
defined by the equations
^0 ^'^ ^z^-^-az^h ' ^" + ^ ^^) " 22T.y TTTh — ^" ^^^- (Pincherle.)
Z -i-aoZ-^Oo Z i-ttn + i-i + On + i
25. Let C be a contour enclosing the point a, and let 0(2) and f{z) be analytic when
z is on or inside C. Let | ^ j be so small that
I ;0 (2) I < 1 2 — a I
when 2 is on the periphery of C.
By expanding
^ni J c' 'z-a-t(f>{z)
in ascending powers of t, shew that it is ecpial to
Hence, by using j5j^6'3, 6-31, obtain Lagrange's theorem.
CHAPTER VIII
ASYMPTOTIC EXPANSIONS AND SUMMABLE SERIES
8"1. Simple example of an asymptotic expansion.
Consider the function f(x) = I t~'^ e^~^ dt, where cc is real and positive,
and the path of integration is the real axis.
By repeated integrations by parts, we obtain
%Aj tAj \Aj tAj J X ^
In connexion with the function /(x), we therefore consider the expression
(_)n-](^_l)!
and we shall write
^ hKy tA^ tAy lAj
in = \
Then we have j Umlum-i ! = (^^ — 1) ^~^ -^ °c as m -^ go . T/ie smes %Um is
therefore divergent for all values of x. In spite of this, however, the series
can be used for the calculation oif{x) ; this can be seen in the following way.
Take any fixed value for the number n, and calculate the value of Sn-
We have
f{x)-S,,{x) = {-Y^-{n^V)V
and therefore, since e*^~*^l
''^e'^-^dt . , , ^,. f^^ dt ni
l/(^-)-'S'„(.Oi = («+l)!j" ~^ < (n + !)!£
For values of x which arc sufficiently large, the right-hand member of this
equation is very small. Thus if we take x ^ 2??, we have
<f(ar)-Sn(xy<—^ ,
8-1, 8-2] ASYMPTOTIC EXPANSIONS 151
which for large values of n is very small. It follows therefore that the value
of the function f (x) can be calculated with great accuracy for large values of x,
by taking the sum of a suitable number of terms of the series "Zum .
Taking even fairly small values of x and n
55(10) = -09152, and 0</(10)-aS5(10)<-00012.
The series is on this account said to be an asymptotic expansion of the
function f(x). The precise definition of an asymptotic expansion will now
be given.
8"2. Definition of an asymptotic expansion.
A divergent series
z z^ z^^
in which the sum of the first {n + 1) terms is Sn {z), is said to be an asymptotic
expansion of a function f{z) for a given range of values of arg^, if the
expression i2„ {z) = z^^ [f{z) — Sn (z)} satisfies the condition
lira Rn (z) =0 (n fixed),
|2r|-»-oo
even though lim i-Rn(^)i = ^ (2^ fixed).
When this is the case, we can make
l^-{/(^)-.Sf,(^)}i<e,
where e is arbitrarily small, by taking \z\ sufficiently large.
We denote the fact that the series is the asymptotic expansion off(z) by
writing
The definition which has just been given is due to Poincare*. Special
asymptotic expansions had, however, been discovered and used in the
eighteenth century by Stirling, Maclaurin and Euler. Asymptotic expan-
sions are of great importance in the theory of Linear Differential Equations,
and in Dynamical Astronomy ; some applications will be given in subsequent
chapters of the present work.
The example discussed in | 8"1 clearly satisfies the definition just
given : for, when x is positive, \x'^ {f{^) — ^n (x)} \ <nl x~^ - ^ 0 as ^' -»• cc .
For the sake of simplicity, in this chapter we shall for the most part consider
asymptotic expansions only in connexion with real positive values of the argument.
The theory for complex values of the argument may be discussed by an extension of the
analysis.
* Acta Mathematica, viii. (1886), pp. 295-344.
152
THE PROCESSES OF ANALYSIS [CHAP. VIII
8 '21. Another example of an asymptotic expansion.
As a second example, consider the function f{x\ represented by the series
where x>0 and 0<c<l.
The ratio of the kih term of this series to the {k- l)th is less than c, and consequently
the series converges for all positive values of x. We shall confine our attention to positive
values of x. We have, when x>k,
1 _1_^ A^_:^4.^_
^+I•~a; x'^ x^ x^ x^
If, therefore, it were allowable* to expand each fraction — ^ in this way, and to
rearrange the series for f{x) in descending powers of x, we should obtain the formal series
X x^ x^^
where J„ = (-)»-iife
fc=i
But this procedure is not legitimate, and in ftict 2 ^„a-" diverges. We can, however,
shew that it is an asymptotic expansion olfix).
Forlet '^»(^) = lr' + ^' + -+^^i-
Then '^"^■^')=,!,t ~ ^' ^ :^ + - + -^
^^y^"^
sothat i/(^)->S„(.-.-)| = | 2 (-^y^'^i<^--2i^nc^.
Now 2 b'd' converges for any given value of n and is equal to C„, say ; and hence
|/(a;)-*S„(A')|<(7n^-"-2_
Consequently f {x) ~ 2 AnX''^.
Example. If f {x)—\ e^ ~ ' dt, where x is ])ositive and the path of integration is the
real axis, prove that
. 1 1_ 1.3 1.3.5
[In fact, it was shewn by Stokes in 1857 that
e^-^dt^+^J^/" (^ ^ 1.3_1.3.5
6 dt~±-,s/nc -(^--^3+-3-._-^5^ + ..,
the upper or lower sign is to be taken according as -|7r<argA'<j7r or ^7r<ai'g.r<f7r.]
8'3. Multiplication of asymptotic expansions.
We shall now shew that two asymptotic expansions, valid for a common
range of values of arg^-, can be multiplied together in the same way as
ordinary series, the result being a new asymptotic expansion.
* It is not allowable, since k > x for all terms of the series after some definite term.
8-21-8-31] ASYMPTOTIC EXPANSIONS 153
For let f{z)'^ 2 AmZ'"",
111 = 0
»t = 0
and let 8n{z) and Tn{z)he the sums of their first (w + 1) terms; so that,
n being fixed,
f{z) - Sn (Z) = 0 (!-"), </, (Z) - Tn {z) = 0 {z^).
Then, if Cm = ^o^»r + ^i 5^.-1 + ... + ^m^o, it is obvious that*
Sr^{z)Tr,{z)= I C,,^— + 0(^-").
m=0
But f(z) </, (^) = {Sn (z) + 0 (z-)] [Tn {z) + 0 (^-")}
= 8n{z)Tn{z)+0{z-)
n
This result being true for any fixed value of n, we see that
f(z)<f>(z)^ i (7^^—.
8*31. Integration of asymptotic expansions.
We shall now shew that it is permissible to integrate an asymptotic
expansion term by term, the resulting series being the asymptotic expansion
of the integral of the function represented by the original series.
00 n
For let /(a;)'^ % AmX-"", and let Sn(x)= t A.nX-"^.
m = 2 w = 2
Then, given any positive number e, we can find Xq such that
\f{x) — Sn{x)\<€\ ^' ]""■ when x > x^,
and therefore
I f{x) dx — I 8n {x) dx\ ^ \ \f{x) — Sn {x) \ dx
<
.n—i
{n - 1 )*•"-!
But /Js„(.)^.4^+^4j-.... + ^__4^,
and therefore 1 f(x)dx'^ S , ,^ .
On the other hand, it is not in general permissible to differentiate an asymptotic
expansion; this may be seen by considering e~'^sin(6'*).
* See § 2-11 ; we use o (2-") to denote any function \f/ (2) such that 2" i/- (2) -*- 0 as I 2 I -* 30 .
154 THE PROCESSES OF ANALYSIS [CHAP. VIII
8"32. Uniqueness of an asymptotic expansion.
A question naturally suggests itself, as to whether a given series can be
the asymptotic expansion of several distinct functions. The answer to this
is in the affirmative. To shew this, we first observe that there are functions
L (x) which are represented asymptotically by a series all of whose terms are
zero, i.e. functions such that lim a;"X (x) = 0 for every fixed value of n. The
a;-*- 00
function e~* is such a function when x is positive. The asymptotic expansion*
of a function J(x) is therefore also the asymptotic expansion of
J {x) + L (x).
On the other hand, a function cannot be represented by more than one distinct
asymptotic expansion over the whole of a given range of values of z ; for if
»i=0 m=0
then ii,^'^(^o+4-^ + - + ^'-^o-f-...-ff) = 0,
which can only be if Aq=Bo; Ai = Bj^, —
Important examples of asymptotic expansions will be discussed later, in connexion
with the Gamma-function (Chapter xii) and Bessel functions (Chapter xvii).
8'4. Methods of ' summing ' series.
We have seen that it is possible to obtain a development of the form
fix) = t A^x-'^ + Rn {x),
m = l
where Rn (x) -^ oc as w -^ cc , and the series X AfnX~'^ does not converge.
m = l
We now consider what meaning, if any, can be attached to the ' sum ' of
a non-convergent series. That is to say, given a set of numbers aj, a^, ..-,
we wish to formulate definite rules by which we can obtain from them a
00 CO
number S such that S = S a^ if X a.^ converges, and such that 8 exists
n==0 n=0
when this series does not converge.
8'41. Borel's'f method of summation.
We have seen (| 7"81) that
00 Too
S UnZ'' = e~f 6 (tz) dt,
« = () Jo
00 ,, fn ,?i.
where (}> (tz) = S — ^— , the equation certainly being true inside the circle
* It has been shewn that when the coefficients in the expansion satisfy certain inequalities,
there is only one analytic function with that asymptotic expansion. See Fhil. Trans. 213, a,
pp. 279-313.
t Borel, Lerons sur les Series Divergeiites.
8-32-8-43] SUMMABLE SERIES 155
of convergence of 2 ftn^". If the integral exists at points z outside this
w = 0
00
circle, we define the ' Borel sum ' of S a^z^ to mean the integral.
M = 0
00
Thus, whenever It{£)<\, the ' Borel sum ' of the series 2 z^^ is
! W = 0
If the * Borel sum ' exists we say that the series is ' summable {B)!
8'42. Euler's* method of summation.
A method, practically due to Euler, is suggested by the theorem of § 3'7l ;
00 00
the ' sum ' of S a„ may be defined as lim 2 a^a;", when this limit exists.
»=0 a;-*-l-0 w=0
Thus the ' sum ' of the series 1 — 1+1 — 1 + ... would be
lim {\ — X ■\-x^ — ...)= lim {1 + x)~^ = \.
a;-»-l-0 a;^l-0
8"43. Cesar os^ method of summation, '
Let 5,1 = «! + tta + ... + an', then if S = lim - (si+s.2+ ... + Sn) exists, we
»l-*-00 ^
00
say that S a„ is 'summable (CI),' and that its sum (CI) is ^. It is
M=l
00
necessary to establish the 'condition of consistency J:,' namely that *S/= S an
when this series is convergent.
00 w
To obtain the required result, let % am = s, X s^ = nSn ; then we have
w = l w=l
to prove that Sn-^s.
Given e, we can choose n such that
so jS — s,jj< e.
Then, if v > n, we have
n+p
< € for all values of p, and
S, = a^ + ajl- -) + ... + ttn (l - ^^^-^) + f'n+i fl - H
1-- +...+aJl
V
Since 1, 1 — v~\ 1 — 2i^ \ ... is a positive decreasing sequence, it follows
from Abel's inequality (§ 2"301) that
"n-ri
l-l) + an^.{l
)+ ...+ajl < l-~
\ V/ \ V / \ V J \ V
* Instit. Calc. Diff. (1755). See Borel, loc. cit. Introduction.
t Bulletin des Sciences Math. s6r. 2, t. xiv. p. 114.
t See the end of § 8-4.
156 THE PROCESSES OF ANALYSIS [CHAP. VIII
Therefore
S^-\a, + aJl--] + ...+an(l- ^~^^]l\< (1 - - ) e.
Making v ^- oo , we see that, if >S^ be any of the limit points (§ 2"21) of S^,
then
8— ^ ami -^e.
Therefore, since | s — 5„ | ^ e, we have
This inequality being true for every positive value of e we infer, as in § 2*2 1,
that S = s; that is to say S^ has the unique limit s ; this is the theorem which
had to be proved.
Example 1. Frame a definition of 'uniform sammability (Cl) of a series of variable
terms.'
Example 2. If bn,v'^bn+i,v'^0 when n<v, and if, when n is fixed, lim 6w,i»=l, and
if 2 am=s, then lim J 2 a,ibn,v\=S.
1)1 = 1 l—^-OO \)l=l j
8 "431. Cemro's general method of srimrnation.
00 V
A series 2 a» is said to be 'simimable (Cr)' if lim 2 anhn,v exists, where
M = 0 v-*-:o w = 0
-1
6o..=i, i,,.= |(^i4-^,^:^J (,i + Mr2^J-A' + r-i
It follows from § 8'43 example 2 that the 'condition of consistency' is satisfied ; in
fact it can be proved* that if a series is summable {Cr') it is also summable {Cr) when
r>r' ; the condition of consistency is the particular case of this result when r=0.
8*44. The method of summation of Riesz\.
A more extended method of ' summing ' a series than the preceding is by means of
lim
2 (1-^) a.
in which X„ is any real function of n which tends to infinity with n. A series for which
this limit exists is said to be 'summable {Rr) with sum function X,,.'
8*5. Hardy's j convergence theorem.
00
Let S tt)i he a series which is summable (Cl). Then if
n = l
an = 0 (l/n),
00
the series % a„ converges.
* Bromvvich, Infinite Series, § 122.
t Comptes Rendiis, cxLix. pp. 18-21.
:;; Proc. London Math. Sac. ser. 2, vol. viii. pp. 302-304. For the proof here given, we are
indebted to Mr Littlewood.
8-431-8-5]
SUMMABLE SERIES
157
Let «„ = ai + tta + . . . + a„ ; then since 2 an is summable (CI), we have
n=l
S1 + S2+ ... + Sn = n{s + o (1)|,
where s is the sum (01) of 2) a„.
M = l
Let
and let
Sm-S=tm, (m= 1, 2, ... W),
*i "r t2 "T • • • 'T *n '''^ ^n •
With this notation, it is sufficient to shew that, if | «« | < K^n~^, where K
is independent of n, and if 0-^ = ^.0(1), then <,i -*- 0 as w -*- 00 .
Suppose first that a^, a^, ... are real. Then if tn does not tend to zero
there is some positive number h such that there are an unlimited number of
the numbers tn which satisfy either (i) tn>h or (ii) tn< — h. We shall shew
that either of these hypotheses implies a contradiction. Take the former*,
and choose n so that L > h.
Then, when r=0, 1, 2, ..
f'n+r
<K/n.
Now plot the points Pr whose coordinates are (r, tn+r) in a Cartesian
diagram. Since tn+r+i — tn+r= ^n+r+i, the slope of the line FyPr+i is less
than 0 = arc tan (K/n).
Therefore the points Pq, Pi. P2, ••• lie above the line y = h — xtan0.
Let Pic be the last of the points Po,Pi, ... which lies on the left of cc — h cot 6,
so that k ^h cot 6.
Draw rectangles as shewn in the figure. The area of these rectangles
exceeds the area of the triangle bounded by y = h — x tan 6 and the axes ;
that is to say
<^n+k ~ ^n—\ = tn + tn+i + • • • + tn+k
> ^h^ cot e = ^h'K-^n.
* The reader will see that the latter hypothesis involves a contradiction by using arguments
of a precisely similar character to those which will be employed in dealing with the former
hypothesis.
158 THE PROCESSES OF ANALYSIS [CHAP. VIII
But I (Tn+k — CTfi-i I ^ I O-n+k I + i 0"n-i |
= (n + k).o(l) + {n-l).o(l),
= n. o(l),
since k ^ hnK~\ and h, K are independent of n.
Therefore, for a set of , values of n tending to infinity,
which is impossible since ^h^K~^ is not o (1) as n ^-^ cc .
This is the contradiction obtained on the hypothesis that lim i„ ^ /i > 0 ;
therefore limifn^O. Similarly, by taking the corresponding case in which
tni^ — h,^'e arrive at the result lim^^^O. Therefore since lim^w^lim^^,
we have lim tn = lim tn ^ 0,
and so tn -^ 0.
00
That is to say s„ -^ s, and so 2 an is convergent and its sum is s.
w = l
If an be complex, we consider R (an) and / (an) separately, and find
OC CO
that 2 Il(an) and S I (an) converge by the theorem just proved, and so
00
S ttn converges.
n-l
The reader will see in Chapter IX that this result is of great importance
in the modern theory of Fourier series.
Corollary. If an{$) be a function of ^ such that 2 «re(|) is uniformly summable {C 1)
throughout a domain of values of ^, and if | «„ (^) |</ui~\ where K is independent of ^,
2 Un ($) converges uniformly throughout the domain.
n=l
For, retaining the notation of the preceding section, if tn{^) does not tend to zero
uniformly, we can find a positive number h independent of n and ^ such that an infinite
sequence of vahies of n can be found for which t„ {^n)>h or tn {^n)< - h for some point |„
of the domain*; the vahie of ^„ depends on the vahie of ?i under consideration.
We then find, as in the original theorem,
^h'^K-hi<n.o{l)
for a set of values of ?i tending to infinity. The contradiction implied in the inequality
shews + that h does not exist, and so tn{^)-*-0 uniformly.
* It is assumed that a,, (^) is real ; the extension to complex variables can be made as in the
former theorem. If no such number )i existed, <„(^) would tend to zero uniformly.
t It is essential to observe that the constants involved in the iuequality do not depencj on !„.
For if, say, K depended on ^„, A'""' would really be a function of n and might be o (1) qua function
of n, and the inequality would not imply a contradiction.
ASYMPTOTIC EXPANSIONS AND SUMMABLE SERIES 159
REFERENCES.
H. PoiNCAR^, Acta Mathematica, vol. viil, (1886), pp. 295-344.
E. BoEEL, Lecona sur leu Series Divergentes.
T. J. I'a. Bromwich, Theory of Infinite Series, Ch. xi.
E. W. Barnes, Phil Trans, of the Royal Society, 206, a, pp. 249-297.
G. H. Hardy and J. E. Littlewood, Proc. London Math. Soc. ser. 2, vol. xi. pp. 1-16*.
G. N. Watson, Phil. Trans, of the Royal Society, 213, a, pp. 279-313.
S. Chapman t, Proc. London Math. Soc. ser. 2, vol. ix. pp. 369-409.
Miscellaneous Examples.
-=«^« 1 2 ! 4 :
— -. dt T + ^
Too g-x« 1 2! 4!
1. Shew that I y^~^' "'^~~- -^+^-
when X is real and positive.
2. Discuss the representation of the function
f{x) = ^'' J{t)e^-dt
(where x is supposed real and positive, and </> is a function subject to certain general con-
ditions) by means of the series
y-.^N <^W <^'(Q) I ^"(Q)
■' ^ X X^ X^
Shew that in certain cases (e.g. (f){t) = e"^) the series is absolutely convergent, and
represents f{x) for large positive values of x ; but that in certain other cases the series is
the asymptotic expansion of f{x).
3. Shew that the divergent series
1 ^ g-l ^ (a-l)(a-2)^ ^
z z^ z^
is the asymptotic expansion of the function
e'z-" I e~''x"~'^dx
• . . J ^
for large positive values of z.
4. Shew that if, when x>0,
/w=/;{iog»+iog(j-i,.)}'^.-,
J.V, J-/ \ 1 -^1 -^2 -^3
Shew also that / (x) can be expanded into an absolutely convergent series of the form
■^^^^=1 ix + l)ix + 2)...ix + ky (Schlomilch.)
5. Shew that if the series 1+0+0-1+0+1+0 + 0-1 + ..., in which two zeros
precede each —1 and one zero precedes each +1, be 'summed' by Cesaro's method,
its sum is f . (Euler, Borel.)
6. Shew that the series 1 - 2 ! + 4 ! — . . . cannot be summed by Borel's method, but the
series 1 + 0 — 2!+0 + 4! + ... can be so summed.
* This paper contains many references to recent developments of the subject.
t A bibliography of the literature of summable series will be found on p. 372 of this
memoir.
CHAPTER IX
FOURIER SERIES
9'1. Definition of Fourier series*.
Series of the type
l-tto + (tti cos x + hi sin x) + (a^ cos 2w + 62 sin 2x) + ...
00
= ^ao + 2 (an cos na; + bn sin wa;),
where an, bn are independent of a;, are of great importance in many investi-
gations. They are called trigonometrical series.
If a function /(^) exists such that I \f{t)\ dt exists as a Riemann integi'al
or as an improper integral, and such that
7ra„= I f(t) cos ntdt, 7rbn= I f(t)sinntdt,
J -TV J —n
then the trigonometrical series is called a Foiirier series.
Trigonometrical series first appeared in analysis in connexion with the investigations
of Daniel Bernoulli on vibrating strings ; d'Alembert had previously solved the equation ot
motion^ = a2-^ in the form y = \ {f{x + at)-\-f{x — at)}, where y=f{x) is the initial shape
of the string starting from rest ; and Bernoulli shewed that a formal solution is
" , . nnx nnat
y— 2 On sm —j— cos — j — ,
11=1 t I
the fixed ends of the string being (0, 0) and (l, 0) ; and he asserted that this was the most
general solution of the problem. This appeared to d'Alembert and Euler to be impossible,
since such a series, having period 21, could not possibly represent such a function ast
cx{l — x) when t = 0. A controversy arose between these mathematicians, of which an
account is given in Hobson's Functions of a Real Variable.
Fourier, in his The'orle de la Chalevr, investigated a number of trigonometrical series
and shewed that in a largo niunber of particular cases a Fourier series actually converged
* Throughout this chapter (except in § 9'11) it is supposed that all the numbers involved are
real.
t This function gives a simple form to the initial shape of the string.
91, 911] FOURIER SERIES 161
to the sum f{x). Poisson attempted a general proof of this theorem. Two proofs were
given by Cauchy; these proofs are concerned with rather particular classes of functions
and one is invalid.
In 1829, Dirichlet gave the first rigorous proof* that, for a general class of functions,
the Fourier series, defined as above, does converge to the sum f{x). A modification of this
proof was given later by Bonnet t.
The result of Dirichlet, stated in modern phraseology, is that if /(.a?) is defined in
the range (-tt, it) and I |/(0 1 ^^ exists and, further, if f{x) is defined by the
equation
f{x + 2n)=f{x)
outside the range ( - tt, tt), then, provided that J
7ra„= I f(t)co8ntdt, 7r6„= I f{t)sinntdt,
the series |ao+ 2 {uncosiix-i-bnsinnx) converges to the sum ^ {/(•^ + 0)+/(^ — 0)} pro-
n=l
vided that these limits exist and that f{x) has limited total fluctuation in an interval of
finite length of which x is an interior point.
Later, Riemarm and Cantor developed the theory of trigonometrical series generally,
while still more recently Hurwitz, F^jer and others have investigated properties of Fourier
series when the series does not necessarily converge. Thus Fejer has proved the re-
markable theorem that a Fourier series (even if not convergent) is 'summable (Cl)'
at all points at which f{x±0) exist, and its sum (Cl) is ^ {/(^ + 0)+/(.r-0)},
provided that I \f{t) \ dt converges. The investigation of the convergence of Fourier
series which we shall give later is based on this result.
For a fuller account of investigations subsequent to Riemann, the reader is referred to
Hobson's Functions of a Real Variable.
9*11. Nature of the region within which a trigonometrical series converges.
Consider the series
^ ao + 2 (a,j cos nz + bn sin nz),
where z may be complex. Writing e'^ = ^, the series becomes
l^ao + J^ 1^ («„ - ib,,) ^ « + 1 (c/„ + ib^) t - «| .
This Laurent series will converge, if it converges at all, in a region in which a^\(\^by
where a, b are positive constants.
But, if z = x-\-iy., ICI = ^ ^ ^^^ ^'^ we get, as the region of convergence of the trigono-
metrical series, the strip in the z plane defined by the inequality
log a ^-y^ log 6.
* Crelle's Journal, Bd. iv.
t Memoires des Sava7its etrangers of the Belgian Academy, vol. xxiii. Bonnet employs the
second mean value theorem directly, while Dirichlet's original proof makes use of arguments
precisely similar to those by which that theorem is proved.
X The numbers a,^, 6„ are called the Fourier constants of /(x). The letters «„, b,i will
be employed iu this sense throughout §^ 9-1-9 "O. It is easily seen that the integrals defining
a„, b,^ converge absolutely if I l/(f) i dt converges.
W. M. A. 11
162 THE PROCESSES OF ANALYSIS [CHAP. IX
The case which is of the greatest importance in practice is that in which a = 6 = l, and
the strip consists of a single line, namely the real axis.
Example 1. Let
/ (3) = sin 2 --sin 22+ -sin Zz- "Sin4z+...,
where z=x-\-iy.
Writing this in the form
/(2)= _ 1 i U^- 1 e2u + l ^3.^ _ _ J^ + 1 j; L-iz^ _ 1 g-2u + 1 e-3i^ - ... j
we notice that the first series converges* only if y ^ 0, and the second only if y ^0.
Writing x in place of z (x being real), we see that by Abel's theorem (§ 3-71),
f (.r) = lim (r sin x- - 7'^ sin 2x + 5 '"^ sin 3x— ...)
-^ lini I - L ^• [re'^ - \ r^e"-'--" + \ r^e^*^ - . . . )
This is the limit of one of the values of
- ^i log ( 1 + re'*) + \i log ( 1 + re - '*),
and as )•-»► 1 (if -it < x < n), this tends to ^x + kn, where k is some integer.
X z' ... Vi ~ 1 sin 71JC 1 n
Now 2 — • converges uniformly (§ 3-35 example 1) and is therefore con-
tinuous in the range —7r + 8^x^Tr~8, where 8 is any positive constant.
Since hx is continuous, k has the same value wherever x lies in the range ; and putting
.v = 0, we see that ^=0.
Therefore, when — tt < a* < tt, f ix) — \x.
But, when tt < x <i Stt,
/(.r)=/(.^-2,r) = |(a7-27r) = ^;r-7r,
and generally if (2ji — 1) tt < .r < (2«+l) tt,
f{x)=-\x-'Mz.
We have thu.s arrived at an example in which fix) is not represented by a single
analytical expression.
It must be observed that this phenomenon can only occur when the strip in which the
Fourier series converges is a single line. For if the strip is not of zero breadth, the
.\ssociated Laurent series converges in an aunulus of non-zero breadth and represents an
analytic function of f in that annulus ; and, since ^ is an analytic function of s, the Fourier
series represents an analytic function of z ; such a series is given by
r sin X — \r'^ sin 2.r+ ^/^ sin 3.r — . . . ,
where 0 < r < 1 ; its sum is arc tan , the arc tan always representing an angle
J. ~j~ V COS oc
between •^\it.
Example 2. W^hen - tt ^ x ^ n,
^ (-)"-icos>i.x' 1 2 1 2
n=i n- 12 4
* The series do converge ii tj = 0, see § 2'31 example 2.
91 2] FOURIER SERIES 163
The series converges only when x is real ; by § 3'34 the convergence is then absolute
and uniform.
Since ^^=sina;-|8in2a;+Jsin3.r- ... ( — w +8 <^ ^ «• — 8, 8 > 0),
and this series converges uniformly, we may integrate term-by-term from 0 to :r (i;^ 4'7),
and consequently
1 9 "(-)"-! (1 -COS wo;)
4 n=l li
That is to say, when —ir + h^x^ir-b,
4 n=i n^
where C is a constant, at present undetermined.
But since the series on the right converges uniformly throughout the range —tr^x^ir,
its sum is a continuous function of x in this extended range ; and so, proceeding to the
limit when x-*- ± tt, we see that the last equation is still true when x= ±ir.
To determine C, integrate each side of the equation (§ 4*7) between the limits — tt, tt ;
and we get
277(7-^773 = 0.
D
^ ,, 1 „ 1 „ "^ (— )"~^C0S7l^ , ,
Consequently —it^—-x^= 2 ^ — ~ ^ ( - tt ^ .r ^ it).
Lai 4 7J = 1 '*'
* Example 3. By writing tt — 2^7 for x in example 2, shew that
sm^nx (=^x {tt —x) (0 ^ x ^ tt),
„=i n^ \ = 1{it\x\-x^\ (-TT^x^ir).
9"12. Values of the coefficients in terms of the sum of a trigonometrical
series.
00
Let the trigonometrical series | Co + S (c„ cos nx + dn sin nx) be uniformly
w = l
convergent in the range (— tt, tt) and let its sum hef(x). Using the obvious
results
f ^ (=0 (mi^n),
I cos mx cos nxax ■{ \ , ^\
j _„ l=-jr {m = ni= 0),
f • ■ , (=0 (m^n), r ^ o
sin mx sin nxax ■{ ; , ^^ I aa; = 27r,
J-^ i=7r (m = ni=0), J _„
00
we find, on multiplying the equation |Co+ S (Cn cos 7iic + o?,i sin wa;) =/(.t)
w = l
by* coswa; or by sinna; and integrating term-by-term (§ 4"7),
7rCn=l f{x)cosnxdx, 7rdn= \ f(x) sin nxdx.
J —n J —n
Corollary. A trigonometrical series uniformly convergent in the range ( - tt, tt) is a
Fourier sseriess.
* Multiplying by these factors does not destroy the uniformity of the convergence.
11—2
164 the processes of analysis [chap. ix
9-2. F^jer's theorem*.
The Fourier series of a function f{x) is summahle-f (01) at all points x
at which the limits f{x-\- 0), f{x - 0) exist. And its sum (0 1) is
i{/(^ + 0)+/(a.-0)}.
In accordance with § 9-1, let/(0 be such that I |/(01 dt converges, and
J —It
let I f (t) cos ntdt= Iran, I f (t) sin ntdt = 7rhn.
J -IT J -It
m
Also, let ^ao = Ao, an cos nx + bnsinnx = An (x), X An{x) = Sm{x).
n=0
Then we have to prove that
lim - {^0 + S, (x) + S,{x)+...+ Sm-, (x)] =^{f(x + 0) +f{x - 0)},
if f(x + 0) and f(x — 0) exist.
It is easy to see that J
m- 1
^0+2 Sn{x) = mAo + {m-l)A,(x) + {m-2)A^{x) + ...+Am-i{x)
1 /■'^
= - [^m + (m - 1) cos (x-t)-^ (m -2)cos2{x- t) + ...
+ cos {m - l){x-t)\f it) dt
sm^m(x-t)^.^^^^^
27rj_^ sm^^{x — t)
1 ['^'^•^ sin" ^m(x — t)
27rj_^+, sin^i(^--0 -^ ^'
if /(O be defined outside the range (- tt, tt) by the equation /(* + Stt) =f(t).
Dividing the range of integration into the ranges {—tt + x, x) and (x, tt+x),
and writing ^ = a; ± 2^ in the respective ranges, we get
A ""v^ o / N 1 f^'^sin^m^ ., ^,, ,. 1 fi^'sin'md . . .
A,+ t Sn («^) = ^ ^.Tr^/ -^ + 2e)de + ^ -^-^ fix - 26) dd.
Consequently it is sufficient to prove that§, if a be independent of m and
0< a'^^TT, then
1 p ^_^'!}0j.^^ ^ 2^ ^^ _ ^^^ 1 p sm'^^^ (^ _ 2^) d^-iTT/ U-0)
m
as wi ^- X
Math. Annalen, Bd. lviii. pp. 51-69. f See § 8-43.
It is obvious tliat, if in the second line we -pwt e»(*-0 = X, then
m + [m - 1) (X + X-i) + (m _ 2) (X2 + X"^) + . . . + (X»»-i + X^-™)
= (l-X)-i {Xi-"' + X'-i-«-f-...+X-i + l-X-X2- ...-X"'}
= (1 - X)--' {Xi-'» - 2X + X»'+i ; = (X* - x-i)-2 (xi™ - x-^'")2.
§ The reason for investigating the integrals with a as their upper limit will be apparent
J fl.O'J
in § 9-23
92] FOURIER SERIUS 166
To prove the first of these results* we observe, on integrating the
equation
1 sm^ = i m + (m - 1 ) cos 2^ + . . . + cos 2 (m - 1 ) ^,
2 sin'^
fi'sin^m^ „ ,
that -. „^- aff = it7rm.
Jq sin^^
Also, if^7r>a>S>0 and 8 is independent of w,
1 r^sin^m^ ,. I f' de . ^ ^. .
— . „^ a0 <— I . -^ < itirmr^ cosec'^o -*► 0 as m -»- oo .
mj s sm'^^ mj s sin^S
Now, given an arbitrary positive number e, we canf choose S ^ a such that
|/(^ + 2^)-/(a; + 0)|<6
whenever 0 < 0^8. This choice of 8 is obviously independent of m.
. Then
1 r* sin^m^ . . -,^, ,^ 1 r^sin^m^, ,. ^^, .. „., ,^
mjo sin^^ -^ ^ mjo sin26^ ^-^ / ^ v /)
1 ^/ /^^ f^^sin'
m-^ ^ Wo sii
1 r* sin^m^
sin^C7
+
m J s sin'
^f{x+2d)de
1 ^/ /^^ r^'^sin^m^ ,^
= I1 + I2 + IS + li, say.
Now I,= i-7rf{x+0),
^1 1 ^ - • ,^ /(^ + 20) - f{x + 0)\de
mjo sin-t^
Also i/3| + i/4i<,-j^,J^'^{i/(^ + 2^)1 + 1/(^ + 0)1^^
-0,
as m ^- oc , since the integral converges and 8 is independent of m.
Consequently, whatever positive value e may have,
1 ( ^^ ^in^ mO ,, ^^, ,^ , .. _J
^'™ \\m\ ^r.^ -f^"" + ^^^ ^^ ~ ^'^f^'' + ^H
^^Tre;
• The proof of the second is precisely similar to that of the first,
t On the assumption that/(ar + 0) exists.
166 THE PROCESSES OF ANALYSIS [CHAP. IX
and consequently, by the definition of a limit,
^^oo w Jo sin-a
Similarly, i{f{oc — 0) exists,
»t^-oo wj 0 sin (7
Therefore, if f(oo + 0), /(^ — 0) both exist, and if / {/(t) \ dt converges,
J —IT
then
If "'-1 )
lim - ^4- S ^n(^) =i{/(^ + 0) +/(«.- 0)}.
m-*-x III' J. n=l J
Corollary. Let /(a*) be continuous in the range a^x^b. Since continuity implies
uniform continuity (§ 3*61), the choice of b, corresponding to any value of x in this range,
is independent of .?7, and the upper bound of /(a?+0) is independent of x. Therefore the
If ™-i 1
upper bounds of | /i ] , | /s | and | /j | are independent of x, and consequently — j Jq + 2 «SV {x) \
tends to the limit f{x) uniformly throughout the interval a^x^b.
9"21. Riemanns lemma*.
Let yjr (6) have limited total fluctuation in the range (a, b). Then, as
X — X , [ V (^) sin {\d) dd is 0 (IjX).
J a
For, by | 3"64 example 2, we may write
where (f>i{0), <j>2{6) are positive increasing bounded functions in the range
(a, h).
Then, by the second mean-value theorem (§ 4-14), we can find | such that
a ^^ ^b and
</>! (0) sin (\e) dO
= <f,,(b) sm{\d)dd
^2x-^<p,(by
Therefore [ (Pr(0)sm(\d)dd,
J a
rb
and snnilarly ,^2 (61) sin {\d) dd
J a
is 0(1/X), and the lemma is then evident.
Corollary. In the same circumstances I ^ (d) cos (X^) dd is 0 (1/X).
Ges. Math. Werlce, p. 241.
9 21-9-23] FOURIER SERIES 107
922. Lebesgue's lemma*.
Let I \'^ {6)\dd exist and let ^ (6) be such that when a finite number of
arbitrarily small intervals 8j, 8^, ... 8^ are omitted from the range (a, b), ^{0)
has limited total fiuctuation in each of the r + 1 surviving portions of the range.
rb
Then, as\-^x>, ^ ((9) sin (\^) dS is o (1).
J a
A function possessing the proi)erties attributed to ^ (6) will be said to satisfy Dirichle^s
conditions in the range (a, h). This phraseology is convenient, although Dirichlet really
contemplated only such functions as were continuous except at a finite number of ordinary
discontinuities and had only a finite number of maxima and minima t.
Let 6 be an arbitrary positive number; take 8^, 8^, ... 8n so small that
"^{ey^ddKe
(w=l, 2, ...r).
If r„ be any one of the r + 1 surviving portions of the range, by Riemann's
lemma we can find a number| N, independent of X but depending on e, such
that
L
M> {d) sin {\d) dd
< NX-\
(n = l,2,...r + l).
)^
Hence
[ ^ (6') sin (\^) c^^ = T f ^(0)sm{\d)dd+ 2 f ^(e)dd
•la n=lJ r„ n=lJ Sn ^
< (r+ l)NX-^ + r€.
Therefore, when X, > N'€~\
'^(e)sm(X0)dd <(2r + l)e,
rb
lim [ ^'(d)sm{\e)d0=O;
and so
this is the result stated.
9-23. Dirichlet's statement of Fourier's theorem.
Let f(t) be a function which satisfies Dirichlet's conditions (§ 9*22) in the
range (— ir, ir). Let f{t) be defined outside the range (— tt, tt) by the equation
fit + 27r) =f(t). Let X be an interior point of an interval {a, b) in which f(t)
has limited total fiuctuation.
Then if Tra^— j f(t) cos ntdt, 7rbn= I f{t)sinntdt,
J -n J —n
00
the series ^ao+ S ((X„ cos nx + 6„sin nx) is convergent and has the sum^ .
^[f(^+0)+f{x-0)}.
* Series Trigonovietriques, Cliapter iii.
t These functions are such as occur in applications of Fourier series in various branches of
Applied Mathematics.
X N may be taken to be four times the fluctuation of ^ {6) in all the ranges ri, r^, ... »V+i-
§ The limits /(x±0) exist by § 3-64 example 3.
168 THE PROCESSES OF ANALYSIS [CHAP. IX
First, suppose that the interval {a, h) may be taken to be the interval
(— TT + a;, TT+ic). Then, by Riemann's lemma (§ 9'21), a^ cos twc + 6„ sin wa;
is 0 (l/n) as w -^ 00 ; and, by Fejer's theorem (§ 9'2), the series
^tto + S (an cos 7ix + bn sin nx)
is summable (Gl) and its sum (C 1) is ^ {f(x + 0) +f(cc — 0)}. Therefore, by
00
Hardy's convergence theorem (§ 8*5), ^ao + S (a,i cos ??« + 6,i sin Aia;) is con-
n = l
vergent awe? its sum is ^ {f(x + 0) +f(x — 0)}.
Secondly, suppose that it is not legitimate to take the interval (a, b) to
be the interval (— ir + x, tt + x), so that — ir + x < a < x <b < ir -\- x.
Let iran =\ f{t) cos ntdt, Trbn = I f{t)smntdt.
Then, by Riemann's lemma (§ 9"21) and Fejer's theorem (§ 9"2),
^ do' + X (un COS nx + 6,i' sin ??a;)
n = l
is convergent and has the sum
hm J, -^—^ f (x + 2d) dd + — ^-^-^f(x-2d)dd[,
and this limit is ^ {/(a; 4- 0) +f{x - 0)|, by Fejer's theorem.
Now consider
TO
I (tto - tto') + S [{an — ttn) COS ??ir + (bn - bn ) siu nx]
= -]| +1 i .{^ + COS (x-t) + COS 2 {x-t)+ ...+ COS m(x-t)]f (t)dt
^ 1 f. sini(2,.+ l)(..-0 ,^ ^ 1 p- -i(2!» +l)i£::_ 0 ^(,) ,,
27rJ_„+^ sm^{x-t) -^ ^ ^ 2ir]b sm^(a-'-^) ^^'
1 /-i- sin (27/1 + 1)6> 1 ri- sin (2m +1)6',, ^ „, „
= - ^-^ — —/(x-26)dd+- — ^-^—a — ~'f{x + 2d)d6.
In the first integral | cosec 6 \ ^ cosec ^ (x - a), and so f(x — 2^) cosec 0
satisfies Dirichlet's conditions in the range (^x - ^a, ^tt); consequently, by
Lebesgue's lemma (§ 9'22),
,. r^'^ sin(2m + l)<9 ^, ^^, ,^ ^
lim -J f{x-2e)dd = 0.
Similarly limf*' "'" <^'" + '^ V(.+ 2^) rfff = 0.
Therefore the series
00
i («o - «o') + S {(a„ - a,/) cos «^ + (bn — bn) sin 7?a;}
n = l
converges and its sum is zero
9*24] FOURIER SERIES 169
Since ^Qo + t {an coanx + bn sin nx] -j^{f(x + 0)-\-f(x -0)},
n=l
we get by addition
00
^ tto + S {dn COS nx + bn sin nx] = ^ {f(x + 0) +f{x - 0)} ;
M = l
and this^s the required result.
Corollary. Since ^ao+ 2 ancosnx is the Fourier series associated with the function
n=l
00 00
i {/(•^■) +/( ~ *')}> ^^6 series ^ao+ 2 «« cos nx converges, and so does the series 2 6» sin no;
n=l n=l
provided that x and - ,r are interior points of intervals in which the conditions concerning
limited total fluctuation are satisfied.
• Example 1. Deduce from Fejer's theorem that, if f{t) satisfies Dirichlet's conditions
and X is an interior point of an interval in which /(^) has limited total fluctuation,
lim
where 0 < a < \ir.
' Example 2. If <^ (6) satisfies suitable conditions in the range (0, n), shew that
lim /•-^in(2m + l)^ ^^^ ^.^ p^sin(2^+l)^ ^^
m^'^.Jo Sine ^^' m — oojo sm^
,. /"J"-sin(2»i + l) ^ , , ., ,.
+ hm — ■ ' ' (b(7r-6)dB
=i7r{(/)(+0) + (^(7r-0)}.
• Example .3. Prove that, if a>0,
,. /"* sin (2/1 + 1)^ zi 7/, 1 ^11
[Shew that
' sin (2)1 + 1)^
aTT.
(Math. Trip. 1894.)
sin d
e-a9
/"""'sin(2?i + l)<9
dB= hm / • T e-aOdd
lim /"" «yLL2'i+l)i (e-ae + e-a(e+7r)+... + e-a(fl+m,r)i,
,^^ J 0 sm<9
't sin(2n + l)(9 e-a^d6
sin ^ 1 - e - ftT '
and use example 2.]
Example 4. Deduce from Lebesgue's lemma that, if a„ and &„ are Fourier constants,
<^», &H-^"0 as »-»oo.
9"24. Uniformity of convergence of Fourier series.
Let /X^) satisfy Dirichlet's conditions in the range (-tt, tt) ; and let a4-S^^'$6-5
where S > 0 and f{t) is continuous and has limited total fluctuation in the range (a, b).
Then the Fourier series associated with the function f {x) converges uniformly to the siC7nf{x)
when a + B <x<b—8.
170 THE PROCESSES OF ANALYSIS [CHAP. IX
fb-S fb-S
For if 7ra„'= I f{t) cos ntdt, irbn'= j f {t) sin ntdt,
J a+S J a+S
by § 9-2 corollary, it follows that |ao'+ 2 (a„'cos na;+bn' sin nx) is uniformly summable ;
n=l
and since
I a„' cos nx-\-hn sin nx \ < (an'^ + 6n'^)*,
which is independent of ^ and is 0 {Ijn) by § 9"21, it follows from i^ 8-5 corollary that
\aQ + 2 {an cos nx + W sin n^)
converges uniformly to the sum / {x).
Now, by § 9-23,
\ («„ - cto') + 2 {{an - an) COS nx-\-{bn — bn) sin nx)
n=\
1 /"i'r sin(2OT + l)^
T j i(a;-a) sin 6
f{x-2e)d6+'- f^" ^^(^^'^^4A)V(,, + 2^)^^;
■^ ^ T^ J i{b-x) smi9
and, as in § 9*22, given e, we can take the intervals yi, y^, ... y,. (in which /(<) is not
bounded) so small that
/ \f{t)\dt<e sin 8 {n=l,2,...r)
J y .
and then if i^ be the range {^x — ^a, ^tt) with the intervals -yi, y^, ...y,- omitted, we have
sin (2??i + l)^
/.,
sin d
^f{x-2e)de <#(2m + l)-i.
where iVis four times the fluctuation of /(«-2^) cosec^ in the range i,. ; and it follows
from § 3-64 example 4 that •
vV^iVoCosec 8,
where N^^ depends only on the values of f{t) in the range ( — tt, n) with the intervals
yi> ■y2)---yr omitted.
rru e I /"*"■ sin(2??i + l)^ ., ^., ,. i {r-\-\) N^
Therefore / ■ Z /(-^ - 2(9) rf^ < ,-r-^^^v- • ^ + '"f
<(2r+l)e
when 2??i + l> A^o cosec S ; since this choice of m is independent of x, the integral tends to
zero uniformly. Applying the same arguments to I ' —^^ a' f{x + 26)dd, we
J Hb-x) smp
see that ^{a,y — ao')+ 2 {{a^ — a^') cos 7ix + {bn- b„') sin nx] converges uniformly to the sum
n=l
zero.
Since the sum of two uniformly convergent series converges uniformly, we see that
^00+ 2 (a„cos?i^+6„sin?i.r) converges imiformly to the sum/(^) when a + 8^x^b — 8.
n=l
Note. It must be observed that no general statement can be made about uniformity
or absoluteness of convergence of Fourier series. Thus the series of § 9'11 example 1
converges imiformly except near x={2n + l)ir but converges absolutely only when x^nn,
whereas the series of 5^ 9'11 example 2 converges uniformly and absolutely for all real
values of x.
9-3, 9-31] FOURIER SERIES 171
9"3. The representation of a function by Fourier series for ranges other
than (— TT, tt).
Consider a function f(a;) satisfying Dirichlet's conditions in the range
Write a; = l(a + b)-l(a-b)'rr-'iv', f{x) = F{x').
Then we have proved that
1 1 "
g [F {x' + 0) + jP {x - 0)} = g tto + S {an cos nx' + &„ sin nx),
^ ^ n=l
and so
|{/(^ + 0)+/(^-0)l
1 5 f nTT (2a; — a — 6) , . nir (2x — a — b))
= ^ao+ S -^ancos 7 + 6„sin — ^ '-y ,
^ n=i [ 0 —a o — a)
where by an obvious transformation
^{b — a) an= j J {x) cos — , ax,
2 (6 — a) bn =\ f{x) sin ^-r dx.
J a 0 a
9*31. The cosine series and the sine series.
Let f(x) be defined in the range (0, I) and satisfy Dirichlet's conditions
in that range. Define f{x) in the range (0, — I) by the equation
f{-a:)=f{x).
Then
- {/(a; + 0)+/(^'-0)} = 2^0+ X ia„cos*-y'^ + 6.rtSin*^ ,
where, by § 9"3,
lan= \ f (t) cos —J- dt = 2 j f(t) cos ^-^cL»;
lK=f f{t)sm^dt^O,
so that when — I ^x^l,
9 l/(^ + 0) +/(^ - 0)} = ^ <Xo + t an cos —r- ;
this is called the cosine series.
If, however, we define /(a?) in the range (0, — I) by the equation
172 THE PROCESSES OF ANALYSIS [CHAP. IX
we get, when —I ^004,1,
I {/(^ + 0) +Ax - 0)} = i 6„ sin ^ ,
where lbn = 2 j f(t) sin -^ dt ;
this is called the sine series.
Thus the series
1 S nTTX 2, , . rnrx
g tto + S a„ cos -J- , Z On sm —j- ,
^ n=l I' w = l ''
where ^lan= \ f (t) cos —j- dt, ^lbn= j f (t) sin— j-dt,
have the same sum when 0 ^x^l; but their sums are numerically equal and
opposite in sign when O^w^ — l.
* Example 1. Expand ^(tt — ^■)sin^ in a cosine series in the range O^x^tt.
[We have, by the formula just obtained,
^{7r—x)sm.v=^aQ+ 2 a^cosw.^,
w=l
where iiran= i ^ (tt — ^) sin ^ cos nx dx.
Jo
But, integrating by parts, if w 4=1,
I 2{Tr — x) sin x cos nxdx
J 0
= I {tt — x) {Hin (n + 1) X - sin {71 — I) x} dx
Jo
_r. _ , Jcos(% + l)^ cos {n — I) xYl"^ f~ jcos(n + l)x cos{n — l)x'\ , ^
~L 1 w + 1 »^^^ J Jo "jot »i+l ^^^i J
/^^ 1_\ _ -277
~ "" V^r n-\)~ (/i + 1 ) (7^ - 1 ) *
Whereas if w=l, we get I 2 (tt — .r)sin.'«7C0SA-(i^= J-tt.
JO
Therefore the required series is
7; + - cos:>7 — - — T cos 2,^■ — - — -cos 3^ — — -~ cos 4^—
24 1.3 2.4 3.5
It will be observed that it is only for values of x between 0 and tt that the sum of this
series is proved to be A (tt — x) sin x ; thus for instance when x has a value between 0 and
— TT, the sum of the series is not ^(7r-^)sin^, but —\ijf\- x) sva. x ; when x has a value
between tt and 27r, the sum of the series happens to be again ^ (tt — a*) sin a-, but this is a
mere coincidence arising from the special function considered, and does not follow from
the general theorem.]
Example 2. Expand 'J.-r (tt — j.') in a sine series, valid when 0 ^.r^i
in 3
"33"
rrr.! • • ■ sin 3.:^ sin bx ^
[The series is sin x H — -^^ 1 — h . . . .]
9-4]
FOURIER SERIES
173
Example 3. Shew that, when 0 ^ :p ^ tt,
1 / ^ V , .. ^ ^ o\ COS 3^ C08 5;»
— TT ( IT -2x){7r^ + 27rx-2x^) = COS x + —:ij- + — ^i- +
yb
3*
5*
[Denoting the left-hand side hy f{x), we have, on integrating by parts and observing
that/(0)=/(7r) = 0,
/ f{x)coanxdx=-\f{x)ii'mnx\ I f (x)ainnxdx
= — , f (x) cos nx \ 5 f" (x) cos nxdx
n^H ^ ' Jo w-'./o-^
= -A f" (x) sin no; + ^ I /'" {x) sin nxdx
«^L Jo ^-Vo
= 4[r
(a7)cosn^ = -|(1 -cosMjr).]
' Example 4. Shew that for values of x between 0 and tt, e** can be expanded in the
cosine series
2s, ,x/l cos2a' cos 4^ \ 2s, ,./cos^ cos 3a;
and draw graphs of the function ^^ and of the sum of the series.
. Example 5. Shew that for values of x between 0 and tt, the function Jn- {■n — 'ix) can
be expanded in the cosine series
cos X + -
cos 3^ cos hx
+
+ ...,
32 ' 5^
and draw graphs of the function \tt (tt — ^x) and of the sum of the series.
9'4. The nature of the coejficients in a Fouriei- series.
Suppose that (as in the numerical examples which have been discussed)
the interval (— tt, tt) can be divided into a finite number of ranges
(— TT, k^), {k^, k.2) ... {kn, tt) such that throughout each range f{x) and all its
differential coefficients are continuous and have limited total fluctuation and
have limits on the right and on the left (§ 3'2) at the end points of these
ranges.
Then
Tram = I / (0 ^os mtdt + I f (t) cos 7ntdt +
+ I f{t) COS mtdt.
Integrating by parts we get
7n~^ f (t) sin mt
+
m~^ f (t) sin mt
+ ...+
A-,
m ^ f (t) sin mt
SO that
where
k, ft,
f'{t) sin mtdt — m~^ I /' {t) sin mtdt —
-TT '' A,
m '
— m
/' (t) sin mtdt,
/>■„
a in —
A.,
m
TrArn = S sin mkr [f{k, - 0) -/(^•, + 0)|,
and hm is a Fourier constant of /' {x).
174 THE PROCESSES OF ANALYSIS [CHAP. IX
bimilarly Om = r
m m
where
TrBm = - i cos mkr [f{kr - 0) -f{kr + 0)} - /cos m {/(tt - 0) -/(- TT + 0)},
r=\
and a^' is a Fourier constant of /' {x).
Similarly, we get
A ' h " Ti ' n "
ttm — > ^wi ^ + J
mm mm
where a^', bm" are the Fourier constants of /" (x) and
n
'rrAj= S sinmA;^{/'(^,.-0)-/(A;, + 0)},
?• = !
n
TrBm' = - S COS mkr {/' (kr - 0) - /' (A;, + 0)}
r=l
- COS m-rr [f (tt - 0) - /' (- tt 4- 0)j.
Therefore
m nv irr m m- iir
Now as m -*» 00 , we see that
AJ = 0{\), Bm'=0(l),
and, by § 9*21, it is evident that
am" = 0(1), Kr = o(l).
Hence if Am= 0, Bm = 0, the Fourier series forf(x) converges absolutely
and uniformly, by § 3'34.
The necessary and sufficient conditions that ^,^ = ^^ = 0 for all values of
m are that
f{kr-0) =f{K + 0), /(tT-O) =/(- TT + 0),
that is to say that */(;») should be continuous for all values of x.
9*41. Differentiation of Fourier series.
The result of differentiating
1 ""
^ ao + % (ttm COS mx + 6,„, sin mx)
^ m = l
00
term by term is 2 {nibm cos mx — ma„i sin mx].
m = l
With the notation of | 9*4, this is the same as
- tto' + 2 (a J cos mx + 6„,' sin mx),
provided that J.,„ = B^^ = 0 and I /' {x) dx — Q;
J — IT
these conditions are satisfied ii f(x) is continuous for all values of x.
* Of course /(a;) is also subject to the conditions stated at the beginning of the section.
9-41-9-5] FOURIER SERIES 176
Consequently sufficient conditions for the legitimacy of differentiating
a Fourier series term by term are that f{x) should be continuous for all
values of x and /' {x) should have only a finite number of points of discon-
tinuity in the range (— tt, tt), both functions having limited total fluctuation
throughout the range.
9'42. Determination of points of discontinuity.
The expressions for a^ and h^ which have been found in § 9"4 can frequently be applied
in practical examples to determine the points at which the sum of a given Fourier series
may be discontinuous. Thus, let it be required to determine the places at which the sum
of the series
sin 2 + J sin 3z-|-^ sin 50+...
is discontinuous.
Assuming that the series is a Fourier series and not any trigonometrical series and
observing that a,„ = 0, 6^=(2m)~*(l — cosmTr), we get on considering the formiila found in
§ 9-4,
Hence if ^i, /(".i, ... are the places at which the analytic character of the sum is broken,
we Ijave
0 = 7rJ^=[sinm^i{/(^i-0)-/(^i+0)} + sinm^2{/(^2-0)-/(^2+0)} + ...].
Since this is true for all values of m, the numbers ^i, ^2> ••• must be multiples of tt ; but
there is only one even multiple of tt in the range — 7r<a;^7r, namely zero. So ^i=0,
and ^2» ^3) ••• do not exist. Substituting ^i = 0 in the equation B^^^^-^cosmn-, we have
7r(^-JcosTO7r)=-[cosm7r{/(7r-0)-/(-7r + 0)}+/(-0)-/( + 0)].
Since this is true for all values of m, we have
A7r=/( + 0)-/(-0), W=/(7r-0)-/(7r+0).
This shews that, if the series is a Fourier series, f{x) has discontinuities at the points
nir (n any integer), and since a^ — b^ = 0, we should expect* /(a;) to be constant in the
open range ( — tt, 0) and to be another constant in the open range (0, tt).
9"5. The Hurwitz-Liapounoff '\ theorem concerning Fourier constants.
Let f{x) he hounded and satisfy Dirichlet's conditions (§ 9"22), so that
1 ""
its Fourier constants an, hn exist. Then s ao" + 2 (a^^ + hn) converges to the
sum
x\r [fixyf-dx.
^ J -IT
For, Sn (x) being defined as in § 9*2, if x is not a point of any one of
r intervals each of arbitrarily small length S, given e we can choose mi
depending on B and e but independent of x, so that, when m > m^,
( 1 '"-I )
( m „=o '
< e;
* In point of fact f{x)=-^ir (-7r<x<0);
/(x) = i7r (0<x<ir).
+ Math. Annalen, lvii. (1903), p. 426. Liapounoff discovered the theorem in 1896.
X This integral converges, by § 412 example 1.
176 THE PROCESSES OF ANALYSIS [CHAP. IX
and if ;» be a point of one of these intervals, it follows from the integral
TO — 1
obtained for 2 /S„(a;) in § 9*2 that if k be the upper bound of \f{x) \, then
«=o
-1
2 ^„ {x) I < mK, and so
M = 0
1 m-l
^ 2 .Sf„(^)-/(^)
<2/e.
Therefore, when m>m-^, we have
I m-\ )2
fix)-- t 8n(x)[ dx < (27r - r8) e^ + 4>K'r8.
Since this is true for all values* of 8 and e no matter how small, it follows
that
rir f \ m-l ]2
lim /(^)-- S ^n{x)i dx = 0.
But
f /(^)--^ S >S«(^) dx^\ \f{x)- t ^~-A^(x)\ dx
= {/(«;)p dx-2 fix) t "^ A,, (x) dx
j An(x)y dx.
But since
m — n\
13a; qxdx = 0, when « =1= o, and since
. cos cos ^ 112
/(ii:) cos wa;c?a; = 7ra„, I /(*') sin ?iA"c?a; = irbn,
we have
1 '""^ )-
/'(ic) 2 Sn(x)> dx
r TT 171— L f^Y^ ,^ 1
J -TT W-1 ''^^ •^
»J - 1
+ 2 {cin^ + h,;^)
m-l
f{x)Y~dx-^7ra^-'n t K^ + 6n^) + — 2 w^ (««' + V)
1
w— 1
2
n = l
>H - 1
f{x) - - cIq - % An {x)\ dx + -— 2 n" (a„" + hn%
IT
Consequently, since the left-hand side tends to zero and both of the
expressions in the last line are positive,
TT '""^
rii ,j = 1
* To make the right-hand side of this inequality into a function of one variable e, we may
take 5 = 6.
9*6, 9-61] FOURIER SERIES ' 177
and therefore, considering the last but one of the above set of equal
expressions,
J^ {f{x)Y dx - TT 1^ a„2 + V {an' + 6„^)| - 0.
This is the theorem stated.
Corollary. ParsevaVs theorem, li f{x), F{x) both satisfy the conditions laid on /(.r)
at the beginning of this section, and if J„, 5„ be the Fourier constants of F{x\ it follows
by subtracting the pair of equations which may be combined in the one form
J" ^ {fix) ± F {x)Y dx = 7rh (ao ± ^o)2 + J ««« ±Anf + {K ± i?0"^}]
that / f{x)F{x)dx=nhaoA^i+-2{anAn + bnBn)\-
9"6. Riemanns theory of trigonometrical series.
The theory of Dirichlet concerning Fourier series is devoted to series
which represent given functions. Important advances in the theory were
made by Riemann, who considered properties of functions defined by a series
1 *
of the type * ^ «o + X (an cos nx + bn sin nx), where it is for the most part
assumed that lim (a„ cos )ix + bn sin nx) = 0. We shall give, as an illustration
of Riemann 's theory, the propositions leading up to the theorem f that if two
trigonometrical series converge and are equal at all points of the range
(— TT, tt) with the possible exception of a finite number of points, corre-
sponding coefficients in the two series are equal.
9'61. Riemanns associated function.
2 GO 00
Let the sum of the series -ao-l- S (a,iCos wa?-f- ^nsinwa;) = ^o + 2 An{x)
at any point x where it converges be denoted hy f{x).
Let F{x)=\a^''- S n-'-An{x).
Then, if the series defining f{x) converges at all points of any finite interval,
the series defining F{x) converges for all values of x.
To obtain this result we need the following Lemma due to Cantor :
Cantoris lemma\. If lim An (^) = 0 for all values of x such that a ^x ^b, then an -»• 0,
bn^O.
For take two points x, x + 8 of the interval. Then, given f, we can find Hq such that§,
when n>nQ
I an cos nx + bn sin nx | < e, | a„ cos n {x + 8) + bn sin n (x + 8)\<€.
Therefore
I cos w8 (a,( cos nx + bn sin nx) + sin 9i8 ( — a„ sin «^ + b„ cos nx) \ < e .
* Throughout §§ 9 '6-9 '632 the letters a,„, b^ do not necessarily denote Fourier constants,
t The proof given is due to G. Cantor, Journal fiir Math, lxxii.
X Riemann appears to have regarded this result as obvious. The proof here given is a
modification of Cantor's proof (Math. Annalen, iv.).
§ The value of ?io depends on x and ou 5.
W. M. A. 12
178 THE PROCESSES OF ANALYSIS [CHAP. IX
Since | cos n8 {an cos nx + hn sin nx) \ < f ,
it follows that | sin nh ( - a« sin nx + 6„ cos nx) \ < 2f ,
and it is obvious that | sin n8 {ctn cos nx + bn sin 7ix) | < 2e.
Therefore, squaring and adding
(a„2^-6„2)4|sin%S|<2fV2.
Now suppose that a„, 6„ have not the unique limit 0; it will be shewn that this
hypothesis involves a contradiction. For, by this hypothesis, some positive number to
exists such that there is an unending increasing sequence tij, n2,... of values of n, for
which
Now let the range of values of S be called the interval /j of length Zj on the real axis.
Take w/ the smallest of the integers n^ such that Wi'Zi>27r ; then sin?^l'y goes through
all its phases in the interval /^ ; call /g that sub-interval* of /j in which sin %i'y> 1/^/2 ;
its length is irj{2ni) = L2. Next take n^ the smallest of the integers nj.{>ni) such that
n2'Ii2>27r, so that sin%'y goes through all its phases in the interval I2 ; call Is that sub-
interval* of I2 in which sin%2'y>l/v/2 ; its length is 7r/(2n2') = L^. We thus get a
sequence of decreasing intervals Ii, I2, ... each contained in all the previous ones. It is
obvious from the definition of an irrational number that there is a certain point a which
is not outside any of these intervals, and sinna^l/v/2 when n = ni, «2'j ••• (^V+i >»*/)•
For these values of n, (aj + b„^)^ sin na>2fQ ^2. But it has been shewn that corresponding
to given numbers a and e we can find «o such that when n>no, {an^ + bn^)h(smna)<2f ^2 ;
since some values of n,.' are greater than ??.(,, the required contradiction has been obtained,
because we may take f <eo ; therefore a„ -^ 0, bn ->- 0.
Assuming that the series defining /(a?) converges at all points of a certain
interval of the real axis, we have just seen that a^^-O, 6„-^0. Then, for all
real values of x, \ an cos nx + b^ sin nx j ^ (a„- + bn^)--^0, and so, by § 3'34, the
CO
series ^A^x'^ — ^ n~^ An{x) = F(x), converges absolutely and uniformly for all
M = l
real values oi x; therefore, (§ 3"32), F(x) is continuous for all real values of x.
9'62. Properties of Riemanns associated function ; Riemann's first lemma.
It is now^ possible to prove Riemann's first lemma that if
F (x + 2a) + F(x-2a)- 2F(x)
G {x, a) =
4a2
then lim G{x, a) =f(x), provided that % An{x) converges for the value of x
under consideration.
Since the series defining F{x), F{x ± 2a) converge absolutely, we may
rearrange them ; and, observing that
cos n {x -f 2a) + cos n {x — 2a) — 2 cos nx = — 4 sin- no. cos nx,
sin n (x + 2a) + sin n {x — 2a) — 2 sin nx = — 4< sin^ wa sin nx,
it is evident that
n / \ A Z /sin naV . , ,
G(x,a) = A,+ 2 -— An(x).
n = i V na J
* If tbere is more than one such sub-interval, take that which lies on the left.
962] FOURIER SERIES 179
It will now be shewn that this series converges uniformly with regard to
00
a for all values of a, provided that 2 ^»(a?) converges. The result required
n = l
/sin 7icc\^
is then an immediate consequence of § 3'32 : for, if/„(o)= f J , (a^O),
and/„(0) = 1, then/„(a) is continuous for all values of a, and so G(x, a) is a
continuous function of a, and therefore, by § 3'2, G(a;, 0) = lim G(x, a).
To prove that the series defining G (x, a) converges uniformly, we employ
the test given in § 3'35 example 2. The expression corresponding to (On{x)
is /n(a), and it is obvious that !/„(«) j^^l; it is therefore sufficient to shew
00
that 1 |/n+i(«)— /n(a) i < K, where K is independent of a.
n = l
In fact* if s be the integer such that s|a|^7r<(s + l)la|, when a=t=0 we have
s — 1 s — 1 sin^ fl sin 5fl
2 I /» + 1 («) -/n (a) I = 2 ( /„ (a) -/„ + 1 (a)) = —7^ —ili- •
Also
2 l/„ + i(«)
-Ma)
= i
n=s+l
[sin2«a/I 1 \] ^ sin2«.a-sin2(7i + l)a
t a2 U"^ {n + l)y] ' (7H-l)2a='
^ * 1 /I 1 \ - |sin2na-sin2(?i + l)a|
■" n=s+l a' W {n + 1 )y ' „=, {n + iy a2
1 ^ 1 sin a sin (2%+ 1) a 1
^{s^+l)
, 1 sin a 1 °° 1
2a2^ a2 „=,+! (71 + 1)2
1 1
<-,+ -
TT-
sin a 1 /"* dr
a' Is i^+iy
<^ + (-
1
'+i)i«r
Therefore
2 l/n.l
(«)-
-/n
^ ^1 sin^a sin^sa fsin^ sa sin2(s + l)a^ 1 1
(«)l^ „2 ,2„2 ^(^,2„2 1 (5+l)2a2; + ,r2 + ^
<1 +
1 -2
- + — ■
TT TT-
Since this expression is independent of a, the result required has been obtained t.
00
Hence, if X An(x) converges, the series defining. G{x, a) converges
uniformly with respect to a for all values of a, and, as stated above,
lim G (x, a) = G (x, 0) = ^0 + i An{x) =f{x).
a^O n=l
Example. 1{ E (x, a, ^) = ^ — - ---^ ^^-^ — ^ ^ ^ shew
that ff{.v, a, ^)-*-f{x) when f{x) converges if a, /3-»-0 in such a way that a//3 and fija
remain finite. (Riemann.)
* Since x~'^ sin x decreases as x increases from 0 to tt.
t This inequality is obviously true when a = 0.
180 THE PROCESSES OF ANALYSIS [CHAP. IX
9-621. Riemanns second lemma. With the notation of §§ 9-6-9-6f, if
a.., 6,^0. th^ lim J-(^+2a) + f(.-2«)-2f(.) ^ ^ ^^ ^„ „^,,,^^ „^ ^^
For la-'[F{x+2oL)+F{x-1a)-^F{x)] = A,a+ ^ -— -^„(^;); but
sin''' na
bv ^ 9-11 example 3, if a > 0, ^ — ^ — = i (tt - a) ; and so, since
•^ "^ n=i wa
, , , ^ sin'^wa . . ^
n = l i m=l ''<' " J
it follows from § 3-35 example 2, that this series converges uniformly with
regard to a for all values of ct ^ 0*.
But lim la-'{F(x+2a) + F(x-2a)-2F{x)]
= lim
Ao{x)a+^{ir-a)A,{x)+ 2 gn(cc) {An+i(x) - An(x)]
n = l
and this limit is the value of the function when a = 0, by § 3-32; and this
value is zero since lim An(x) = 0. By symmetry we see that lim = lim.
9'63. Riemanns theorem-f on trigonometrical series.
Two trigonometrical series which converge and are equal at all points of
the range (— tt, tt), ^uith the exception of a finite number of points, must have
corresponding coefficients equal.
An immediate deduction from this theorem is that a function of the type considered
in § 9-23 cannot be represented by any trigonometrical series in the range ( - tt, tt) other
than the Fourier series. This fact was first noticed by Du Bois Reymond.
We observe that it is certainly possible to haVe other expansions of (say) the form
CO
Oq + 2 (oto cos I mx + /3,„ sin | mx),
which represent f{x) between 0 and 2it ; for write .^' = 2^, and consider a function (^ (^),
which is such that </)(^)=/(2|) when -^7r<^<^7r, and 4>i^)=9ii) when -7r<|< ^^ir,
and when ^7r<^<7r, where g {^) is any function satisfying Dirichlet's conditions. Then on
expanding 0 (^) in a Fourier series of the form
ao+ 2 (a„iCOS m£ + /3„iC0sm^),
this expansion represents /(.x-) when - 7r<.^•<7^ ; and clearly by choosing the function c/ (^)
in different ways an unlimited number of such expansions can be obtained.
The question now at issue is, whether other series })roceeding in sines and cosines of
integral multiples of x exist, which differ from Fourier's expansion and yet represent f{x)
between - rr and n.
"* Sill'' tJlCL
* If we define f7„ (a) by the equations (/„ (a) = ^ (tt- a) - 2i ,-, , (a4=0), and r/,^(0) = i7r,
then (/,i(a) is continuous when a > 0, and (;„_,., (a) c< gn(a).
+ The proof we give is due to Cantor, Journal fur M<ith. lxxii.
9-621-9 632] Fourier series 181
If possible, let there be two trigonometrical series satisfying the given
conditions, and let their difference be the trigonometrical series
»=i
Then f{x) = 0 at all points of the range (— tt, tt) with a finite number of
exceptions; let ^i, ^2 be a consecutive pair of these exceptional points, and
let F{x) be Riemann's associated function. We proceed to establish a
lemma concerning the value of F{oi;) when ^i<x< ^2-
9'631, Schwartz' lemma*. In the range |i<^<|2j ^{^) ^^ <^ linear function of x,
if f{x) = 0 in this range.
For if ^=1 orif ^=-1
<f^ix) = d^F{x)-F{^,)--^-^'^{F{^,)-Fi$,y^-ih\x-^0i$2-:^)
is a continuous function of ^ in the range |^i<a7<^2) ^^^ <t>{^i) — ^ {$2) = ^-
If the first term of 4>{x) is not zero throughout the range t there will be some point
x=c at wliich it is not zero. Choose the sign of 6 so that the first term is positive at c,
and then choose h so small that <^ (c) is still positive.
Since ^(^) is continuous it attains its upper bound (§ 3*62), and this upper bound is
positive since cp (c)>0. Let (j) (x) attain its upper bound at Ci, so that Ci + ^i, Ci + ^-i-
Then, by Riemann's first lemma,
a-*0 «^
But (f) (Cj + a) ^(j) (ci), (^ (Cj — a) ^ <^ {Cj), so this limit must be negative or zero.
Hence, by supposing that the first term of ^ (x) is not everywhere zero in the range
(li, ^2)) we have arrived at a contradiction. Therefore it is zero ; and consequently F{x) is
a linear function of x in the range |i<.^'<^2- The lemma is therefore proved.
9"632. Proof of Riemann's Theorem.
We see that, in the circumstances under consideration, the curve y = F{x)
represents a series of segments of straight lines, the beginning and end of
each line corresponding to an exceptional point ; and as F{x), being uniformly
convergent, is a continuous function of x, these lines must be connected.
But, by Riemann's second lemma, even if ^ be an exceptional point,
lim F^^-^^) + J'(^-^)-^F{^) ^ ^
Now the fraction involved in this limit is the difference of the slopes of
the two segments which meet at that point whose abscissa is ^ ; therefore the
two segments are continuous in direction, so the equation, y = F{x) represents
* Quoted by G. Cantor, Journal fiir Math, lxxii.
+■ If it is zero throughout the range F (x) is a linear function of .v.
182 THE PROCESSES OF ANALYSIS [CHAP. IX
a single line. If then we write F{x)=^cx + c', it follows that c and c' have
the same values for all values of x. Thus
00
^A(,x'^ — cx — c'= ^ n—^An{x),
the right-hand side of this equation being periodic, with period 27r.
The left-hand side of this equation must therefore be periodic, with period
27r. Hence
A, = 0, c = 0,
00
and — c'= X n-^An{x).
M=l
Now the right-hand side of this equation converges uniformly, so we can
multiply by cos nx or by sin nx and integrate.
This process gives
■an = — c' I cos nxdx — 0,
7rn~
'7rn~- 6„ = — c' I sin nxdx = 0.
Therefore all the coefficients vanish, and therefore the two trigonometrical
00
series whose difference is J-o -f- 2 An{x) have corresponding coefficients equal.
n=l
This is the result stated in § 9"63.
9"7. Fourier s representation of a function by an integral''^.
It follows from § 9"23 example 1 that, if f{x) be continuous except at a
finite number of discontinuities and if it have limited total fluctuation
in the range (— oo , oo ), then, if x be any internal point of the range (— a, /3),
lim [^ sm(2m-H)(^-a;)^^^^^^ ^ ^.^ ^^^_^ ^.^ ^ ,^^^ ^ 2^) +y(^ _ 2^)}.
Mt-*-ooJ -a (t — X) e_^o
Now let X be any real number, and choose the integer m so that
X = 2m -I- 1 + 277 where 0 ^ 77 < 1.
Then [ {sin X (i -«)- sin (2m + 1) («-«;)} (^-a;)-\/"(^) tZ^
J —a
2 {cos (2m -^ 1 -F 7;) {t - x)] . {sin 'n{t~ x)] {t - x)-^f(t) dt
as m-*QO by § 9-21, since (t - x)-^ f {t) simj (t - x) has limited total
fluctuation.
* Theorie Anaiytique de la Chaleur, Ch. ix.
9*7] FOURIER SERIES 183
Consequently, from the proof of Lebesgue's lemma (§ 9'22), it is obvious
that if / \f{t)\dt and 1 \f{t)dt\ converge, then*
J 0 J -co
limf ^'^)Stlf^f(^t)dt = ^7r{f(x + 0)+f{a^-0)],
and so
lim f I Tcos w (« - ^) du\f{t) dt = ^tt {/(a; + 0) +f{x - 0)}.
To obtain Fourier's result, we must reverse the order of integration in
this repeated integral.
For any given value of A. and any arbitrary value of e there exists a
number ^ such that
r\f{t)\dt<he/x;
J p
writing cos u{t- x) .f{t) = j> (t, u), we havef
I \j(f>(t,u) dii> dt- \j 4){t, u) dt\ dii
= I \i (f) {t, ii) du i dt + i \i (f) {t, u) dul dt
-j \j (f>{t,n)dt\du-l \j <f>(t,u)dtldi
= I I \l(f}{t,u) dul dt- I \ I j>{t, u) dt\ du
< 1 \ I \(f>{t, u)\dur dt+ I \<j> (t, u) I dt^du
<2X \f{t)\dt<€.
J s
Since this is true for all values of e, no matter how small, we infer that
• — 00 /"A. /"A r-aa
/•<» rK rK /•<» r-
= ; similarly
J 0 J 0 J 0 J 0 Jo
0 J Q J 0
to lim
means the double limit lim I . If this limit exists, it is of course equal
p^-oo,(r-*-x J -P
t The equation | i = l I is easily justified by § 4*3, by considering the ranges within
J 0 ; 0 ./ 0 7 0
which /(a;) is continuous.
184 THE PROCESSES OF ANALYSIS [CHAP. IX
Hence ^tt [f{x + 0) +f{x - 0)} = lim [ [ cos u (t - x)f{t) dt du
^00 fOO
= j I COS Ji {t — x) f (t) dt du.
J 0 J -00
This result is known as Fourier's integral theorem* .
Example. Verify Fourier's integral theorem directly (i) for the function
(ii) for the function defined by the equations
fix) = l, {-l<x<l); f{x)=0, {\x\>l). (Rayleigh.)
REFERENCES.
G. F. B. RiEMANN, Ges. Werke, pp. 213-250.
E. W. HoBSON, Functions of a Real Variable, Chaj^. vii.
H. Lebesgue, Lecons sur les Series Trigonometriques. (Borel tract.)
C. J. DE LA Vall^e Podssin, Cours d' Analyse, t. ii. Chap. iv.
Miscellaneous Examples.
. 1. Obtain the expansions
1 — r cos 2 , « ^
(a) r, = 1+ r cos z + r^ cos 2z+ ...,
^ ' l-2rG0iiz + r^
(b) - log (1 — 2r cos z + r^) = —r cos z — ^r^ cos 2z — ~r^ cos 3z— .,,,
, , r sin z . 1 o • ^ ^ n ■ ^
(c) arc tan , =rsinz + --r^ 8in2z + --')-^iiin3z+...,
' 1 - }• cos s 2 3
, ,, I , 2r sin z . 1 o • „ 1 r • ,-
{a}/ arc tan .j ^ = r sin z + -r^ sm 3z + - r^ sm 02 4- ... ,
and shew that, when | r | < 1, they are convergent for all values of z in certain strips
parallel to the real axis in the 2-plane.
2. Expand x'-^ and x in Fourier sine series valid when — tt <x<jr; and hence find
the value of the series
1 • o 1 ■ o 1 • .
sin .r - — , sin 2x + ^, sin Sx - —, sin ix + ...,
2-i ,^' 4^
for all values of x. (Jesus, 1902.)
3. Shew that the function of x represented by 2 n~'^ sin nx sin- na, is constant
j(=i
(0<.^'<2a) and zero {2a<x <tv), and draw a graph of the function.
(Pembroke, 1907.)
4. Find the cosine series re})rosenting f {x) where
_/' {x) — ii\nx-\- cos X {0<x-^\n) ■
/ (.r) = (siiK-i? — cos 07 {\TT^x<ir). (Peterhouse, 1906.)
* For a proof of the theorem when f{x) is subject to less stringent restrictions, see
Hobson, Functions of a Real Variable, §§ 492-493. The reader should observe that, although
[^ f\ /'"" f /"°° 1
bra I j exists, the repeated integral I \ I sin ?f (f -a;) ci?/ -/(f) dt does not.
A^x./ -X ./ 0 j _x (,_/ 0 J
FOURIER SERIES
185
5. Shew that
sin Snx sin bnx ' sin Inx . , ,
HinTTX-i — H — H i-... = ^Tr[x],
3 5 7
where [x] denotes + 1 or - 1 according as the integer next inferior to x is even or uneven,
and is zero if x is an integer. (Trinity, 1895.)
6. Shew that the expansions
and
log
log
2 cos ^ X
2 sin X X
= cos o;- - cos 2a; + .j cos ^x.
-COS X— r cos 207-= cos 3a;...
2t o
are valid for all real values of x, except multiples of tt,
7. Obtain the expansion
« ( _ )"» cos tnx
and find the range of values of x for which it is applicable.
8. Prove that, if 0<a?<27r, then
= (cos X + cos 2.r) log ( 2 cos -x\-\--x (sin 1x + sin x) — cos x,
(Trinity, 1898.)
sin X 2 sin 'ix 3 sin 2x _Tr sinh a{iT -x)
aHT2 "^ a2 + 22 "*" a2:r32 "^ "• ~ 2 sinlTa^i^ '
(Trinity, 1895.)
9. Shew that between the values - n and + tt of .r the following expansions hold :
2 . / sin X 2 sin 2x 3 sin 3.^
sin 7nx=- sni tott ~ — „ ~ -^ •> + "52 J ~
/ 1
cos mX=- sin JJITT ( ;;^ — H
TO cos X m cos 2a; m cos 3.r
+
V2m ^ 12 - r)f' 22 - ^2 3^ - m^
gmx^g-mx 2/1 mcoax m COS 2a; m cos 3.*'
10. Let 07 be a real variable between 0 and 1, and let n be an odd number ^ 3.
Shew that
, ,, 1 2*1, 7nit
( — lY=--\ — 2 — tan cos 2?tt7ra7,
if X is not a multiple of - , where s is the greatest integer contained in nx ; but
(Berger.)
^ 1 2 =^^ 1 , mrr
0= -H — 2 - tan cos 27mrx,
11 n m=\ "i '*
if X is an integer multiple of \jn,
11. Shew that the sum of the series
^ + 47r ~ 1 2 m~^ sin § mir cos 2»i7ra7
/« = i
is 1 when 0<x<}^, and when *<.^■<l, and is — 1 when ]^<x<'i.
12. If
(Trinity, 1901.)
shew that, when - l<a-'<l.
ae«» _ =^ a" l'^ {x)
cos47r.t; cos67r.r , ,, ,22»-ijr2''
22n ' 32n
. ^ sin 47r.r sin67r.'«; , ,, ,,
sm27ra;+ ^,„,^- + g.^.^ +. .. = (-)"+!
F2„(^),
^2,.,l(.^•).
2h :
22it^'2» + l
2?J +T!
(Math. Trip. 1896.)
186 THE PROCESSES OF ANALYSIS [CHAP. IX
13. If m is an integer, shew that, for all real values of x,
^ 1.3.5...(2m-l) fl , OT „ , m{m-\)
cos2»»^ = 2 ^ ,~cr^^ 1 o + — TT ^^^ 2^7 + , — -\^, — ~ COS Ax
2.4.6 ...2m [2 m + l (TO+l)(m + 2)
m(m — l)(m — 2) "j
"^ (;n+'l)(m + 2)(m + 3) ^^^ ^-^ + • • • j' '
COS'""- '07
4 2.4.6...(2m-2) fl 2to-1 ^ (2m-l)(2m-3) , )
-^lT3.5..:(2m-l)i2 + 2^;^^"^'^+(2/n+l)(2m + 3)^"'^^+--r
14. A point moves in a straight line with a velocity which is initially u, and which
receives constant increments, each equal to u, at equal intervals r. Prove that the velocity
at any time t after the beginning of the motion is
u ut u ^ I . 2m7rt
- + - + - 2 - sin
2 T -T jn=im T
(Trinity, 1894.)
and that the distance traversed is
ut , . UT Mr * 1 2riint\
2r '12 27r'^M=i»i t
15. If
f(x)= 2 p^^sill(6J^-3).^'-2 2 -^ sin(2«- 1) .r
Z JZ [ . sin hx sin Ix sin 1 \x
+ — jsm^-- ^ + -,^, 112- +•
shew that /( + 0)=/(7r-0)= -Jtt,
and /(J^ + 0)-/(l^-0)=-|7r, /(|^+0)-/(f7r-0)=j7r.
Observing that the last series is
6 * sin^(2w- 1) TT sin (2/1- l):r
^ „!i ' (2^rri)2 '
draw the graph of/(j;). (Math. Trip. 1893.)
16. Shew that, when 0<^<7r,
/ W = -^ ( cos.r-- cos 5.r+^ cos Ix-^^ cos 11^ + ... 1
= sin 2x + ^ sin 4a" + j sin 8a' + ^ sin 10:*;+... ♦
where /(■^) = ;|7r (0<.r<i7r),
/(■*')=0 (l7r<.r<§7r),
f{x)^\iv (§7r<a.-<7r).
Find the sum of each series when .r = 0, ^tt, fw, tt, and for all other values of x.
(Trinity, 1908.)
17. Prove that the locus represented by
X (_)«-i ,
2 — -, — sm nx sm ny — 0
«=] n"
is two systems of lines at right angles, dividing the coordinate plane into squares of
area tt^. (Math. Trip. 1895.)
18. Shew that the equation
2 ^—^ , -^ =0
FOURIER SERIES 187
represents the lines y= ±ni-n^ {m=0, I, 2, ...) together with a set of arcs of ellipses whose
axes are n and 3~* tt, the arcs being placed in squares of area rr^. Draw a diagram of the
locus. (Trinity, 1903.)
19. Shew that, if the point (x, y, z) lies inside the octahedron bounded by the planes
±^±y±2=»r, then
* , , , sin nx sin ny sin nz 1
(Math. Trip. 1904.)
20. Circles of radius a are drawn having their centres at the alternate angular points
of a regular hexagon of side a. Shew that the equation of the trefoil formed by the outer
arcs of the circles can be put in the form
""^ =1 + J^, cosSd-J^ COS ee + K^T.coaQe- ...,
6^3a 2^2.4 5.7 "'8.10
the initial line being taken to pass through the centre of one of the circles.
(Pembroke, 1902.)
21. Draw the graph represented by
r ^ 2m . TT (1 * ( — )"coswTO^1
a
m[2'^„ti 'l-{nmf j'
where m is an integer. (Jesus, 1908.)
22. With each vertex of a regular hexagon of side 2a as centre the arc of a circle of
radius 2a lying within the hexagon is drawn. Shew that the equation of the figure
formed by the six arcs is
2a ^ „=i {6n — l)(6n+l)
the prime vector bisecting a petal. (Trinity, 1905.)
23. Shew that if c>0,
lim f"
' 1 1
e~'^ cot x sin {2n + l) x . dx=^ir tanh ^ cir.
(Trinity, 1894.)
24. Shew that
sin(2» + l)a; dx 1 . , ,
^— ^ — -- . = -7rC0thl.
0 sm.r l+x^ 2
(King's, 1901.)
25. Shew that when - 1 <a;<l and a is real
sin (2n+l) 6 sin {\-\-x)6 6 ^/)_ 1 sinha.r
sin^ a^ + 6^ ~ ~2^ sinh a '
(Math. Trip. 1905.)
lim /""
M-*-00 J 0
lim ["
26. Assuming the possibility of expanding f{x) in a uniformly convergent series of
the form lAfiSinkx, where ^ is a root of the equation ^cosa^ + 6 sina^=0 and the
summation is extended to all positive roots of this equation, determine the constants A^.
(Math. Trip. 1898.)
1 *
27. If /(■^) = H«o+ 2 (a„cos ;ia;+6,isin?«^)
is a Fourier series, shew that, iif{x) satisfies certain general conditions,
4 /"* 1 dt 4 /"^ 1 dt
an=-P I f {t) cos nt tan - t -— , b,^=- I f (t) sin nt tan- t -- .
IT J 0 ' 2, t '"'Jo 2 t
(Beau.)
CHAPTER X
LINEAR DIFFERENTIAL EQUATIONS
lO'l. Linear Differential Equations*. Ordinary points and singular points.
In some of the later chapters of this work, we shall be concerned with the
investigation of extensive and important classes of functions which satisfy
linear differential equations of the second order. Accordingly, it is desirable
that we should now establish some general results concerning solutions of
such differential equations.
The standard form of the linear differential equation of the second order
will be taken to be
dru / ^du . . . , . .
S.+i>«3- + 2W" = 0 (A).
and it will be assumed that there is a domain aS' in which both p (z), q (z) are
analytic except at a finite number of poles.
Any point of >S^ at which p (z), q (z) are both analytic will be called an
ordinary point of the equation ; other points of S will be called singular
'points.
10'2. Solution-f of a differential equation valid in the vicinity of an
ordinary point.
Let h be an ordinary point of the differential equation, and let Si be the
domain formed by a circle of radius r^, whose centre is h, and its interior, the
radius of the circle being such that every point of S^ is a point of 8, and is
an ordinary point of the equation.
Let z be a variable point of S^.
In the equation write u^v exp \- ^\ p(^)d^i , and it becomes
]"
^J b
£ + ^(-)^ = 0 (B),
where /(.) = ^(.) _ ^ ^^ _ i j^(,)|,
* The analysis contained in this chapter is mainly theoretical ; it consists, for the most part,
of existence theorems. It is assumed that the reader has some knowledge of practical methods
of solving differential equations ; these methods are given in works exclusively devoted to the
subject, such as Forsyth, xi Treatise on Differential Equations (1914).
t This method is only applicable to equations of the second order. For a method applicable
to equations of any order, see Forsyth, Differential Equations, vol. iv. (1902), Chap. i.
101, 10-2] LINEAR DIFFERENTIAL EQUATIONS 189
It is easily seen (§ 5*22) that an ordinary point of equation (A) is also
an ordinary point of equation (B).
Now consider the sequence of functions Vniz), analytic in Sb, defined by
the equations
Vq (z) = «„ 4- «! (^ - h),
where ao, a^ are arbitrary constants.
Let M, fihe the upper bounds of \J{z)\ and |vo('2')| in the domain St.
Then at all points of this domain
I Vn (z) I ^ /A^" \z-b {"^/(n !).
For this inequality is true when n = 0; if it is true when n = 0, 1, . , . m — 1,
we have, by taking the path of integration to be a straight line,
Vm (^) I =
{^-z)J(OVm-r{Od^
b
j;^^J'^\^-z\.\j(o\f^Mm-^\^-b'r-.^^^^^
1 r\z-b\
{■m-l)y 'Jo
1
m
< — f^M"^ \z-b l""^.
Therefore, by induction, the inequality holds for all values of n.
Also, since | Vn (z) \ ^ , — when z is in S^ and 2 /Ai¥'V^^'7(^ 0 con-
n ! n=o
00
verges, it follows (§ 3*34) that v{z)= S v„ (z) is a series of analytic functions
n=0
uniformly convergent in Si ; while, from the definition of Vn (z),
^^Vn{z) = -jy{OVn-^(Odt (W=l, 2, 3, ...)
it follows (§ 5-3) that
b
-T-2 Vn (z) = -J{z) Vn-i (z),
d^{z) ^ d%{z) ^ dh!n{z)
dzr dz- H^i dz^
= -J(z)v(z).
Therefore v{z) is a function of z, analytic in S^, ivhich satisfies the
differential equation
d-v(z) T, ^ r s rv
az-
190 THE PROCESSES OF ANALYSIS [CHAP. X
and, from the value obtained for -r- Vn{z), it is evident that
ivhere a^, a^ are arbitrary.
10'21. Uniqueness of the solution.
If there were two analytic solutions of the equation for v, say v^{z) and v^{z)
such that Vi(6) = v.^Q)) = a„ v/{b) = v^ (b) = a^, then, writing w{2) = v^{z) - v^{z),
we should have
Differentiating this equation n-2 times and putting ^^ = 6, we get
w<") (b) + J{b) w<«-^' (b) + n-2 C\ J' (6) w<*-^) (6) + . . . + J"*-^' (6) w (6) = 0.
Putting w = 2, 3, 4, . , . in succession, we see that all the differential coefficients
of w{z) vanish at b ; and so, by Taylor's theorem, w{z) = 0 ; that is to say the
two solutions Vi(z), v^iz) are identical.
Writing u{z) = v(z)exip \-^j p{^)dd,
we infer without difficulty that u{z) is the only analytic solution of (A) such
that u{b) = Ao, u' (b) ^ A^, where
Ao = ao, Ai = ai-^p(b)ao.
Now that we know that a solution of (A) exists which is analytic in S^
and such that u{b), u (b) have the arbitrary values A^, A-^, the simplest
method of obtaining the solution in the form of a Taylor's series is to assume
00
u{z)= 2 An{z — hY, substitute this series in the differential equation and
M = 0
equate coefficients of successive powers oi z — b to zero (| 3"73) to determine
in order the values o{ A^, A^, ... in terms of ^o> A-y.
[Note. In practice, in carrying out this process of substitution, the reader will find
it much more simple to have the equation 'cleai-ed of fractions' rather than in the
canonical form (A) of § 10-1. Thus the equations in examples 1 and 2 below should
be treated in the form in which they stand ; the factors 1 -z'^, (■'-2) (2 — 3) should not be
divided out. The same remark applies to the examples of §§ 10"3, 10'32.]
From the general theory of analytic continuation (§ 5'5) it follows that
the solution obtained is analytic at all points of 8 except at singularities
of the differential equation. The solution however is not, in general,
' analytic throughout 8 ' (| 5'2 cor. 2, footnote), except at these points, as it
may not be one-valued ; i.e. it may not return to the same value when z
describes a circuit surrounding one or more singularities of the equation.
10-21, 10-3] LINEAR DIFFERENTIAL EQUATIONS 191
[The property that the solution of a linear differential equation is analytic
except at singularities of the coefficients of the equation is common to linear
equations of all orders.]
When two particular solutions of an equation of the second order are not
constant multiples of each other, they are said to form a fundamental system.
* Example 1. Shew that the equation
(l-«2)tt"-23%' + |M=0
has the fundamental system of solutions
Determine the general coefficient in each series, and shew that the radius of con-
vergence of each series is 1.
Example 2. Discuss the equation
(2-2)(3-3)?<"-(22-5) ic'-\-2u = Q
in a manner similar to that of example 1.
10'3. Points which are regular for a differential equation.
Suppose that a point c of ^ is such that, although p (z) or q (2) or both
have poles at c, the poles are of such orders that (z — c)p(z), {z — cfq{z) are
analytic at c. Such a point is called a regular point* for the differential
equation. Any poles oi p{z) or of q{z) which are not of this nature are called
irregular points. The reason for making the distinction will become apparent
in the course of this section.
If c be a regular point, the equation may be written f ,
where P{z — c), Q{z — c) are analytic at c; hence, by Taylor's theorem,
P{z-c)=po + Pi{z- c)+p^{z-cy-ir ...
Q{z-c) = q, -\-q^{z-c)+qo{z-cf+ ...,
where po, Pi, ...,5'o, qi, ■■• are constants; and these series converge in the
domain Sc formed by a circle of radius r (centre c) and its interior, where r is
so small that c is the only singular point of the equation which is in Sg.
Let us assume as a formal solution of the equation
u = {z — cY
1 + S an{z - cf
n = l
where a, aj, a2, ... are constants to be determined.
* The name 'regular point' is due to Thome, CrelWg Journal, Bd. lxxv. (1872), p. 266.
Fuchs had previously used the phrase 'point of determinateuess.'
t Frobenius calls this the normal form of the equation.
192 THE PROCESSES OF ANALYSIS [CHAP. X
Substituting in the differential equation (assuming that the term-by-term
differentiations and multiplications of series are legitimate) we get
a((x-l)+ t an(0L + n)(a+n-l){z-c)"]
{z-cy
n=l
+ (z- cY P{z-c).
a-f- X an{oL-^n){z-cy
-\-{z-cYQ{z-c)
l + tan{z-cf
= 0.
Substituting the series for P (z — c), Q(z — c), multiplying out and equating
to zero the coefficients of successive powers of z — c, we obtain the following
sequence of equations :
a, {{a+iy + {po-l)(a. + l) + q,] + ap^ + q^ = 0,
a^ {{a + 2y + (po - 1) (a + 2) + q,} + a, {(a + l)p, + q,] + ap.^ + q,= 0,
iin {(a -I- Tif + {po-l){a. + n) + q,]
n-l
+ 2 an-m {(a + n- m) p^ + qm] + ^Pn + 5n = 0.
The first of these equations, called the indicial equation*, determines two
values (which may, however, be equal) for a. The reader will easily convince
himself that if c were an irregular point, the indicial equation would have
been (at most) of the first degree ; and he will now appreciate the distinction
made between regular and irregular singular points.
Let a = pi, a = p„ be the roots f of the indicial equation
F{a) ^ce + {p,-l)a. + q, = 0;
then the succeeding equations (when a has been chosen) determine a^, a.^, ...,
in order, uniquely, provided that ^(o -|- ri) does not vanish when w = 1, 2, 3, . . . ;
that is to say, if a = pi, that p, is not one of the numbers p-^-\-\, p-^-\-1, ...\
and, if a = p^, that pi is not one of the numbers /32 + 1, P2 + 2,
Hence, if the difference of the exponents is not zero or an integer, it is
always possible to obtain two distinct series which formally satisfy the
equation.
• Example. Shew that, if m is not zero or an integer, the equation
is formally satisfied by two series whose leading terms are
determine the coefficient of the general term in each series, and shew that the series
converge for all values of z.
* The name is due to Cayley, Quarterly Journal, vol. xxi. (1886), p. 326.
t The roots pi, p> of the indicial equation are called the exjjoiients of the differential
equation at the point c.
10-31]
LINEAR DIFFERENTIAL EQUATIONS
193
10*31. Convergence of the eaypansion of § 10*3.
If the exponents p^, p^ are not equal, let p^ be that one whose real part is
not inferior to the real part of the other, and let pi — Pi = 8; then
F{pi + n)=n{s + n).
Now, by § 5*23, we can find a positive number M such that
\pn\< Mr-"", \qn\< Mr-"", \p^Pn + qn I < Mr-"",
where M is independent of w ; it is convenient to take M^l.
Taking a = pj , we see that
„ .iPiPi + qil^
M
M
F{p^ + l)\ r|s + l| r'
since |s + 1 1 ^ 1.
If now we assume | «« I < M^r~^ when w = 1, 2, ... m — 1, we get
ni-l
ttm, —
t a^-t {{pi + m-t)pt + qt]+ piPm + Qr
^ipi + m)
m—l m-1
S I a^_t ! . I Pipt + qt\ + \ piPm + ^m I + S (m-t)\ arn-t \\pt\
7n\s -\- ni
t=i
< ^-^ ,J ..
m'^\\ + sni ^ I
Since 1 1 + sm~'^ | ^ 1, because R (s) is not negative, we get
m + 1
O'm. "^^
2m
J\^m,f,—m ^ J^m^—m
and so, by induction, | a,i | < M^r~'^ for all values oi n.
If the values of the coefficients corresponding to the exponent p^ be
<-h\ (-h, ••■ we should obtain, by a similar induction,
\an\< M'^K'^r-'^,
where k is the upper bound of jl— s|~\ \l—^s^-\ jl— ^s|-^...; this
bound exists when s is not a positive integer.
We have thus obtained two formal series
w, (z) = {z- c)"' 1 + S a„ (2 - cy
w.^(z) = (z- cf'
l+%an'(z-cY
The first, however, is a uniformly convergent series of analytic functions
when \z — c\< rM~\ as is also the second when \z — c\< rM~^K~\ provided
W. M. A. ' 13
194 THE PROCESSES OF ANALYSIS [CHAP. X
in each case that arg (z — c) is restricted in such a way that the series are
one-valued; consequently, the formal substitution of these series into the
left-hand side of the differential equation is justified, and each of the series is
a solution of the equation ; provided always that p^ — p^ is not a positive
integer or zero*.
With this exception, we have therefore obtained a fundamental system of
solutions valid in the vicinity of a regular singular point. And by the theory
of analytic continuation, we see that if all the singularities in >S^ of the equation
are regular points, each member of a pair of fundamental solutions is analytic
at all points of S except at the singularities of the equation, which are branch-
points of the solution.
10*32. Derivation of a second solution in the case when the difference
of the exponents is an integer or zero.
In the case when pi — p2 = s is a positive integer or zero, the solution
W2{z) found in § lO'Sl may break downf or coincide with Wi(z).
If we write u =Wi (z) ^, the equation to determine ^ is
of which the general solution is
= A + Bj(z- c)-Po-^p^ g {z) dz,
where A, B are arbitrary constants and g{z) is analytic throughout the
interior of any circle whose centre is c, which does not contain any singu-
larities of P (2: — c) or singularities or zeros of {z — c)~p^ w^ (z) ; also g (c) = 1.
Let g{z) = l + :ig,,.{z-cy\
Then, if 5 :jfe 0,
l;=A+BJ'i^l+i{z- c)4 (z - c)-^-> dz
= A+B
-l(z- c)- - t' -^ (z - c)"- + g.. log (z - c)
4- 5 -^" {z-cy-
n = s^\ n — s
* If Pi - P2 is a positive integer, k does not exist ; if Pi = p2 , the two solutions are the same.
t The coetlieient «/ may be imletermiuate or it may be infinite ; in the former case 202 (z)
will be a solution containing two arbitrary constants a^' and a/ ; the series of which a/ is a
factor will be a constant multiple of Wi{~).
10-32] LINEAR DIFFERENTIAL EQUATIONS 195
Therefore the general solution of the differential equation, which is
analytic at all points of C (c excepted), is
A w, (z) + B[g,Wi (z) log {z-c) + w (z)] ,
where, by § 2-53, w (z) ==.(z - cyA - - + t hn(z - c)""] ,
the coefficients A„ being constants.
When 5 = 0, the corresponding form of the solution is
00
A Wi (z) + B Wi (z) log (z-c) + {z- cY"- 1 hn(z- cY
L M=i
The statement made at the end of § 1031 is now seen to hold in the
exceptional case when s is zero or a positive integer.
In the special case when g8 = 0, the second solution does not involve
a logarithm.
The solutions obtained, which are valid in the vicinity of a regular point
of the equation, are called regular integrals.
Integrals of an equation valid near a regular point c may be obtained
practically by first obtaining w^ {z), and then determining the coefficients in
CO
a function w^ {z) = X bn(z — c/^+'S by substituting w^ (z) log {z — c) + w^ (z) in
the left-hand side of the equation and equating to zero the coefficients of the
various powers of ^ — c in the resulting expression. An alternative method
due to Frobenius* is given by Forsyth, Treatise on Differential Equations,
pp. 243-258.
» Example 1. Shew that integrals of the equation
d^u I du „
-y~, H -. — m^u = 0
dz^ z dz
regular near 3 = 0 are
, , , , «= m2'*32n /l 1 1\
and wM^ogz-^^^ ^,,^, (^_ + _ + ...+-J.
Verify that these series converge for all values of z.
♦ Example 2. Shew that integrals of the equation
, , , d^u ,^ ^.dtil ^
.(.-l)^ + (2.-l)^ + _«=0
regular near 2=0 are
, , ,, ^ ^ /1.3...27i-l\2 /111 1\
Verify that these series converge when |2|<1 and obtain integrals regular near 2 = 1.
* Crelle, lxxvi. pp. 214-224.
13—2
196 THE PROCESSES OF ANALYSIS [CHAP. X
Example 3. Shew that the hypergeometric equation
is satisfied by the hypergeometric series of § 2-38.
Obtain the complete solution of the equation when c = l.
10'4. Solutions valid for large values of \z\.
Let ^=l/^'i; then a solution of the differential equation is said to be
valid for ' large values of | ^^ | ' if it is valid for sufficiently small values of \zi\;
and it is said that ' the point at infinity is an ordinary (or regular or irregular)
point of the equation ' when the point z^ = 0 is an ordinary (or regular or
irregular) point of the equation when it has been transformed so that z^ is
the independent variable.
Since
d?u
dz^
, . du , . , dhi . (^ ., , [1\] du /1\
we see that the conditions that the point z = cc should be (i) an ordinary
point, (ii) a regular point, are (i) that 2z — z'^p{z), z*q(z) should be analytic
at infinity (§ 5"62) and (ii) that zp (z), z^q (z) should be analytic at infinity.
Example 1. Shew that every point (including infinity) is either an ordinary point or
a regular point for each of the equations
(jL 1L d'iL
0(l-z)^-^ + {c -(« + & + 1)2} -r-ahu = ^,
dz^ ' dz
„, d'^u r^ du ,
where a, h, c, n are constants.
Example 2. Shew that every point except infinity is either an ordinary point or a
regular point for the equation
dH du
dz'' dz
where n is a constant.
Example 3. Shew that the equation
has the two solutions
dz^ dz
1 1,3^ 1 3.4.5.6 1
3' s-i ■'■2.7 25 + 27477:9?"^'
the latter converging when | s i> 1.
10"5. Irregular singularities and confluence.
Near a point which is not a regular point, an equation of the second order
cannot have two regular integrals, for the indicial equation is at most of
the first degree ; there may be one regular integral or there may be none.
We shall see later (e.g. § 16-3) what is the nature of the solution near
10-4-106] LINEAR DIFFERENTIAL EQUATIONS 197
such points in some simple cases. A general investigation of such solutions*
is beyond the scope of this book.
It frequently happens that a differential equation may be derived from
another differential equation by making two or more singularities of the
latter tend to coincidence. Such a limiting process is called conflixence;
and the former equation is called a confluent foi'm of the latter. It will be
seen in § 10"6 that the singularities of the former equation may be of a more
complicated nature than those of the latter.
10"6. The differential equations of mathematical physics.
The most general differential equation of the second order which has
every point except a^, a^, a^, ^4 and 00 as an ordinary point, these five points
being regular points with exponents a^, /3r at ar{r = 1, 2, 3, 4) and exponents
/*], /Lt2 at 00 , may be verified f to be
dz^' ir=i z-CLr ] dz u=i(^ -««•)' U (z - a,) ^
r=l
where A is such that J /Aj and ^2 are the roots of
fju' + fllt (ar + /3r) -S\+l ttr^r + A = 0,
[r=l } r = l
and B, G are constants.
The remarkable theorem has been proved by Klein § and B6cher|| that
all the linear differential equations which occur in certain branches of
Mathematical Physics are confluent forms of the special equation of this
type in which the difference of the two exponents at each singularity is \ ;
a brief investigation of these forms will now be given.
If we put /S,. = a,. + I, (r = 1, 2, 3, 4) and write ^ in place of z, the last
written equation becomes
dhi { 4 ^ - 2a,| du ( I a,K + i) AC' + 2B^-^G\ ^
r=l
^* Some elementary investigations are given in Forsyth's Differential Equations (1914).
Complete investigations are given in his Theory of Diff'erential Equations, Vol. iv.
t The coeflELcients of — and u must be rational or they would have an essential singularity
4 4
at some point; the denominators of p(z),q{z) must be IT (2 -a,.), II (2 - a,.)- respectively;
putting p (z) and q{z) into partial fractions and remembering that j) {^) = 0{z~^),q{z) = 0{z~-)
as ] J I -*■ 00 , the required result follows without difficulty.
4
X It will be observed that fj-i, 1J.2 are connected by the relation-/ij + /x2+ 2 (0^4-/3,.)= 3.
r=l
§ Ueber lineare Differentialgleichungen der zweiter Ordnung, p. 40; see also Vorlesuny
ilher Lame'schen Funktionen.
IS Ueber die Reihenentwickeluugen der Potentialtheorie, p. 193.
198
THE PEOCESSES OF ANALYSIS
[chap. X
where (on account of the condition /ij — /ij = |)
This differential equation is called the generalised Lam6 equation.
It is evident, on writing a^^a^ throughout the equation, that the
confluence of the two singularities a^, a^ yields a singularity at which the
exponents a, ^ are given by the equations
a + y8 = 2 (ai + a^), a/3 = Wj («! + i) + ^2 (a2 + i) + D,
where D = (^ai^ + 'IBa^ + G)l[(a^ - as) (a^ - a^)].
Therefore the exponent-difference at the confluent singularity is not ^,
but it may have any assigned value by suitable choice of B and C. In like
manner, by the confluence of three or more singularities, we can obtain
one irregular singularity.
By suitable confluences of the live singularities at our disposal, we can
obtain six types of equations, which may be classified according to (a) the
number of their singularities with exponent-difference ^, (b) the number of
their other regular singularities, (c) the number of their irregular singu-
larities, by means of the following scheme, which is easily seen to be
exhaustive* :
(a)
(&)
{c)
(I)
3
1
0
Lam^
(11)
2
0
1
Mathieu
(III)
1
2
0
Legendre
(IV)
0
1
1
Bessel
(V)
1
0
1
Weber, Hermite
(VI)
0
0
1
Stokes t
These equations are usually known by the names of the mathematicians
in the last column. Speaking generally, the later an equation comes in
this scheme, the more simple are the properties of its solution. The
solutions of (II) — (VI) are discussed | in Chapters XV — Xix of this work.
The derivation of the standard forms of the equations from the generalised
Lame equation is indicated by the following examples :
* For instance the arrangement (a) 3, (h) 0, (c) 1 is inadmissible as it would necessitate six
initial singularities.
t The equation of this type was considered by Stokes in his researches on Diffraction
(Collected Papers, ii.); it is, however, easily transformed into a particular case of Bessel's
equation (example 6, below).
X For properties of equations of type (I), the reader is referred to the works of Klein and
Forsyth cited at the end of this chapter ; also to Todhunter, The Functions of Laplace, Lame
and Bessel.
106] LINEAR DIFFERENTIAL EQUATIONS 199
• Example 1. Obtain Lamp's equation
(where h and n are constants) by taking
01 = 02=03 = 04 = 0, 85 = /i(n + l)a4, ^G=ha^^
and making a4 -^ » ,
• Example 2. Obtain the equation
dhi .(ij , i \ c?« , (a-16y + 32g0w_
^f' Vc f-i/^C 4C(f-l)
(where a and ^ are constants) by taking ai = 0, a2=l> ^^^^ making a3 = a4-*oo. Derive
Mathieu's equation (§ 19"1)
T-^ + (a + 1 6^ cos 22) 24 = 0
by the substitution ^^cos^ z.
Example 3. Obtain the equation
by taking
«i = «?=0, a3=a^=l, 01 = 02=03=0, 04 = J.
Dei'ive Legendre's equation (§§ 15*13, 15'5)
by the substitution (=z~'^.
' Example 4. By taking ai = a2=0, 01 = 02 = 03=04 = 0, and making a3=a4-*-x, obtain
the equation
Derive Bessel's equation (§ 17"11)
,^ _
dz'^ dz
od^a du , „ „,
z^ -r^ + z -^ +{z^ - n^) u = 0
by the substitution f =2^.
^ Example 5. By taking ai = 0, 01 = 02 = 03 = 04 = 0, and making a2==«3 = '^4^ ^ 1 obtain
the equation
Derive Weber's equation (§ 165)
dz'^
by the substitution f =2^.
^ + (.i + i-i22)« = 0
Example 6. By taking 0^ = 0, and making a^.^'cc {r = l, 2, 3, 4), obtain the equation
By taking
dh/
u=^{B,C+C,)^v, Z?iC+Ci = (5A)*4
shew that
,,d^v dv , ., ^.
''d?^'dz^^':-^^'=''-
200 THE PROCESSES OF ANALYSIS [CHAJ*. X
Example 7. Shew that the general form of the generalised Lamd equation is un-
altered (i) by any homographic change of independent variable such that oc is a singular
point of the transformed equation, (ii) by any change of dependent variable of the type
u = {z — a,.)^ V.
Example 8. Deduce from example 7 that the various confluent forms of the
generalised Lam^ equation may always be reduced to the forms given in examples 1-6.
[Note that a suitable homographic change of variable will transform any three distinct
points into the points 0, 1, c» .]
10'7. Linear differential equations with three singularities.
Let -^^+p(,)- + q(,)u = 0
have three, and only three singularities *, a, b, c ; let these points be regular
points, the exponents thereat being a, a' ; ^, l3' ; y, <y'.
Then p {z) is a rational function with simple poles at a, h, c, its residues at
these poles being 1 — a — a', 1 — /3 — yS', 1 — 7 — 7' ; and as ^^ -* 00 , p{z) — 2z~^
is 0(0- Therefore
z — a z — 0 z — c
andf a + a'+yS + )S' + 7 + 7' = 1.
In a similar manner
g (A = K(«-fe)(o^-c) ^ ^^'(b-c)(b-a) _^ yy'{c-a)(c-b)]
\ z — a z — b z — c j
{z - a){z — b) {z — c)'
and hence the differential equation is
^ + [!-«-« I- ^-^' 1 - 7 - 7 ) dM
dz^ \ z - a z - b z — c ] dz
+ |««^(«-^)(ft-c) ^ I3^'{b-c){b-a) ^ yy'{c-a){c-b)]
( z — a z — b z — c J
u
(z — a){z — b) {z — c)
This equation was first given by PapperitzJ.
To express the fact that u satisfies an equation of this type (which will be
called Riemann's P-equation), Riemann§ wrote
{a b c \
«' ^' 7 J
* The point at infinity is to be an ordinary point.
t This relation must be satisfied by the exponents.
X Math. Ann. xxv. (1885), p. 213.
§ Abhandlungen d. K. Gesell. d. Wiss. zu Gottingen, vii. (1857). It will be seen from this
memoir that, although Eiemann did not apparently construct the equation, he must have inferred
its existence from the hypergeometric equation.
107, 10*71] LINEAR DIFFERENTIAL EQUATIONS 201
The singular points of the equation are placed in the first row with the
corresponding exponents directly beneath them, and the independent variable
is placed in the fourth column.
4 Example. Shew that the hypergeometric equation
,, . d^u , , , ,. ,du
z{l-z)^, + {c-{a + b+l)z}-^^
is defined by the scheme
10"71. Transformations of Riemanns P-equation.
The two transformations which are typified by the equations
(a b c \ (a h c
(I) (t^J (t-7J ^ « ^ 7 ^ =P «+A; ^-k-l y+l A,
[a ^' y J [o! + k ^'-k-l y+l J
(a b c
a /3 7
(where z-^, a^, b^, c^ are derived from z, a, b, c by the same homographic
transformation) are of great importance. They may be derived by direct
transformation of the differential equation of Papperitz and Riemann by
suitable changes in the dependent and independent variables respectively ;
but the truth of the results of the transformations may be seen intuitively
when we consider that Riemann's P-equation is determined uniquely by a
knowledge of the three singularities and their exponents, and (I) that if
I* /% — r\^
then «i = f 7 j ( 7 J XI satisfies a differential equation of the second
order with the same three singular points and exponents a 4- h, a' + A' ;
^ — k — I, ^' — k — I; y + I, y' + I ', and that the sum of the exponents is 1.
Also (II) if we write z = j^ ^^ , the equation in z^ is a linear equation
of the second order with singularities at the points derived from a, b, c by this
homogi-aphic transformation, and exponents a, a ; /y, l3' ; y, y thereat.
202 THE PROCESSES OF ANALYSIS [CHAP. X
10*72. The connexion of Riemanns P-equation with the hypergeometric
equation.
By means of the results of | 10'71 it follows that
(ahc^ (a h '^l
Pa /. , 4 = (:-^:nH7P O ^^a^, , ^\
[a! /3' y J [oi'-OL /3' + a + 7 7-7 J
r 0 00 1 \
[oL —a ^' + oi + <y 7' - 7 J
where x = -. r^— ^ .
{z — b){c — a)
Hence, by § 10'7 example, the solution of Riemann's P-equation can
always be obtained in terms of the solution of the hypergeometric equation
whose elements a, 6, c, a? are a. -\- B -^ <y, a-\- B' + <y, 1 + a — a', )- yf^ ~
' r- /, ' (^z — b) {c — a)
respectively.
10"8. Linear differential equations with two singularities.
If, in § 10*7, we make the point c a regular point, we must have
1 /A ' n . aoi'(a-b)(a-c) /3/3' (b - c) (b - a) ^ ,
1—7-7=0, 77 =0 and — ^ — + '-^-— ^ ^-y^ ^ must be
z — a z — 0
divisible hy z — c, in order that p {z) and q {z) may be analytic at c.
Hence a + a' + /S + /3' = 0, aa' = ySyS', and the equation is
dru il-a-a! 1 -f- « + g'] da aa {a - b)" u _
dz' ^ I z'- a ^ z-b I d^ "•■ {z-af(7^'by ~ '
of which the solution is
. fz — aY ^ (z — a
u = A\ J + B
^z — b) \z — b/
that is to say, the solution involves elementary functions only.
When a = a', the solution is
REFERENCES.
L. FucHs, Crelle, lxvi.
L. W. Thom^, Crelle, Lxxv, Lxxxvii.
L. ScHLESiNGER, ffandhuch der linearen Differentialgleichungen.
G. Frobknius, Crelle, Lxxvi.
G. F. B. RiEMANN, Ges. Math. Werke, pp. 67-87.
F. C. Klein, Ueher lineare Differentialgleichungen der zweiter Ordnung.
A. R. Forsyth, Theori/ of Differeiitial Equations, Part iii. (Vol. iv.).
T. Craig, Differential Equations.
10-72] LINEAR DIFFERENTIAL EQUATIONS 203
Miscellaneous Examples.
1. Shew that two solutions of the equation
are « - j^ «* + ... , 1 - J2^ + . . ., and investigate the region of convergence of these series.
2. Obtain integrals of the equation
regular near z=0, in the form
16'
U2 = Ui log Z-^ + ...
3. Shew that the equation
has the solutions
0?2tt / 1 1
^. + (^ + 2-4^'^^ = ^
'' l2~^+ 80 ""'"•••'
and that these series converge for all values of z.
4. Shew that the equation
-,-5 + ^2 ^—^7 J- + 1 2 7-^T2 + 2 ^ V ?< = 0,
az^ [r=i Z—ttj. ) az (.r=l (■2 — «r) r=l ^ - (^r)
where
2 (a^+/3r) = n-2, 2 i)r = 0, 2 (a,.Z>^ + a,/3,.) = 0, 2 (cf^2^r + 2a,.a^/3^) = 0,
r=l r=l r=l r=l
is the most general equation for which all points (including qc ), except ai, ag) ••• ^») ^^^
ordinary points, and the points a^ are regular points with exponents a,., /S^ respectively.
(Klein.)
5. Shew that, if ^ + y + /3' + y' = ^ , then
pJo ^ y s^l^pjy 2/3 y zi. (Riemann.)
U ^' y ) [ i 2^' i J
[The diflferential equation in each case is
6. Shew that if y + y' = i and if w, w^ are the complex cube roots of unity, then
ro 00 1 \ ri CO 0)2 ^
PJo 0 y 3n = P-|y y y sj. (Riemann.)
IJ \ y J
[The differential equation in each case is
cPu 2z^ du Qyy' zu
'd^^W~\ ^5 ■*■ (sS'^ 1)2
204
THE PROCESSES OF ANALYSIS
[chap. X
7. Shew that the equation
a-z')'^,-{'-<' + ^)^'^,+nin + 2a)u = 0
is defined by the scheme
and that the equation
{l+CV^, + n{n + 2)u=0
may be obtained from it by taking a=l and changing the independent variable.
(Halm.)
8. Discuss the solutions of the equation
(Cunningham.)
valid near z = 0 and those valid near 2 = oo .
9. Discuss the solutions of the equation
d^u 2u die ^ du ^ , . ^
^ + 75^-2^^. + '^"-^)"="
valid near 2 = 0 and those valid near 2 = x .
Consider the following special cases :
(i) /^=-i-) (ii) M = i, (iii) ^ + " = 3.
10. Prove that the equation
,, , d^u 1 _ ^ . dii , ,,
has two particular integrals the product of which is a single-valued transcendental
function. Under what circumstances are these two particular integrals coincident ?
If their product be F [z)^ prove that the particular integrals are
(Curzon.)
'1, u-2 = '^'F{z) . exp \±C \
dz
F{z)\'z{\-z)
where C is a determinate constant.
(Lindemann ; see § 19'5.)
11. Prove that the general linear difierential equation of the third order whose
singularities are 0, 1, 00 , which has all its integrals regular near each singularity (the
exponents at each singularity being 1, 1, - 1) is
(Pu f 2 ^ I d'hi _ J 1 3 1 \ du
dz^ ■•" Xz "*" ^1 j ~cP (_7^ V{z- 1) + '{z^^\ 'dz
j\__ 3 cos'^g _ 3sih^a 1
,2(,_1) ,(2_1)2 ■ (,_1);
where a may have any constant value.
u = 0,
(Math. Trip. 1912.)
CHAPTER XI
INTEGRAL EQUATIONS
ll'l. Definition of an integral equation.
An integral equation is one which involves an unknown function under
the sign of integration ; and the process of determining the unknown function
is called solving the equation*.
The introduction of integral equations into analysis is due to Laplace
(1782) who considered the equations
f{x) = je-' </> (t) dt, g {x) ^jt^-' <f> (t) dt
(where in each case ^ represents the unknown function), in connexion with
the solution of differential equations. The first integral equation of which
a solution was obtained, was Fourier's equation
f{x) = I cos {xt) ^{t) dt,
J - OD
of which, in certain circumstances, a solution isf
2 f^
(f) {x) = — I cos (ux)f{u) du,
f{x) being an even function of x, since cos {xt) is an even function.
Later, Abel was led to an integral equation in connexion with a mechanical
problem and obtained two solutions of it ; after this, Liouville investigated an
integral equation which arose in the course of his researches on differential
equations and discovered an important method for solving integral equations,
which will be discussed in § 11 '4.
In recent years, the subject of integral equations has become of some
importance in various branches of Mathematics ; such equations (in physical
problems) frequently involve repeated integrals and the investigation of them
naturally presents greater difficulties than do those elementary equations
which will be treated in this chapter.
To render the analysis as easy as possible, we shall suppose throughout
that the constants a, h and the variables x, y, | are real and further that
* Except in the case of Fourier's integral (§ 9-7) we practically always need continuous
solutions of integral equations.
t If this value of 0 be substituted in the equation we obtain a result which is, effectively, that
of §9-7.
206
THE PROCESSES OF ANALYSIS
[chap. XI
a^x,y, ^^b; also that the given function* K(w, y), which occurs under the
integral sign in the majority of equations considered, is a real function of
X and y and either (i) it is a continuous function of both variables in the
range {a^x-^h, a^y ^h), or (ii) it is a continuous function of both variables
in the range a^y ^x ^b and K (x, y) = 0 when y > x; in the latter case
K(x, y) has its discontinuities regularly distributed, and in either case it is
easily proved that, i(f(y) is continuous when a^y ^b,
continuous function of x when a<.x ^b.
f{y) K(x,y)dy is &
ll'll. An algebraical lemma.
The algebraical result which will now be obtained is of great importance in Fredholm's
theory of integral equations.
Let {xi, yi, 2i), (^2) y2> ■^2)) (■^3> y^i ■23) be three points at unit distance from the origin.
The greatest (numerical) value of the volume of the parallelepiped, of which the lines
joining the origin to these points are conterminous edges, is +1, the edges then being
perpendicular. Therefore, if Xy?+yr^ + z^?=\ (r = l, 2, 3), the upper and lower bounds of
the determinant
^1
yi
Z\
y-i
z%
yz
zz
*'3
are ±1.
A lemma due to Hadamardt generalises this result.
Let
«ii,
^12) ••• %jj
0!22, ... a^n
(t-nli
= A
where a,„,. is real and 2 a\i,.= 1 (m = l, 2, ... w) ; let A^^ be the cofactor of «,„,. in D and
let A be the determinant whose elements are -4,„,., so that, by a well-known theorem J,
Since 2) is a continuous function of its elements, and is obviously bounded, the
ordinary theory of maxima and minima is applicable, and if we consider variations in
n 92>
air(r=l, 2, ... n) only, D is stationary for such variations if 2 ^ — 8ai,. = 0, where Saj,., ...
n
are variations subject to the sole condition 2 ax,.Sai,. = 0; therefore §
n
but 2 airAyr=D, and so \2d\T.-= D ; therefore ^ir = Z)aj,..
r=\
* Bocher in his important work on integral equations {Caiitb. Math. Tracts, No. 10), always
considers the more general case in which K{x, ?/) has discontinuities regularly distributed,
i.e. the discontinuities are of the nature described in Chapter iv, example 16. The reader will
see from that example that the results of this chapter can almost all be generalised in this
way. To make this chapter more simple we shall not consider such generalisations.
t Bulletin des Sciences Math. ser. 2, t. xvii.
X Burnside and Panton, Theonj of Equations, Vol. 11. p. 40.
§ By the ordinary theory of undetermined multipliers.
1111, 112] INTEGRAL EQUATIONS 207
Considering variations in the other elements of Z), we see that D is stationary for
variations in all elements when Amr = Dumr (w=l, 2, ... n ; r=l, 2, ... n). Consequently
A=D^ . D, and so Z)" ■*" ' = Z)"~ ^ Hence the maximum and minimum values of D are + 1.
Corollary. If a^^ be real and subject only to the condition | a^^ | <My since
r=l
we easily see that the maximum value of | Z) | is {n^ MY = n^^ M^.
11*2. Fredholms* equation and its tentative solution.
An important integral equation of a general type is
J a
where f{x) is a given continuous function, X is a parameter (in general
complex) and K{x, |) is subject to the conditionsf laid down in § 111.
K {x, ^) is called the nucleusl of the equation.
This integral equation is known as Fredholm's equation or the integral
equation of the second kind (see § 11 "S). It was observed by Volterra that an
equation of this type could be regarded as a limiting form of a system
of linear equations. Fredholm's investigation involved the tentative carrying
out of a similar limiting process, and justifying it by the reasoning given
below in § 11'21. Hilbert {Gott. Nach. 1904) justified the limiting process
directly.
We now proceed to write down the system of linear equations in question,
and shall then investigate Fredholm's method of justifying the passage to
the limit.
The integral equation is the limiting form (when S-*-0) of the equation
n
^ (^) =/ (■*■) + >*■ 2 K{x, x^ (f) (Xq) 8,
9=1
where Xg - a', _ i = S, XQ = a, x^—h.
Since this equation is to be true when a^x^h, it is true when x takes the values
Xi, X2, ... x^; and so
- XS 2 K{xp, Xg) 4> {Xg)+(}i (Xj,) =f{Xp) (p= 1, 2, ... n).
9=1
* Fredholm's first paper on the subject appeared in the Proceedings of the Sicedish Academy,
Vol. Lvii. (1900). His researches are also given in Acta Mathematica, Vol. xxvii. (1903).
+ The reader will observe that if K (x, ^) — 0{^> x), the equation may be written
<p{x)=f{x)+\ [^ K{x,^)4,i^)d^.
This is called an equation with variable upper limit.
J Another term is kernel ; French noyau, German Kern.
208
THE PROCESSES OF ANALYSIS
[chap. XI
This system of equations for (\){Xp), {p = \,2, ...n) has a unique solution if the
determinant formed by the coefficients of ^ (^'p) does not vanish. This determinant is
\-\bK{xx,Xi) -\bK{x-^, X2) ... -X8K{xi,Xn)
— \8K{x2, Xi) 1 -\8K {x2, X2) ... —X8K{x2, x^
I>nW-
-X8K {Xn, Xj) —\8K (Xn ,X2) ... l — X8K{Xn, Xn)
A. (Xpj Xp) A {^Xpj Xqj
:1-X 2 8K{xp,Xp) + ^^ 2 82
X3 n
K {Xq, Xp) K \Xq, Xq)
li. \Xp, Xp) A [Xpi Xq) A \Xp, Xy.)
K{Xq,Xp) K(Xq,Xq) K{Xq,Xr)
on expanding* in powers of X.
Making S -*- 0, % -»- qo , and writing the summations as integrations, we are thus led to
consider the series
X2 Cb fb
i)(X) = l-X j
K{^u^i)d^i +
2!
d^i d^2 -
Further, if Dn{x^, Xv) is the cofactor of the term in Z>„(X) which involves K{xv, a>),
the solution of the system of linear equations is
(p (Xf^) =
Z>„(X)
Now it is easily seen that the appropriate limitingTform to be considered in association
with D„ (.i>, x^) is D (X) ; also that, if /i + i/,
D, (.r^, x,)^\8 \ K{x^, .Vy) - XS 2
1
" ■ V, '7 = 1
A \X^^ Xv) A (•^fi, Xp)
A \Xp^ Xv) A \Xpj Xp)
A (.^'(X) Xv) A \Xp_^ Xp) A (Xfj^^ Xq)
A (Xp^ Xv) A \Xp^ Xp) A (Xp^ Xq)
A (Xq^ Xp) A (Xqj Xp) A (Xq^ Xq)
So that the limiting form for 8-ii>(.x>, Xv) to be considered t is
K{xp.,Xv) K{xp.,^^) j^^
A' (.^^ , ^i,) K (xp. , ^1 ) A' (.»> , I2)
i> (.^> , Xv ; X) = X A (% , .r^) - X^ I
^:-/:/:
(i^l rf^2
A(|,,^.) A(ii,^,) a: (^1,^2)
A(6,^.) A(^„^,) a: (^2,12)
Consequently we are led to consider the possibility of the equation
ct> (^)=/(^) +^ fy (•^■> ^ ; ^)/(^) ^^>
giving the solution of the intcgnxl equation.
* The factorials appear because each determinant of .s rows and columns occurs s I times as
p, q, ... take all the vakies 1, 2, ... «, whereas it appears only once in the original deter-
minant for I>„(X).
t The law of formation of successive terms is obvious from those written down.
11-21] INTEGRAL EQUATIONS 209
Example 1. Shew that, in the case of the equation
<f){x) = x + \l xy4){2i)dy,
we have
D{\) = \-\\, D{x,y; \) = \xy
and a solution is
Example 2. Shew that, in the case of the equation
(\>{x) = x->r\\ {xy + ?/2) 0 (^) dy,
we have
D{x,y; \) = 'k{xy+y'^)+\Wxy-lxy-ly^ + ky),
and obtain a sohition of the equation.
11 21. Investigation of Fredholms solution.
So far the construction of the solution has been purely tentative ; we nov^^
start ah initio and verify that we actually do get a solution of the equation ;
to do this we consider the two functions D(\), B{x, y; X) arrived at in § 11-2.
We write the series, by which D (X) was defined in § 11 "2, in the form
1+ 2 ^ so that
since K (x, y) is continuous and therefore bounded, we have \K{x, y)\< M,
where M is independent of x and y] since K{x, y) is real, we may employ
Hadamard's lemma (§ ll'll) and we see at once that
\an\ <'n- M .{o — a) .
Write 71* il/" (b - a)'' = n ! 6„ ; then
n^* «^x (71+ 1)5 (V Wy J
f-, '^y
since I 1 + - 1 ^- e.
The series 2 6„A," is therefore absolutely convergent for all values of \ ;
n = l
and so (§ 2'34) the series 1 + S ^' converges for all values of X and there-
71=1 nl
fore (§ 5"64) represents an integral function of \.
Now write the series for D (x, y:\) in the form 2 '^ ' . .
W. M. A. 14
210
THE PROCESSES OF ANALYSIS
[chap. XI
Then, by Hadamard's lemma (§ ll'll),
W-dx,y)\<n'^''M\h-af-\
Vn (^, y)
and hence "J^^^-L^ < c„, where c„ is independent of x and y and 2 c„X«+i is
n ! «=o
absolutely convergent.
Therefore I){x, y; \) is an integral function of X and the series for
D{x,y;\)- \K{x, y) is a uniformly convergent (§ 3-34) series of continuous*
functions of x and y when a^x^b, a^y^h.
Now pick out the coefficient of K {x, y) in D{x,y;\); and we get
D(x,y;\) = \D (\) J^ (x, y)+t (-)-X"+^
Qn («, y)
where
rb rb rb
Qn{x,y)=\ ...
■I a J a J a
0, K{x, fo, ir(^-,|2X...^(^,^n)
d^i...d^n-
Expanding in minors of the first column, we get Qn {oo, y) equal to the
integral of the sum of n determinants; writing ^i, ^^, ... ^m-i, ^, ^m, ••■ fn-i
in place of |i, fg, ••• ^n in the mth of them, we see that the integrals of all
the determinants f are equal and so
where
Q, (x, y) = -nr ('...['k (f, y) Pnd^, . . . d^n-i,
J a J a J a
P„ =
K{x,^), K(x,^,),
■ ■ Kix, ^n-i)
It follows at once that
D{x,y; \)^\D{X)K{x, y) + \ j' D {x, |; \) K {^, y)d^.
J a
Now take the equation
<^{^)=f{^) + ^\' K{ly)<l>{y)dy,
multiply by D{x, f ; X) and integrate, and we get
= rct>(^)D(x, f ; \)d^-xt' r D(x, ^; X)K(^, y)ct>(y)dyd^, ,
•/a J a J a
the integrations in the repeated integral being in either order.
* It is easy to verify that every term (except possibly the first) of the series for D {x, y ; X)
is a continuous function under either hypothesis (i) or hypothesis (ii) of § 11-1.
t The order of integration is immaterial (§ 4-3).
11'21] INTEGRAL EQUATIONS 211
That is to say
J a
= f<f> (f ) D (x, ^; X) d^ - f * {D (X, y;\)-\D (X) K{w, y)\ c}> (y) dy
J a J a
^\D{\)\'K{x,y)4>{y)dy
■I a
=^D{\)[<\>{x)-f{x)],
in virtue of the given equation.
Therefore if D (X) 4= 0 and if Fredholm's equation has a solution it can be
none other than
^(.)=/(.,+/V(f)^-§lp<^f;
D(\)
and, by actual substitution of this value of ^{x) in the integral equation,
we see that it actually is a solution.
This is, therefore, the unique continuous solution of the equation if
D (X) 4= 0.
Corollary. If we put f{x) = 0, the 'homogeneous' equation
J a
has no continuous solution except (f)(x) = 0 unless Z>(X) = 0.
Example 1. By expanding the determinant involved in Q„ {x, y) in minors of its first
row, shew that
D{x,y;\) = \D{\)K{x,y)^\f' K{x,^)D{^,y;\)d^.
J a
Example 2. By using the formulae
D{\) = 1+ I ^^-, D{x,y;\) = \D{\)K{x,y)+ I ( - )» ^^'^li^ii^'-^) ,
shew that / D{^,^;\) d^= -\
J «
dD{\)
dX
Examples. If K{x,y) = l (y^x), K{x,y) = 0 {y>x),
shew that i> (X) = 1 - (6 - a) X.
Example 4. Shew that, if K {x, y) =/i {x) . f-i (y ), and if
/ f\{x)h{x)dx==A,
.' a
then
i>(X) = l-^X, D(x,y;\) = \f,{x)f,{y),
and the solution of the corresponding integral equation is
<t> (■■^)=/(^-)+|4-jx jj^^^^' ^^^ '^^■
U— 2
212 THE PROCESSES OF ANALYSIS [CHAP. XI
Example 5. Shew that if
K {x, y) =/i {x) gi {y) +/2 {x) g<i, (y),
then D (X) and D{x,y; X) are quadratic in X ; and, more generally, if
n
K{x,y)= 2 fm{x)gm{y\
OT=1
then D (X) and D {x, y, X) are polynomials of degree n in X.
11*22. Volterra's reciprocal functions.
Two functions K {x, y), k{x,y; X) are said to be reciprocal if they are
bounded in the ranges a-^x, y ^h, if any discontinuities they may have are
regularly distributed (§ ll'l, footnote), and if
K{x,y) + k{x,y;\) = \\ k{x, ^•,\)K{ly)dl
•I a
We observe that, since the right-hand side is continuous* the sum of two reciprocal
functions is continuous.
Also, a function K{x,y) can only have one reciprocal if Z)(X)=t=0; for if there were two,
their difference k^ {x, y) would be a continuous solution of the homogeneous equation
h 0^, 3/ ; ^) = ^ \ J^i {x, ^•,~^)K (I, y) d^,
(where x is to be regarded as a parameter), and by § 11-21 corollary, the only continuous
solution of this equation is zero.
By the use of reciprocal functions, Volterra has obtained an elegant
reciprocal relation between pairs of equations of Fredholm's type.
We first observe, from the relation
D{x,y; \) = \D(\)K(x,y) + \l D{x,^; \)K{ly)d^,
J a
proved in § 11 "21 that the value of k (x, y;X) is
-D{x,y;\)/\\D{X)},
and from § 11-21 example 1, the equation
k(x,y;\) + K{x,y)^xrK{x,^)k(^,y;\)d^
J a
is evidently true.
Then, if we take the integral equation
cf>(x)=f(x) + xrK{x, B<i>{^)dl
J a
when a-^x^ h, we have, on multiplying the equation
J a
* By example 10 at the end of Chapter iv.
11-22, 11-23] INTEGRAL EQUATIONS 213
by k{x, ^ ; \) and integrating,
■' a
= fkix, ^;\)f{^)d^ + \f' \'k{x, ^; X)K{1 ^,) <\> {^d d^.d^
J a J a J a
Reversing the order of integration* in the repeated integral and making
use of the relation defining reciprocal functions, we get
\''k{x,^;\)<f>{^)d^
J a
= I'k {X, ^ ; X)/(|) d^ + r [K {X, ^0 + k{x,^; \)] </, (f,) d^„
J a J a
and so \ f^k (x, ^ ; X.)/(|) d^=-\l^ K {x, ^,) 0 (f ,) d^,
J a J —a
= -<t>(x)+f{x).
Hence f(x) = ^ {x) + X ['k{x, ^; X)/(^) d^;
J a
similarly, from this equation we can derive the equation
cl>{x)=f(x) + xfK{x,^)cf>{^)dl
J a
so that either of these equations with reciprocal nuclei may be regarded as
the solution of the other.
11-23. Homogeneous integral equations.
The equation (ji(x) = X I K (x, ^) (f) (^) d^ is called a homogeneous integral
■J a
equation. We have seen (§11-21 corollary) that the only continuous solution
of the homogeneous equation, when D (X) 4= 0, is 0 (x) = 0.
The roots of the equation D (X) = 0 are therefore of considerable
importance in the theory of the integral equation. They are called the
characteristic numbers f of the nucleus.
It will now be shewn that, when D (X) = 0, a solution which is not
identically zero can be obtained.
Let:J: X = Xo be a root m times repeated of the equation D (X) = 0.
Since D (X) is an integral function, we may expand it into the convergent
series
D (X) = c,n (X - Xor + c„,+, (X - Xo)'«+' + . . . {m>0, c,n + 0).
* The reader will have no difficulty in extending the result of § 4-3 to the integral under
consideration.
t French valeurs caracteristiquen, German Eigenwerthe.
X It will be proved in § 11-51 that, if K {x,ij)=K{y,x), the equation D (\) — 0 has at least one
root.
214 THE PROCESSES OF ANALYSIS [CHAP. XI
Similarly, since D {x, y; X) is an integral function of \ there exists
a Taylor series of the form
by § 3 '34 it is easily verified that the series defining gn{x, y), {n = l, l-\-\, ...)
converge absolutely and uniformly when a^x ^.h, a^y ^b, and thence that
the series for D{x, y; X) converges absolutely and uniformly in the same
domain of values of x and y.
But, by § 11*21 example 2,
J a
dX
now the right-hand side has a zero of order w — 1 at X.O) while the left-hand
side has a zero of order at least I, and so we have m — l^L
Substituting the series just given for D (\) and D {x, y;\)\n the result of
§ 11 '21 example 2, viz.
D{x,y;\) = \D(X)K(x, y) + xf'K(x, ^)D{1 y; X)d^,
J a
dividing by (X — X^f and making X ^ Xq. we get
gi («> y) = \ K (x, f ) gi (^, y) d^.
J a
Hence if y have any constant value, gi(x, y) satisfies the homogeneous
integral equation, and any linear combination of such solutions, obtained by
giving y various values, is a solution.
Corollary. The equation
(^(^)=/(^)-f-Xo fV(^,^)</,(^)rf^
J a
has no sohition or an infinite number. For if 0 {x) is a solution, so is ^ {x) + 2 CyQi {x, y),
where Cy may be any function of y.
Example 1. Shew that solutions of
4>{x) = \ r cos" {X~^)(li (I) d^
are ^ (.2^) = cos (n - 2r) .r, and (ii{x)=iim{n-2r)x ; where r assumes all positive integral
values (zero included) not exceeding hi.
Example 2. Shew that
<^ (:r) = X J ^ ^ cos" {x + ^)ct>(^) di
has the same solutions as those given in example 1, and shew that the corresponding
values of X give all the roots of Z)(X) = 0.
11-3-1 1-4] INTEGRAL EQUATIONS 215
H*3. Integral equations of the first and second kinds.
Fredholm's equation is sometimes called an integral equation of the second
kind ; while the equation
f{x) = \\'K{x,^)<^{^)d^
J a
is called the integral equation of the first kind.
In the case when K(x, ^) = 0 if ^ > x, we may write the equations of the
first and second kinds in the respective forms
f{a^) = \\^K{x,^)4>{^)dl
J a
c}>(x)=f(x) + \rK(x,^)cf>(^)d^.
J a
These are described as equations with variable upper limits.
11*31. Volterra's equation.
The equation of the first kind with variable upper limit is frequently
known as Volterra's equation. The problem of solving it has been reduced
by that writer to the solution of Fredholm's equation.
Assuming that K(x, |) is a continuous function of both variables when
^ ^x, we have
f(x)=^\rK{x,^)ci>(^)di
J a
The right-hand side has a differential coefficient (§ 4*2 example 1) if
^r— exists and is continuous, and so
ox
f(x) = \K{x,x)(f>{x) + \f^-^cf>(^)d^.
This is an equation of Fredholm's type. If we denote its solution by
(f>{x), we get on integrating from a to x,
f{x)-f{a) = \\'' K{x,^)4>{^)dl
■J a
and so the solution of the Fredholm equation gives a solution of Volterra's
equation if f{a) = 0.
The solution of the equation of the first kind with constant upper limit
can frequently be obtained in the form of a series*.
11"4. The Liouville- Neumann method of successive substitutions f.
A method of solving the equation
cj>(x)=^f(x) + \f'K(x,^)<f>(^)d^,
•' a
which is of historical importance, is due to Liouville.
* See example 7, p. 225 ; a solution valid under fewer restrictions is given by Bocher.
t Liouville's Journal, ii. (1837), iii. (1838). K. Neumann's investigations were later (1870) ;
see his Untersuchu7igen iiber das logarithinische and Neicton'sche Potential.
216 THE PROCESSES OF ANALYSIS [CHAP. XI
It consists in continually substituting the value of <f>(x) given by the
right-hand side in the expression ^ (^) which occurs on the right-hand side.
This gives the series
J a w=2 ■' a J a
...Ik (^rn-i, ^m)f(^m) d^m - " d^i + ■ - • ■
J a
Since \K(x, y) \ and \f{x)\ are bounded, let their upper bounds be M, M'.
Then the modulus of the general term of the series does not exceed
\X\mM'''M'{h-ay.
The series for 8 {x) therefore converges uniformly when
\\\<M-^{h- a)-i ;
and, by actual substitution, it satisfies the integral equation.
If K{x, y)=0 when y'> x^ we find by induction that the modulus of the general
term in the series for S{x) does not exceed
I X '^M'^M' {x-af\{in !)< | X j™ Jlf^i^' (6 - aY'lmK,
and so the series converges uniformly for all values of X ; and we iefer that in this case
Fredholm's solution is an integral function of X.
It is obvious from the form of the solution that when | A- 1 < M~^ (b — a)~\
the reciprocal function k(x, ^ ; X) may be written in the form
k(x,^;X) = -K(x,^)- t X— ^ rK(x, |,) fV(|„ ^,)
TO = 2 J a J a
... i K {^rn-i , I) d^m-i d^,n-2 ...d^„
J a
for with this definition of k (x, ^ ; X), we see that
S{x)=f(x)-\fk(x,^; X)f{^)dl
J a
so that k{x, ^; X) is a reciprocal function, and by § 11-22 there is only one
reciprocal function.
Write
K (X, I) = K, (x, ^), ['k^x, r) Kn (f ', I) ^r - ^n+x (^, ^),
■' a
and then we have
-k{x,^;X)=t X'^K^^+M^),
m=0
while f V„, (x, r ) Kn (r, f ) d^' = K,,+n (^ n
J a
as may be seen at once on writing each side as an (m + n - l)-tuple integral.
The functions K,r, (x, f ) are called iterated functions.
11-5, 11-51] INTEGRAL EQUATIONS 217
11'5. Symmetric nuclei.
Let K^ {x, y) = Ki (y, x) ; then the nucleus K (x, y) is said to be symmetric.
The iterated functions of such a nucleus are also symmetric, i.e.
Kn{x, y)= Kniy, x) for all values of n; for, if Kn{x, y) is symmetric, then
Kn^, {X, y) = f V, {x, I) Kn {I y) d^ = f V, (I, x) K^{y, f) d^
J a J a
= f Kn (y, ^) K, (l x) d^ = Kn^, {y, x),
J a
and the required result follows by induction.
Also, none of the iterated functions are identically zero ; for, if possible, let
Kp (x, 3/) = 0 ; let ?i be chosen so that 2"~^ < jo ^ 2", and, since Kp (x, y) = 0, it
follows that K^ (x, y) = 0, from the recurrence formula.
But then 0 = Kqu {x, x) = I K^n-i (x, f) K^n-i (^, x) d^
J a
==l\K,n-x{x,^)Ydl
J a
and so K^-i{x, ^) = 0; carrying on this argument, we get ultimately that
K^ (x, y) = 0, and the integral equation is trivial.
11"51. Schmidt's* theorem. If the nucleus is symmetric, the equation
B (A,) = 0 has at least one root.
To prove this theorem, let
Un= \ Kn {X, x) dx,
SO that, when | A, | < i¥~^ (6 — a)~S we have, by § 11-21 example 2 and § 11-4,
D{\) dX n=i "" •
Now since I I {fxKn+i (x, ^) + Kn-i (x, ^)Y' d^dx ^ 0
J a J a
for all real values of fi, we have
A*" ^211+2 I ^f^ ^271 ">" '-^ 2n—2 ^ ">
and so t^2n+2 ^Zn—2 ^ ^271 ) ^2W— 2 -^ ^•
Therefore U.^, 17^, ... are all positive, and if V'JU'2 = v, it follows, by in-
duction from the inequality U^n+oU^n-^^ U^n, that 11271+2/ U'271 ^ ^"'•
00
Therefore when | X^ | ^ v~^, the terms of "Z ?7,iX"~^ do not tend to zero ;
w = l
and so, by § 5-4, the function yr—-- ^~~ has a singularity inside or on the
* The proof given is due to Kneser, Palermo liendiconti, xxii.
218 THE PROCESSES OF ANALYSIS [CHAP. XI
circle \\\ = v~^; but since D (\) is an integral function, the only possible
singularities of 7^ r t -7 are at zeros of D (X) ; therefore D (X) has a zero
inside or on the circle \\\ = v~'^.
[Note. By § 11 '21, D (X) is either an integral function or else a mere polynomial ; in
tbe latter case, it has a zero by § 6'31 example 1 ; the point of the theorem is that in
the former case D (X) cannot be such a function as e^\ which has no zeros.]
11"6. Orthogonal functions.
The real continuous functions ^1 (x), (fj^ (x), . . . are said to be orthogonal
and normal* for the range (a, b) if
j (f>,„(a;)(f)n{x)dx (=0 (m^n),
{— 1 (ni = n).
If we are given n real continuous linearly independent functions
Ui (x), U2 {x), ... Un (x), we can form n linear combinations of them which
are orthogonal.
For suppose we can construct m — 1 orthogonal functions (/>i, ... <^,„_i such
that </)p is a linear combination of u^, u^, ... tip (where p = l, 2, ... m— 1);
we shall now shew how to construct the function (f>m such that (f>i, ^o. ••• (f>m
are all normal and orthogonal.
Let ,4>,n (x) = Ci^m <^l (^) + C2, m (^2 (^) + . . . + C„i_.i (^„^_i (x) + W,„ {x),
SO that i(^,rt is a function of u^, u^, ... Um,.
Then, multiplying by <^p and integrating,
I 10m (^) 0^ (^) dx = Cp^ ,n + I Um (x) (f)p (x) dx (p < Vl).
J a .'a
Hence I j^^^ (x) (f)p (x) dx = 0
J a
if Cp^ m = — I U,-n {x) <f)p (x) dx ]
a function ^^„^ {x), orthogonal to 0i (.r), ^.^ {x), . . . 0„,_i (x), is therefore con-
structed.
Now choose a so that a- I {i0,„ (x)]'^dx= 1 ;
and take (^„, (x) = a. i0,„ (x).
Then / </>„, (x) (f)p (x) dx (=0 (p< m),
J a J
[=1 (p = m).
We can thus obtain the functions 0,, ^o, ... in order.
* They are said to be orthogonal if the first equation only is satisfied ; the systematic study
of such functions is due to Murphy, Camh. Phil. Trans., Vols. iv. and v.
ire, 11-61] INTEGRAL EQUATIONS 219
The members of a finite set of orthogonal functions are linearly inde-
pendent. For if
ai</>i (^0 + «202 (a;)+ ...+ an<f>n (x) = 0,
we should get, on multiplying by <f>p(x) and integrating, a^, = 0 ; therefore all
the coefficients Op vanish and the relation is nugatory.
It is obvious that tt ~ ^ cos mx, ir~^ sin mx form a set of normal orthogonal functions
for the range ( - tt, it).
Example 1. From the functions 1, x, x\... construct the following set of functions
which are orthogonal (but not normal) for the range - 1, 1 :
1, .r, *2_^, ^_a^, ^_^^2 + ^^....
Example 2. From the functions 1, x, x^, ... construct a set of functions
which are orthogonal (but not normal) for the range a, b ; where
[A similar investigation is given in § 15-14.]
11'61. The connexion of orthogonal functions with homogeneous integral
equations.
Consider the homogeneous equation
4>{x) = \A'<l>{^)K{x,^)dl
J a
where \q is a real * characteristic number for K (x, f ) ; we have already seen
how to construct solutions of the equation ; let n linearly independent solutions
be taken and construct from them n orthogonal functions ^i, (f).2, ... ^n-
Then since the functions <^^ are orthogonal
J a [_m = l J a J 7n = \J a \_ J a
and it is easily seen that the expression on the right may be written in the
form
'b
1 = 1 (J a
on performing the integration with regard to y ; and this is the same as
t \'k{x, y)<f>,n(y)dy J'k(x, B^.n{^)dl
w=l -a . n
Therefore if we write K for K (.r, y) and A for
n rb '
% cf>,„(y) K{x,^)c^,„{^)dl
'b
4>.a (y) I
m=l
It will be seen immediately that the characteristic numbers of a symmetric nucleus are all
real.
220 THE PROCESSES OF ANALYSIS [CHAP. XI
we have I A?dy=\ KAdy,
J a J a
and so [ A'dy= I K^-dy - ( (K - Afdy.
Therefore
rb ( n p rb
I < t \(l>m(y)(f>m(x)\ dy^ [K{x,y)Ydy,
n rb
and SO Xo S {(f>m(x)Y^ [K{x, y)Ydy.
7ft=l J a
Integrating, we get
n^Xo-' \ I {K{x,y)Ydydx.
J a J a
This formula gives an upper limit to the number, n, of orthogonal functions
corresponding to any characteristic number \q.
These n orthogonal functions are called characteristic functiofis (or auto-
f unctions) corresponding to X,,.
Now let ^'''* {x), ^<i' {x) be characteristic functions corresponding to
different characteristic numbers X^, Xj.
Then </)(«> {x) <^(i' {x) =\A K {x, |) (/><»> {x) <^(i> (^) d^,
J a
and so
[ <f>^'>^(x)(f>^'^(x)dx = \r I K(x,^)cf>^o^(x)<f>^'^{^)d^dx (1),
J a J a J a
and similarly
[ (^(«) (x) (/)»' (x) dx = \J ( K(x, ^) <^'») {^) </)W (x) d^dx
= \ f rK(^,x)4>i^^{x)<f>^H^)dxd^ (2),
on interchanging x and ^.
We infer from (1) and (2) that if \ =^ X^ and if K(x, |) = K(^, x),
(f>^°^(x)(f>^'^{x)dx = 0,
J a
and so the functions ^c* (x), 0<i' (x) are mutually orthogonal.
If therefore the nucleus he symmetric and if corresponding to each
characteristic number we construct the complete system of orthogonal
functions, all the functions so obtained- will be orthogonal.
Further, if the nucleus be symmetric all the characteristic numbers are
i-eal ; for if Xy, Xi be conjugate complex roots and if* u„ (x) = v (x) + iw (x) be
* V {x) and w (x) beiog real.
117] INTEGRAL EQUATIONS 221
a solution for the characteristic number Xo, then u^ (x) = v (x) — iw (x) is
a solution for the characteristic number \ ; replacing 0'"' (x), ^"' (x) in the
equation
0«'>(a;)</>("(a!)rfa; = O
J a
by V (x) + iw {x), v (x) — iw {x), (which is obviously permissible), we get
[[v{x)Y-^{w{x)Y\dx=^0,
which implies v{x) = w {x) = 0, so that the integral equation has no solution
except zero corresponding to the characteristic numbers Xo, Xj ; this is
contrary to § 11*23; hence, if the nucleus be symmetric, the characteristic
numbers are real.
11"7. The development* of a symmetric nucleus.
Let ^1 {x\ <f>2 (x), ^3 (x), ... be a complete set of orthogonal functions
satisfying the homogeneous integral equation with symmetric nucleus
4>(x)^xfK(x,^)cf>{^)d^,
J a
the corresponding characteristic numbers being f Xj, Xg, X3, ....
°° d> (x^ (b (v)
Now supposel that the series S rrn\yj ^^ uniformly convergent
when a ^x ■^b, a% y ^b. Then it will be shewn that
K(x,y)^t^'^^^l
n=l f^n
For consider the symmetric nucleus
<i>n i^) <i>n (y)
H{x,y) = K{x,y)- t
X~
If this nucleus is not identically zero, it will possess (§ 11 "51) at least one
characteristic number /j,.
Let -^ (x) be any solution of the equation
^lr{x)^f.l'H{x,^)ir(^)d^,
■1 a
which does not vanish identically.
Multiply by 0,i {x) and integrate and we get
f V (^) c/>n {x) dx = t.\' f \k {x, B - i ^^i-l^H ^ (^) <^, (.r) dxd^ ;
• a J a ■ a [ n = l ^n I
* This investigation is due to Schmidt, the result to Hilbert.
t These numbers are not all different if there is more than one orthogonal function to each
characteristic number.
:J: The supposition is, of course, a matter for verification with any particular equation.
222 THE PROCESSES OF ANALYSIS [CHAP. XI
since the series converges uniformly, we may integrate term by term and get
f V {x) <^n (^) dx = ^ f V (^) </>- (^) d^-^I^M^)^ (B d^
J a, '^w J a '^n J a
= 0.
Therefore yfr (x) is orthogonal to <^i (x), (f)^ (x), . . . ; and so taking the
equation
J a [ n=l '^n J
we have ^fr{x) = /J^,| K (x, ^) yjr (^) d^.
J a
Therefore //- is a characteristic number of K (x, y), and so t/t {x) must be
a linear combination of the (finite number of) functions </>«(;») corresponding
to this number ; let
■s^{x) = ^a^<^rn{sc).
m
Multiply by ^^ {^) ^'^d integrate ; then since i/r (x) is orthogonal to all the
functions 0„ {x), we see that a„^ = 0, so, contrary to hypothesis, -»/r {x) = 0.
The contradiction implies that the nucleus H {x, y) must be identically
zero ; that is to say, K {x, y) can be expanded in the given series, if it
is uniformly convergent.
Example. Shew that, if \ be a characteristic number, the equation
0 {x) =/ {x) + Xo \" K {X, I) (/> (^) di
J a
certainly has no solution unless f{x) is orthogonal to all the characteristic functions
corresponding to Xq.
11 71. The solution of Fredholms equation by a series.
Retaining the notation of § 11 '7, consider the integral equation
^ (x) =f{x) + \['k(x,^)cP (I) d^,
■J a
where K (x, |) is symmetric.
If we assume that <t*(|) can be expanded into a uniformly convergent
00
series X «n</>n(f)> we have
n = l
:£ an <^„(.x) =/(*:)+ S r-an<^« («-■)>
M=l »=1 ^n
SO that/(A') can be expanded in the series
S an ~ (pn (^)-
ij = 1 /^n
Hence if the function f(x) can be expanded into the convergent series
<» ^ b X
S bn<pni^), then the series S :. *' \ <l>n (■^^')> if '^^ converges uniformly in
71 = 1 M = l X,,j A/
the 7'ange (a, b), is the solution of Fredhohn's equation.
11-71-11-81] INTEGRAL EQUATIONS 223
ao
To determine the coefficients 6„ we observe that ^ 6„^„ {x) converges uni-
formly by § 3"35* ; then, multiplying by 0„ {x) and integrating, we get
K=\ 4>n {x)f{x) dx.
J a
11 "S. Solution, of AbePs integral equation.
This equation is of the form
/(^)=r/''-%^^ (0</x<l, a^x^b),
where/' (x) is continuous and/(a) = 0; we proceed to find a continuous solution u (x).
Let 0 (^) = I 11 ii) d^i and take the formula t
TT _ p dx
sin (/iijr) ~ J ((z- ^)i-M (x - 1)*^ '
multiply by u{^) and integrate, and we get, on using Dirichlet's formula (§ 4-51 corollary),
-4-^ {<!> {z) - ^ («)} = r d^ r — ^#^^
= {'dx r ^^^^)^^
J a J a{z-x)'^-f^{x-^Y
^f'^ f{x)dx_
J a{z-xY-l^'
Since the original expression has a continuous derivate, so has the final one ; therefore the
continuous solution, if it exist, can be none other than
and it can be verified by substitution J that this function actually is a solution.
11 "SI. Schlomilch's^ integral equation.
Let f{x) have a continuous differential coefficient when —tt^x^tt. Then the equation
2 Ti"-
fix) =- ^{xsva.6)de
'^ J 0
has one solution with a continuous differential coefficient when — tt^^^tt, namely
fin
cf) (x) =^f{0)+x f (x sin d) d6.
From § 4'2 it follows that
2 A'^
/' (.r) = - " sin (90' {x sin 6) d6
■a I n
(so that we have 0 (0)=/(0), 0' (0) = ^7r/ (0)).
* Since the numbers X„ are all real we may arrange them in two sets, one negative the
other positive, the members in each set being in order of magnitude ; then, when [ X„ | ^ \, it is
evident that X„/(X„- X) is a monotonic sequence in the case of either set.
t This follows from § 6-'2-4 example 1, by writing (z - x)j{x - f) in place of x.
X For the details we refer to Bocher's tract.
§ Zeitschrift filr Math. ii. (1857). The reader will easily see that this is reducible to a ease
of Volterra's equation with a discontinuous nucleus.
224 THE PROCESSES OF ANALYSIS [CHAP. XI
Write X sin ^ for x^ and we have on multiplying by x and integrating
{\t 2x /"i'r r Att ■)
xl f (xsin-^) d-^= — / ij sin ^0 (x sm $ am xJa) dd^ d^)r.
Change the order of integration in the repeated integral (§ 4"3) and take a new variable x
in place of yjf, defined by the equation sin _;(^ = sin ^ sin ■^.
fi^ „ , • , ^ 7 , 2.r fh-^ ( p d)' (x sin y) cos v dv] , .
Then xj^ fixs.n^)d^=-j^ |j/- "^^ I '^^^
Changing the order of integration again (§ 4-51),
/"^'"r// ■ i\^i 2^ f*^ f Z"*'^ ^' (-'^ sin x) cos X sin ^ 1
X f {X sin V^) d^^r= — ] ^^' _A=. — d6\ dx.
fi^ aindde f . /cos^Nli'^ ,
But I , --= —arc sin =2'^,
Jx \/c082;^-C0s''i^ L \COS X/ J X
fin fin
and so * I /' i^ sin yj/) d^ = x j cf)' {x sin x) cos x^x
= cl>{x)-cf>{0).
Since 0(O)=/(O), we must have
0 (.r) = f{0) + x j^f {X sin V.) df ■
J 0
and it can be verified by substitution that this function actually is a solution.
REFERENCES.
H. Bateman, Report to the British Association*, 1910.
M. BocHER, Introduction to Integral Equations (Cambridge Math. Tracts, No. 10).
H. B. Heywood et M. Fr^chet, L' Equation de Fredholm.
V. VoLTERRA, Lecons sur les equations integrates et les eqvations integro-differentielles
(Borel Tracts).
T. Lalesco, Introduction a la theorie des eqiiatioas integrates.
I. Fredholm, Acta Mathematica, xxvii. pp. 365-390.
D. HiLBERT, Grundzuge einer allgemeinen Theorie der linearen Integralgleichungen
(Leipzig, 1912).
E. Schmidt, Math. Annalen, Lxiii. pp. 433-476.
Miscellaneous Examples.
1. Shew that if the time of descent of a particle down a smooth curve to its lowest
})oint is independent of the starting point (the particle starting from rest) the curve is a
cycloid. (Abel.)
* The reader will find a more complete bibliography in this Eeport than it is possible to give
here.
INTEGRAL EQUATIONS 226
2. Shew that, if/(^) is continuous, the solution of
(f) (x) =/ (^) + X I cos (2x8) (f) («) ds
Jo
f{x) + \ I / («) cos (2X8) cU
'' *« i^TA^ •
assuming the legitimacy of a certain change of order of integration.
3. Shew that the Weber-Hermite functions
satisfy <f)ix)=\ j e**'*^ <^ (s) ds
for the characteristic values of X. (A, Milne.)
4. Shew that periodic solutions (with period 27r) of the differential equation
satisfy the integral equation
^^) + (a2 + peos2.^)0(^) = O
0 {.v) = X P^e^ °°^ x cos 8 ^ ^^^ ^^ (Whittaker ; see § 19-21.)
J 0
6. Shew that the characteristic functions of the equation
</) (a;) = X /^^ j-TT-i (^-y)2- - |a;- y || 0 (^) o?y
are ^ (x) — cos mx, sin mx,
where X = m ~ ^ and m. is any integer.
6. Shew that <t> {^')= j'' f~^ (l>{i) d^
has the discontinuous solution (f) {x) = kx^~'^. (Bocher.)
7. Shew that a solution of the integral equation with a symmetric nucleus
f{x)=f"K{x,^)(l,{$)di
J a
is ^{^)= 2 an\n<i>n{^)i
provided that this series converges uniformly, wliere Xn, <^b (*') are the characteristic
numbers and functions of K{x, ^) and 2 «„^,i (^) is the expansion off{x).
8. Shew that, if | A | < 1, the characteristic functions of the equation
'^ ^^^ = 2^ jo 1 -2A cos (^--^y+A^'^ (^^ ^^
are 1, cos 7nx, sin nix, the corresponding characteristic numbers being 1, A'", A'"; where
m takes all positive integral values.
W. M. A. 15
PAKT II
THE TRANSCENDENTAL FUNCTIONS
15-
CHAPTER XII
THE GAMMA FUNCTION
12"1. Definitions of the Gamma-function. The Weierstrassian product.
Historically, the Gamma-function* V{z) was first defined by Euler as the
infinite integral I t^~^e~*dt] but in developing the theory of the function, it
Jo
is more convenient to define it by means of an infinite product.
Consider the product ze"^^ n|n+-jenl
where 7= lim i-+7^+...+ log ml = 0-5772157....
[The constant y is known as Euler's or Mascheroni's constant ; to prove that it
exists we observe that, if
w,, = I — -, c dt = — Ioq; ,
Jo n{n + t) n ° n '
/^ dt 1 °^
„ = -5 ; therefore 2 u,^ converges, and
0 ^ '^" ji=i
hm \- + -+...-\r--\ogm\= hm \ 2 u,Mog—-\= 2 u„.
The value of y has been calculated by J. C. Adams to 260 places of decimals.]
The product under consideration represents an analytic function of z, for
all values of z ; for if N be an integer such that | z j ^^N, we havef, if n > N,
liogfiH-l-- =
! \ nj n
=
Xz" Iz'
2n^'^Sn^ '"
\zf\^ \z\ \z'\ "1
]
INU, 1 1
tlz^r + 2 + 2~^ +
) IN"-
■••\^2n'
* The notation T {z) was introduced by Legendre in 1814.
t Taking the principal value of log (l + zjn).
230 THE TRANSCENDENTAL FUNCTIONS [CHAP. XII
00
Since the series S {N^/{2n^)] converges, it follows that, when | ^ | ^ JiV,
n=N+l
% •|log(l+-) \ is an absolutely and uniformly convergent series
of analytic functions, and so it is an analytic function (§ 5"3) ; con-
sequently its exponential 11 -^(1 +-)e «[ is an analytic function, and
so zey^ U. \(l + -] e "i is an analytic function when \2;\ ^^N, w^here N is
any integer ; that is to say, the product is analytic for all finite values of z.
The Gamma-function was defined by Weierstrass* by the equation
^ = 2ey^U \(l + -)e~TX;
r(^;) „=i [V nJ j
from this equation it is apparent that V (z) is analytic except at the points
2 = 0, —1, —2, ..., luhere it has simple poles.
Proofs have been published by Holder +, Moore J, and Barnes § of a theorem known to
Weierstrass that the Gamma-function does not satisfy any differential equation with
rational coefficients.
• Example 1. Prove that
r(i)=i, r'(i)=-y,
where y is Euler's constant.
[Justify differentiating logarithmically the equation
r4:)='-«'i{('+s)«'"}
by § 4"7, and put z = \ after the diflPerentiations have been performed.]
• Example 2. Shew that
,11 I r l-(l-0" 7
and hence that Euler's constant y is given by||
lim
«-*-00
[/:{-(-mT'-/:(-,o"T']
Example 3. Shew that
e>^r(s + l
>.=ilV -- + »A ]~T{z-x+\y
* Crelle, li. (1856). This formula for T (z) had been obtained from Euler's formula (§ 12-11)
in 1848 by F. W. Newman, Cambridge and Dublin Journal, iii.
t Math. Annalen, Bd. xxviii. p. 1.
X Math. Annalen, Bd. xlviii. p. 49.
§ Messenger of Mathematics, Vol. xxix. p. 64.
li The reader will see later (§ 12-2 example 4) that this limit may be written
Wl-e-^)''-" ^"^"^^^
1211, 1212] THE GAMMA FUNCTION
12*11. Euler's formula for the Gamma-function.
By the definition of an infinite product we have
231
^{^)
lim e
1 [lim n |fl4--V"[
~ [1 +- + ... + \ogm]z m [f z\ ~ Z.Y
= z\im eV 2 m 7 n (l + -)e "
= z lim
= z lim
m-*-w
= ^: lim
>»-*30
m-^ n 1 + -
n=l V W
n f 1 + -) n ( 1 + ^
n=l\ W/ n=lV '^^
Hence
«=i (V n]\ nj ]\ ml y
Zn=\\\ n) \ n
This formula is due to Euler* ; it is valid except when 2 = 0, — 1, — 2, ... ,
Example. Prove that
,, ,. 1.2...(»-1)
r (Z) = lim —, -r ) ^-— T '.
(Euler.)
12*12. The difference equation satisfied hy the Gamma-function.
We shall now shew that the function V {z) satisfies the difference equation
T{z + l) = zV{z).
For, by Euler's formula, if z is not a negative integer,
r(^+i)/r(^) =
z+l
lim n
(-ri
-=-
n
1 + ^y
Z jre-*.3c n=l 1 I "^
Hm U y. ^ .— r
1 w^x M=i I 2 + n + 1 )
1 . 771+1
= z lim 7 = 2^.
m^co 2; + m + 1
This is one of the most important properties of the Gamma-function.
Since F (1) = 1, it follows that, if ^ is a positive integer, F (^) = (2; - 1) !.
* It was given in 1729 in a letter to Goldbach, printed in Fuss' Corresp. Math. It was
deduced by Euler from his definition of V (2) as an infinite integral. See § 12-2.
232 THE TRANSCENDENTAL FUNCTIONS [CHAP. XII
Example. Prove that
1 , 1 , 1 , =_A./1 1J_4.1J__ 1
V{z->r\yv{z^^yV{z^'Sy"' V{z)\z~ \\z-^\^^\z-\-^ '"]•
[Consider the expression
111 1
Z 3(2+1) 2(3+1) (0 + 2) 2 (0+1)... (2 + m)"
It can be expressed in partial fractions in the form 2 • — ~ , where
n=0 S + 'i-
^i! i 1! 2! (m-»)!j )l ! ^ r=m-n+l»"U
°° 1 e "'-( — )" 1 r =^' 1 "I
Noting that 2 —,< , r-r^., prove that 2 -^ — '- \ 2 — tV^O as
j-=m-7i+l»'! (m-% + 1)! n=0 »i! 2 + 'i lr=«i-H+l '' U
m ^- 00 when 2 is not a negative integer.]
12*13. The evaluation of a general class of infinite products.
By means of the Gamma-function, it is possible to evaluate the general
class of infinite products of the form
00
n Un,
M = l
where Un is any rational function of the index n.
For, resolving Un into its factors, we can write the product in the form
1^ {A (n -a^{n-ao^ ...{n- a^)]
«=i| {n-h;}...{n-hi)
and it is supposed that no factor in the denominator vanishes.
In order that this product may converge, the number of factors in the
numerator must clearly be the same as the number of factors in the
denominator, and also A=l\ for, otherwise, the general factor of the product
would not tend to the value unity as n tends to infinity.
We have therefore k = l, and, denoting the product by P, we may write
1=1 \{n-\)...{n-hk)] '
The general term in this product can be written
(-t)-(-?)(-r-(-r
_ -, _ ai + g, + . . . + gfc - 61 - . . ■ - ?)fc ,
n
where A^ is 0 {n~^) when n is large.
In order that the infinite product may be absolutely convergent, it is
therefore necessary further (§ 2*7) that
tti + . . . + ttfc — 61 - . . . - 6/; = 0.
12-13, 12-14] THE GAMMA FUNCTION 233
We can therefore introduce the factor
exp {nr^{ai + ... + ajc — hi— ... — hk)]
into the general factor of the product, without altering its value ; and thus
we have
p= n \
(l_|^)e"(l-^)6«...(l-^)e"
['-W'-i'-iY
But it is obvious from the Weierstrassian definition of the Gamma-
function that
z
1
and so
n
p =
^ n)^] -zV{-z)e-y'''
a^V{-(h)-..(ii,r{-aic)
a formula which expresses the general infinite product P in terms of the
Gamma-function.
Example 1. Prove that
n
«(a-|-6 + s) r(a + l)r(6 + l)
8=1 {a + s){b + s) r(a + 6 + l) '
Example 2. Shew that, if a = cos {'injn) + i sin {2ir/n), then
^(l-^)(^i-|„y.. = {-r(-^-)r(-a^^)...r(-a»-i^^)}-«.
12-14. Connexion between the Gamma-function and the circular functions.
We now proceed to establish another most important property of the
Gamma-function, expressed by the equation
sm7r2
We have, by the definition of Weierstrass (§ 12-1),
2
1
T(.)T(-.)^-^^Uj[l^lU
n
I-",.-
Z Sm TTZ
by § 7*5 example 1. Since, by § 12-12,
T{l-z) = -zV{-z)
we have the result stated.
234 THE TRANSCENDENTAL FUNCTIONS [CHAP. XII
Corollary 1. If we assign to z the value ^, this formula gives {r(J)}2 = 7r ; since, by
the formula of Weierstrass, r (|) is positive, we have
r(i) = 7ri
Corollary 2. If ^ {z) =T'{z)lT (z), then yjr {I — z) - -^ {z) = tt cot ttz.
12*15. The multiplication-theorem of Oauss* and Legendre.
We shall next obtain the result
r(.)r(. + 1) r (. + ?)... r (. + ^) = (2.)*("-^)n^-rM.
Jb or let 6 (2) = nr — ^ •
^ ^ '^ nl {nz)
Then we have, by Euler's formula (§12'11 example),
^-^ ,. 1.2 ... (m - 1) . m^+'""»
w^^ n hm ^ ^
^^^^ , . 1.2... (n^n - 1) . (mnY^
n lim
??2;(w2'+ 1) ... (nz+ nm — 1)
j(?n — 1)!| m ^ 'n
^^^ {nm-l)\{nm)
= lim
|(m — 1) !| Tn^^ 'n
m-^oo (nw-l)!
It is evident from this last equation that </) (z) is independent of z.
Thus ^ (2^) is equal to the value which it has when z= -; and so
Therefore (^ ^^ = n; { F Q F (l - ^-)}
. TT . 27r . (?^— l)7r
sm — sm — ... sm
n n n
Thus, since <^ (w~i) is positive.
I.e.
r (z) r(z + l^ ... r (^ + ^) = 7i*-"^(27r)*("-')r(n4
Corollary. Taking «=2, we have
22^-ir(2)r(2 + *) = 7r^r(22).
This is called the duplication formula.
* Werke, Bd. iii. p. 149. The case in which n = 2 was given by Legendre.
12-15-12'2] THE GAMMA FUNCTION 235
Example. If jtt^ n\ r(y)r(g)
shew that
B{p,q)B{p^.\,qy..B{p + '^,q)
Bi^np,nq)=n-^ i^(?, q)Bm, q)...B{{n-l)q, q\ ■
I
12*16. Expansions for the logarithmic derivates of the Gamma-function.
We have |r (2 + 1)} " » = e^^ n jA + 0 e' "I .
Differentiating logarithmically (§ 4-7), this gives
d\ogT{z + \)_ z z z
dz ^"'"1(2+1) "''2 (2 + 2) "^3 (2+3) ■''•••■
Therefore, since log r (2 + l) = log2+T' (2), we have
d 1*1
-J- logr(2)=-y-- + 2 2 ■ , ..
dz ° ^ ' ' Z n=lM'(2 + 7l)
d^ d ( z
Diflerentiating again, ^logr(2 + l) = ^ |^^
1) 2(2 + 2)
1 1
~(MnO=^(M^2 + --
These expansions are occasionally vised in applications of the theory.
12"2. Euler's expression of V {z) as an infinite integral.
The infinite integral / e'H'^'^dt represents an analytic function of z when ■
0
the real part of z is positive (§ 5'32) ; it is called the Eulerian Integral of the
Second Kindf. It will now be shewn that, when R (z) > 0, the integral is
equal to T(z). Denoting the real part of z by x, we have x>0. Now, if:|:
rn / f\n
we have II{z, n) = n^ j (1 — t)"t^~^c^t,
Jo
if we write t = nT; it is easily shewn by repeated integrations by parts that,
when x>0 and w is a positive integer,
ri ri 11 n fi
(1-t)«t^-1cZt= -t^(I-t)'^ +- (1 - t)^-1t^c?t
Jo \_^ Jo 2: J Q
z(z + l) ...(z + n-l)J(i
and so 11 (z, ??) = ~ — ^-^^ r n^.
Hence, by the example of § 12"11, 11 {z, n) -* T (z) as n -^ 00 .
* If the real part of z is not positive the integral does not converge on account of the singu-
larity of the integrand at f = 0.
t The name was given by Legendre ; see § 12"4 for the Eulerian Integral of the First Kind.
:J: The many-valued function t^~'^ is made precise by the equation (^~i = e(^~^' '"»*, log t being
purely real.
236
THE TRANSCENDENTAL FUNCTIONS
[chap. XII
Consequently r(^)=lim (l--)t^-'dt.
T,(z)= f e-H'^-'dt,
Jo
And so, if
we have
T^{z)-T(2)= lim
[/:f-'-(
nj
/
J n
t^-^dt + e-H^-^dt
Now lim [ e-H''-^dt = 0,
since I e~H^~^dt converges.
Jo
To shew that zero is the limit of the first of the two integrals in the
formula for Fj (z) — F (z) we observe that
0<e-
-(-
< n-H^e'
[To establish these inequalities, we proceed as follows : when O^y < 1,
from the series for e^/ and (I -?/)~i. Writing t/n for y, we have
and so
(l + ^)-%.-^(l
0^e-t_(l-iY
n) '
<e-Mi-{i-5.
A"
Now, if O^a^l, (l-a^'^l-na by induction when na<l and obviously when
?2a^l ; and, writing fijn^ for a, we get
\ n^J n
and so* 0<e-«-(l j ^e-H^jn,
which is the required result.]
From the inequalities, it follows at once that
je-' - (l - -)"1 t'-'dt j < rn-'e-H^'+'dt
<n-M e-H''+^dt^O,
as n. -^ oc , since the last integral converges.
* This analysis is a modification of that given by Schlomilch, Compendium der hoheren
Analysis, ii. p. 243. A simple method of obtaining a less precise inequality (which is sufiicient
for the object required) is given by Bromwich, Infinite Series, p. 459.
12-21] THE GAMMA B'UNCTION 237
Consequently Fj (z) = F (z) when the integral, by which Fj (z) is defined,
converges ; that is to say that, when the real part of z is positive,
Jo
e-H'-'dt
And so, when the real part of z is positive, F (z) may be defined either by
this integral or by the Weierstrassian product.
' Example 1. Prove that when R (z) is positive
Example 2. Prove that, if R{k)>0 and R (s) > 0,
T(s)
/
0
« Example 3. Prove that, if R{z)>0 and R(s) > I,
1 1 1 _ 1 pe-x^^-i
• Example 4. From § 12-1 example 2, by using the inequality
deduce that
l_e-«_e-i/« ,
a;.
■=/:
12 '21. Extension of the infinite integral to the case in which the argument of the
Qammafunction is negative.
The formula of the last article is no longer applicable when the real part of z is
negative. Saalschutz* has shewn however that, for negative arguments, an analogous
theorem exists. This can be obtained in the following way.
Consider the function
where k is the integer so chosen that —k>x> - /t — 1, x being the real part of z.
By partial integration we have, when z<-\,
r.(.)=[f(«--i+^-,4+... + (-r^;^,)'
^k-l
The integrated part tends to zero at each limit, since x-\-k is negative and x-^k + 1 is
positive : so we have
Ti{z) = \T.^{z+\).
The same proof applies when x lies between 0 and — 1, and leads to the result
r(2+i) = 2r2(2) (0>.i->-i).
The last equation shews that, between the values 0 and — 1 of .r,
r2(2)=r(2).
* Zeitschriftfilr Math, und Phys. xxxii, xxxiii.
238 THE TRANSCENDENTAL FUNCTIONS [CHAP. XII
The preceding equation then shews that T^ {z) is the same as r {z) for all negative
values of R (?) less than - 1. Thus for all negative values of R {£), we have the result of
Saalschiitz
where k is the integer next less than - R (2). z K (^) /
Example. If a function F (/x) be such that for positive values of fi we have
Jo
and if for negative values of yx we define Pi (fx) by the equation
Fii,x)=j\v'^-'(e-^-l+x-... + {-)^*'y^dx,
where k is the integer next less than — fj., shew that
AW = i'(M)-^+rT(^-- + (-)^-^n(^- (Saalschutz.)
12*22. Hankel's expression of V (z) as a contour integral.
The integrals obtained for T{z) in §§ 12-2, 12-21 are members of a large
class of definite integrals by which the Gamma- function can be defined.
The most general integral of the class in question is due to Hankel * ; this
integral will now be investigated.
Let D be a contour which starts from a point p on the real axis, encircles
the origin once counter-clockwise and returns to p.
Consider I {— tf''^ e~^ dt, when the real part of z is positive and z is not
an integer.
The many-valued function (— 0^~^ is to be made definite by the convention
that (— ty~'^ = e(«-i)iog(-0 and log(- t) is purely real when t is on the negative
part of the real axis, so that, on D, — tt ^ arg (— t) ^ tt.
The integrand is not analytic inside D, but, by § 5-2 corollary 1, the path
of integration may be deformed (without affecting the value of the integral)
into the path of integration which starts fi:om p, proceeds along the real axis
to h, describes a circle of radius 8 counter-clockwise round the origin and
returns to p along the real axis.
On the real axis in the first part of this new path we have arg {—t) = - tt,
so that (— ■^)^-i = e-''^(«-i)^~-i (where \ogt is purely real); and on the last
part of the new path (- ^^-i = e"^(^-i)^«-i.
On the circle we write -t = 8e^^ ; then we get
J J) J p J -w
+ \ e^^'^^-^H^-^e-Ht
= - 2i sin (ttz) ("t'-'e-^dt + ih' [' e''^-^ (co,e+i^ne> ^q
* Math. Annalen, i.
12-22J THE GAMMA FUNCTION 239
This is true for all positive values of 8 ^p; now make S -► 0 ; then B'-^O
and j ewfl-sc^ose+i^inoc^^-* j e'^HO since the integrand tends to its limit
J —IT J -IT
uniformly.
We consequently infer that
/.
(- ty-^e-Ht = - 2t sin {irz) ft^'-'e-^dt.
D Jo
This is true for all positive values of p ; make p^ <x> , and let C be the
limit of the contour D.
Then [ (- tf-' e"' dt=- 2i sin (ttz) ( t'-' e"' dt
J c Jo
Therefore T(z) = - ^ f (- tf-' e"' dt.
zismirz J c
Now, since the contour G does not pass through the point t = 0, there
is no need longer to stipulate that the real part of z is positive ; and
I {—ty~^e~*dt is a one- valued analytic function of z for all values of z.
J c
Hence, by § 5*5, the equation, just proved when the real part of z is positive,
persists for all values of z with the exception of the values 0, ±1, ±2,
Consequently, for all except integer values of z,
T(z) = - —i f (- ty-' e-' dt
^ ' 2i sm iTz] c
This is Hankel's formula ; if we write \ — z for z and make use of § 12-14,
we get the further result that
/•(0+) /• . ,
We shall write for , meaning thereby that the path of inte-
Jco J C
gration starts at ' infinity ' on the real axis, encircles the origin in the positive
direction and returns to the starting point.
Example 1. Shew that if the real part of z be positive, I ( - t)~^e~*dt tends to
zero as
p-*.Qo, when the path of integration is either of the quadrants of circles of radius p + 1
with centres at —1, the end points of one quadrant being p + 1, —l+i{p + l), and of the
other p + 1, — 1 -i(p + l).
240 THE TRANSCENDENTAL FUNCTIONS [CHAP. XII
■ Deduce that ■ lim I ~^\- t)-'e-*dt= lim j {-t)-^e-*dt,
p-*-ao J -1+ip p-*-oo
and putting t= -l+i tan 6 in the first integral shew that
r
L = 1 r " cos (tane-zd) cos^ ~^edB.
(z) ^ Jo
Example 2. By taking as contour of integration a parabola whose focus is the origin,
shew that, if a > 0, then
Y (z) = ^'i^ e-a«2 ( 1 + fi)z-i cos {2at + ( 22 - 1 ) arc tan t} dt.
^ ' SHI TTS j 0
(Bourguet.)
Example 3. Investigate the values of x for which the integral
- I ^^-isin^o^it
TT y 0
converges ; for such values of x express it in terms of Ganama-functions, and thence shew
that it is equal to
(St John's, 1902.)
Example 4. Prove that | (log ^)"» dt converges when m > 0, and, by means
jo i
of example 3, evaluate it when m = l and when ?«- = 2,
(St John's, 1902.)
12'3. Gauss expression for the logarithmic derivate of the Gamma-function
as an infinite integral*.
We shall now express the function j- log F (z) = -pj^^ as an infinite
integral when the real part of z is positive ; the function in question is
frequently written 1^(2^). We first need a new formula for 7.
Take the formula (§ 12-2 example 4)
-/:^*-/:T'--{/:f-/:T*-}
where A = l — e , since
/'? = logr^,.^,-OasS-.0.
t ''l-e-s
Writing ^ = 1 — e~"in the first of these integrals and then replacing w by ^ we have
y= lim i I , -dt- i ~ dti
^ -^Xe-^dt.
\\-e-i t
This is the formula for y which was required.
Werke, Bd. in. p. 159.
12-3]
THE OAMMA FUNCTION
241
To get Gauss' formula, take the equation (§ 1216)
r'(^) 1 ,. « /I 1 \
^-J =-7 4-lim S
r(^) ' Z »— 00 m = i \"^ z + mj
and write
1 f^
+ r>i Jo
this is permissible when vi = 0, 1, 2, ... if the real part of z is positive.
It follows that
p , =-7- e-'Ult + lim t (e-"**-e-<"*+^'*)(^^
^ K^) Jo n-*-oo Jo m = l
= — 7+ lim
g-« _ g— zt _ g— (n+1) « r g— (z+n+i) f
1-e-
t/^
io \ ^ 1 - e-V n^oo Jo
0 \ ^ 1 - e-
1 — ^-^^
1 ^ — t
n-*-oo Jo ••• ^
is a bounded function of t whose limit as t^O is finite ;
Now, when 0<t^l,
and when i;>l, 1^ — ' — ^ < " ' '" — —<■
^ ' I l-e-« 1-e-i 1-e-i
Therefore we can find a number K independent of t such that, on the path of integration,
l + le-
and so
1 —a-^t I
1 — e 'I
1-e-
We have thus proved the formula
fW = ;|iogr(.)=/;(^-l'-ji^)*
which is Gauss' expression of i/r (^) as an infinite integral. It may be
remarked that this is the first integral which we have encountered connected
with the Gamma-function in which the integrand is a single-valued function.
Writing i: = log (1 -\-x) in Gauss' result, we get, if A = e* - 1,
r(^) 6^0 ys \t
1-e-tj
\-dt
dx
A x{\-\-xy
= lim
5^0
{/:
dt
0< --dt<\ — = log -— -^OasS-^0.
Hence
so that
r'{z)
= lim
1
r (2) A^O J A
T'{z) = r{z) I se
an equation due to Dirichlet*.
dx
(l+.ttJ "?'
Werke, Bd. i. p. 275.
W. M. A.
16
242 THE TRANSCENDENTAL FUNCTIONS
Example 1. Prove that, if the real part of z is positive,
*(^)=/:{^4-rS/
dt.
Example 2. Shew that
{{\+t)-^-e-i}t-^dt.
[chap. XII
(Gauss.)
(Dirichlet.)
12'31. Binet's first expression for log V {z) in terms of an infinite integral.
Binet* has given two expressions for logr(2^) which are of great
importance as shewing the way in which log V {z) behaves as | ^ | -* x . To
obtain the first of these expressions, we observe that, when the real part of
2 is positive.
=/:It-;^i-.
T{z + \)
writing z-\-l for ^ in | 12'3.
Now, by § 6*222 example 6, we have
log ^ = M
"^ «-« _ c-tZ
and so, since
we have
d
0 t
"1
dt,
(2^)-i= ^e-^'dt,
0 ^
^^ log r (. + 1) = ^^ + log . - j^ 1^ - ^ + ^^} e-" dt.
The integrand in the last integral is continuous as ^^0; and since
111
h - — - is bounded as t-*oo , it follows without difficulty that the
2 ^ e* - 1 ' -^
integral converges uniformly when the real part of z is positive ; we may
consequently integrate from 1 to z under the sign of integration (§ 4'44) and
we getf
logr(^+l) = U + ^ log 2^-^ + 1 -I-
1 1
K-T +
dt.
^^"^M2-^ + e^lU
is continuous as ^ ^ 0 by § 7"2, and since
Ave have
log r (^ + 1) = log 2- + log r {z),
iogr(2) = ('^--] iog2-^ + i + I L-T +
t
dt
2-7 + e^-iiT^'-
* Journal de VEcole Poly technique, xvi. (1839), pp. 123-143.
t Log r (2 + 1) means the sum of the principal values of the logarithms in the factors of
the Weierstrassian product.
12-31] THE GAMMA FUNCTION 243
To evaluate the second of these integrals, let*
so that, taking «=^ in the last expression for log T{z), we get
Ai • 7 /""/l 2 1 \ «-i<
Also, since 1= I h^--+ ,. — ——at, we have
7o \2 « e*'-l/ « • ■
~ jo V « e'-lj « •
And .so J=\ \ — -. + 4e-< \. - — -I _
jo 1 t e'-l 2 t ^e«-lj t
f^ {e-ht-e-* , A dt
-j. |-T — *n-T
=jo {-ja— r-;---7— --27^^
=L r-Jo+*Jo -t--''
Consequently /= 1 - 1 log (27r).
We therefore have Binet's result that, when the real part of z is positive,
If z = x + iy, we see that, if the upper bound of ( ^ — - 4- ^ — ^ ) 7 ■ ^'^'^ "^^^^
values of t is K, then
|logr(^)-(^-^)log^ + ^-^log(27r): <k\ e-'^dt
< Kx-\
so that, when x is large, the terms iz — ^\ log ^ — ^^ + ^ log {2ir) furnish an
approximate expression for log V {z).
Example 1. Prove that, when li{z)>0,
log r {z) = JJ {'7-T-' ' + (-" - 1) «-'} 7 • (Malm,sten.)
Example 2. Prove that, when /^ (z) > 0,
* This artifice is due to Pringsheim, Math. Annalen, xxxi.
IG— 2
244 THE TRANSCENDENTAL FUNCTIONS [CHAP. XII
Example 3. From the formula of § 12-14, shew that, if 0 < ^ < 1,
2logr(^-)-log-+logsm.^- = j^ [-^^ (1-2^0^- jy
(Kummer.)
Example 4. By expanding sinh {\ — x)t and 1 - 2.1? in Fourier sine series, shew from
example 3 that, if 0 < .r < 1,
log r(;r) = -^ log TT--^ log sin 77^ + 2 2 a„sin2/i7r^,
n=\
Deduce from example 2 that
a,i = 2 — (y + log27r + logn). (Kummer.)
12*32. Binet's second expression for log F (z) in terms of an infinite
integral.
Consider the application of example 7 of Chapter vii (p. 145) to the
equation (§ 12'16)
-f-„iogr(^)= 2 7-^,.
The conditions there stated as sufficient for the transformation of a
series into integrals are obviously satisfied by the function (f) (^) = ^ 77. ., ,
if the real part of ^ be positive ; and we have
where 2iq (t) = -. r-r., — , — .-rz .
^ ^ ^ (z + ity (z - itf
Since \q(t, z -\- n) \ is easily seen to be less than Kit/n, where A''! is inde-
pendent of t and ??, it follows that the limit of the last integral is zero.
Hence |,log T (.) = ^^^ + J +/^ ~y, ^-^ .
Since ' ^ — -' does not exceed K (where K depends only on 8) when the
Z "T~ V I
real part of z exceeds 8, the integral converges uniformly and we may
integrate under the integral sign (§ 4'44) from 1 to z.
We get
f Z , „ , , 1 , X, ^ f "* tdt
-J- log r (^) = - — - + log ^ + C - 2 ~ -.'^-.-irr, — ,-s ,
dz ^ ^ ^ 1z ^ Jo {z" + t^){e-''^ -1}
where 0 is a constant. Integrating again,
log r (.) = (. - ^) log . + (C - 1 ) . + c + 2 JJ •'"■'^i:^^^ dt,
where C is a constant.
12-32, 12-33] THE GAMMA FUNCTION 245
Now if z is real 0 < arc tan t\z ^ tjz,
and so
dt.
\ogV{z)-{z-^^\ogz-{C-\)z-C'\<\\^^^-^
But it has been shewn in § 12*31 that
logr(^)-(^-|)log^ + ^-^log(27r)Uo,
as 2^ ^ 00 through real values. Comparing these results we see that C = 0,
C' = hog(27r).
2
Hence for all values of z whose real part is positive,
logr(^) = (z-i)log^-0 + llog(27r) + 2J'J
where arc tan u is defined by the equation
arc tan (tlz) ,,
dt
arc tan u =
'o 1 + ^-^'
in which the path of integration is a straight line.
This is Binet's second expression for log T (z).
* Example. Justify differentiating with regard to z under the sign of integration, so as
to get the equation
r'(2) 1 p tdt
-''^'-h-'jl
T(z) ^ 2z Jo {t^ + z^){e'i^t-i)-
12*33. The asymptotic expansion of the logarithm of the Gamma-function
(Stirling's series).
We can now obtain an expansion which represents the function log T (z)
asymptotically (§ 8*2) for large values of \z\, and which is used in the
calculation of the Gamma-function.
Let us assume that, if z = x + iy, then x'^h >0; and we have, by Binet's
second formula,
log r {z) =[z- -^\ogz-z + - log 27r + <^ {z\
where (f){z) = 2 1
Jo
arc tan (t/z) ,,
dt.
2nt _ I
Now
, . , t It' It' (_)n-i pi-i (-)n rf u^n^lf^
arc tan (t z) = -- ^ -, + ^ -,-■•■ +1 , ^;^, + V-fr -—-..■
^ ' ^ z S z^ 5 z^ 2?i — I 2-" 1 Z^*' ^ J 0 u- + z-
Substituting and remembering (§ 7*2) that
r^-^ dt _ Bn
' 0 e-''' - 1 ~ 4n '
y 0
246 THE TRANSCENDENTAL FUNCTIONS [CHAP. XII
where Bi, Bo, ... are Bernoulli's numbers, we have
(-)^ r I f^u^'^du) dt
'''-' Jo [JoU^+ n e-'^'^-l'
for positive values of uhe K^.
^2
Let the upper bound* of
Then
du] dt
^ ( rt u'^n
0 li' + z""] e^'^* - 1
ic^ + ^■■'
^ -ST, I ^ 1
"^ ( rt \ (H
u^^du[ ~--7 — -
g-int _ I
0 \,J 0
J^z Bn+1
Hence
4(n + ])(2?i + l)j^|2"
2(-y
0 uo
'li^'du) dt
y2 ^ ^2 g2,rt _ I
K,Bn
2(?i + l)(27H-l)|^p"+i'
and it is obvious that this tends to zero uniformly as | ^ 1 ^ oo if | arg ^ j ^ ^tt — A,
where :|7r > A > 0, so that K^ ^ cosec 2A.
Also it is clear that if ' arg ^ j ^ :^7r (so that Kz= 1) the error in taking the
first n terms of the series
,.=i2r(2r-l)2-'-i
as an approximation to 0 (z) is numerically less than the (n + l)th term.
Since, if j arg z\^^7r — A,
\ z^^-' U {z) - I -/:;t^{-J \ < cosec^ 2 A . ^"+^
,=i2r(2r-l)
as ^ ^ X , it is clear that
2(w+l)(2n + l)
0,
5, B^
1.2.Z 3 . 4 . ^« ' 5 . 6 . ^^
is the asymptotic expansion i* (§ 8-2) of ^{z).
We see therefore that the series
1\ 1 °° (—y-^B
z - ^\\og z - z + -\o^ 2ir + X '
,.=i2r(2r-l)2^'-i
is the asymptotic expansion of log V {£) when | arg z\i^\'k — A.
* 7u-- is the lower bound of ^-^ ^^' ~ -''^^ ^" + ^'"'"•^^
4.r'-')/-
(.i-^ + T/'-i)^
and is consequently equal to
(x'^ + y
.,, or 1 as a;2 < (/2 or .t2>?/2.
t The development is asymptotic ; for if it converged when | 2 | ^ p, we could find K, by § 2-6,
3h that B^ < (2« - 1) 2nKp^'' ; anc
function ; this is contrary to § 7'2.
such that B^ < (2n - 1) 2nKp-" ; and then the series S - ,0 Ti " ^ would define an integral
n=i (^'0 !
12-4] THE GAMMA FUNCTION 247
This is generally known as Stirling's series. In Chapter xiii, it will be
established over the extended range | arg ^^ | ^ tt — A.
In particular when z is positive (= x), we have
v?^dn^ dt Bn+i
-^/:i0
+ ar*] e=*' - 1 2 (w + 1) {2n + l)a^
Hence, when ooO, the value of (f>{x) always lies between the sum of
n terms and the sum of n + 1 terms of the series for all values of n.
n Q
In particular 0 < 4>{x)< ri — —■ , so that <f> (x) = --^, where 0 < ^ < 1.
Hence T (*•) = ^"^ ^ e " ^ {^irf /^^''^l
Also, taking the exponential of Stirling's series, we get
_., .,_i xf 1 1 139 571 ^^/IM,
I(^)-^. X - (2-)^ |1 + 12^ + 288^. -51840^,- 2488320^ + ^t^^^
This is an asymptotic formula for the Gammaf unction. In conjunction
with the formula F (x + 1) = xT (x), it is very useful for the purpose of com-
puting the numerical value of the function for real values of x.
Tables of the function logr(.r), correct to 12 decimal places, for values of x between
1 and 2, were constructed in this way by Legendre, and published in his Exercices de
Calcul Integral, Tome ii. p. 85, in 1817, and his Traite des fonctions elliptiques (1826),
p. 489.
It may be observed that r {x) has one minimum for positive values of .r, when
jr = l-4616321..., the value of log r(:r) then being T-9472391....
Example. Obtain the expansion, convergent when R {z) > 0,
log T{z) = (z-^)\ogz-z + i log (27r) + J {z),
where
^' - \2+1^2(z+l)(2 + 2)^3(0 + l)(2 + 2)(s + 3)^"'J '
in which
^1 = ^5 ^2=3, ^3 = 1^^, C4 = -T(f5
and generally
c^= I (.r + 1) (A- + 2) ... {x + n-\y{2x-l)xdx. (Binet.)
J 0
12"4. The Eulerian. Integrxil of the First Kind.
The name Eulerian Integral of tJte First Kind was given by Legendre to
the integral
B(p,q)=[ xP-' ( 1 - x)i-^ dx,
Jo
which was first studied by Euler and Legendre*. In this integral, the real
parts of p and q are supposed to be positive; and xP~'^, (1 — x)i~'^ are to be
understood to mean those values of e*^~'""*''^ and e(7-i)iog(i-a;) which correspond
to the real determinations of the logarithms.
* Euler, Petrop. N. Coinm. xvi. (1772); Legendre, Exerclces, i. p. 221.
248 THE TRANSCENDENTAL FUNCTIONS [CHAP. XII
With these stipulations, it is easily seen that B {p, q) exists, as a (possibly
improper) integral (§ 4-5 example 2).
We have, on wi-iting {1 — x) for x,
B{p,q) = Biq,p).
Also, integrating by parts,
f xP-^ (1 - xy dx = [ — ^^ ~ '^^^ V 2 C ^'^ (1 - ^Y-' dx,
Jo L i^ Jo pJo ^ '
or B\p,q + \) = 'i^B{p-^\,q).
P
• Example 1. Shew that
' Example 2. Deduce from example 1 that
' Example 3. Prove that if n is a positive integer,
1.2...W
^to»+i)=^(p+i)...(^+„,
' Example 4. Prove that
Example 5. Prove that
^(-'^)=/,°"(r^.^«-
r(0)= lim n' B {z, n).
12-41. Expression of the Eulerian Integral of the First Kind in terms of
the Gamma function.
We shall now establish the important theorem
B {m, n) = ~-~r^^-^' .
1 (m + 7? )
First let the real parts of in and n exceed ^ ; then
r {m) V (n) = I e-=^ .^'«-i ^Z^- x I e~y y''~^ dy.
■f 0 Jo
On writing a;^ for x, and ?/- for ?/, this gives
[R rR
V ( m) r {n) = 4 hm I g-^- a'-'^-i dx x g-^/' y^^-i (it/
rll rB
= 4 lim 1 I e-^'^'+y-^ x-"'-Uf'''-^dxdy.
Now for the values of m and ?i under consideration the integrand is
continuous over the range of mtegration, and so the integral may be con-
sidered as a double integral taken over a square S^. Calling the integrand
12-41] THE GAMMA FUNCTION 249
f{x, y), and calling Qjt the quadrant with centre at the origin and radius R,
we have, if Tji be the part oi 8r outside Qr,
I f(^> y) dxdy - 1 1 fix, y) dxdy '
JJSr JjQji
= 1 fi^>y)d^dy\
^ I \f(^>y)\d^dy
</[ \f(^>y)\da;dy- \f{x,y)dxdy\
-*0 as R-* CO ,
since 1 1 \f{^, y) I dxdy converges to a limit, namely
2 I g-^' I a;2'»-i j cZiT X 2 I e"!'' j y''''-^ \ dy.
Jo Jo
Therefore
lim / / f(x, y) dxdy = lim f{x, y) dxdy.
Changing to polar* coordinates {x = r cos 6, y = r sin 6), we have
f(x, y)dxdy =^ |
Qr
Hence
[f /('^j y) dxdy =i f e-'- (r cos df-' (r sin (9)-"-^ rdrdd.
J J Qn ^ J 0 J 0
r (m) r (7^) = 4 e-'V^ (m+n)-i ^^ cos^"*-i 6 sin^^-^ ^c?(9
Jo J i)
2r (7?i + n) cos^'"-! (9 sin-"-i (9£^^.
Writing cos^ 0 = u we at once get
r (m) r (/?) = r (m + ?i) . i? (r?i, n).
This has only been proved when the real parts of in and n exceed | ; but
it can obviously be deduced when these are less than 2 ^Y § ^^'^ example 2.
This result connects the Eulerian Integral of the First Kind with the
Gamma-function.
Example 1. Shew that
J -1 ^ ^{P + 3)
* It is easily proved by the methods of § 4-11 that the areas Am.iJ. of § 4-3 need not be rect-
angles provided only that their greatest diameters can be made arbitrarily small by taking the
number of areas sufficiently large ; so the areas may be taken to be the regions bounded
by radii vectores and circular arcs.
250 THE TRANSCENDENTAL FUNCTIONS [CHAP, XII
Example 2. Shew that if
/^^,y;-_^ -^07 + 1+ 2r~^ + 2 3! ^ + 3+-
then
/(^.y)=/(y + l, ^-1),
where x and y have such values that the series are convergent. (Jesus, 1901.)
Example 3. Prove that
r [ f{^xy){\-xT-^y^{\-yY-^dxdy = ^-^$^ [ f{z){\-zr^^-Uz.
J 0 J 0 i V/^ + ''} y 0
(Math. Trip. 1894.)
12*42. Evaluation of trigonometrical integrals in terms of the Gamma-
function,
We can now evaluate the integral I coB'^~^xsm'^~'^ xdx, where m and n
Jo
are not restricted to be integers, but have their real parts positive.
For writing cos^r = t, we have, as in § 12'41,
, ' cos--^ sin->^- dx = I ^:(M^^ .
0 2 r{^m + ^n)
The well-known elementary formulae for the case in which m and n are
integers can be at once derived from this.
Example. Prove that, when \k\<\,
fm-\-\\ Ai+1\
/'^cos"'(9sin»(9c^(9_ ^ V 2 / ^ V"2~/ [l cos'" + " (9 c^(9
U Ti^^I^'^T* " ./.r('i±|^) ^" (ilTW?'
(Trinity, 1898.)
12"43. Pochhammers* extension of the Eiderian Integral of the First
Kind.
We have seen in § 12"22 that it is possible to replace the second Eulerian
integral for T {z) by a contour integral which converges for all values of z.
A similar process has been carried out by Pochhammer for Eulerian integrals
of the first kind.
Let P be any point on the real axis between 0 and 1 ; consider the
integral
ra + , Of, l-,0-)
e-^' (»+^> r-i (1 - tf-' dt = € (a, 13).
.' p
The notation employed is that introduced at the end of § 12-22 and
means that the path of integration starts from P, encircles the point 1 in the
positive (counter-clockwise) direction and returns to P, then encircles the
origin in the positive direction and returns to P, and so on.
* Math. Annalen, Bd. xxxv. p. 495.
12-42, 12-43]
THE GAMMA FUNCTION
251
At the starting-point the arguments of t and 1 —t are both zero ; after
the circuit (1 +) they are 0 and 27r ; after the circuit (0 +) they are 27r and
2'7r ; after the circuit (1 — ) they are 27r and 0 and after the circuit (0 — ) they
are both zero, so that the final value of the integrand is the same as the
initial value.
It is easily seen that, since the path of integration may be deformed in
any way so long as it does not pass over the branch points 0, 1 of the
integrand, the path may be taken to be that shewn in the figure, wherein
the four parallel lines are supposed to coincide with the real axis.
If the real parts of a and /3 are positive the integrals round the circles
tend to zero as the radii of the circles tend to zero * ; the integrands on the
paths marked a, h, c, d are
i»-i (1 - ty-\ ^-Hi - 0^~' e^"' *^~'^
^a—lg2TTi(a—l) Q _ A3-1 g27ri(/3-l) ^a-1 g27rt (a-1) Q _ A(3-T
respectively, the arguments of t and 1 — ^ now being zero in each case.
Hence we may write e(a, /3) as the sum of four (possibly improper)
integrals, thus :
e(a, ;Q)-e-"'''+^)
[ ^*-i (1 - tf-' dt+( ^-1 (1 - ty-' e'^'^ dt
Jo .' 1
dt
Hence
+ r f^-' (1 - ty-' e-^' <'^+^» dt + ( ^-1 (1 - 0^-'
Jo -1
€ (a, /3) = e-'^ <"+^' (1 - e''*'<^) (1 - e'^'^) f t''-' (1 - ty-' dt
Jo
... . . ._ ,r(a)r(^)
= - 4 sm (avr) sm {/dir) YToT+Jj
_ -47r'
"r(l-a)r(l-/3)r(a + /3)-
Now 6 (a, /3) and this last expression are analytic functions of a and of /3
for all values of a and /3. So, by the theory of analytic continuation, this
equality, proved when the real parts of a and /S are positive, holds for all
values of a and /3. Hence for all values of a and /3 lue have proved that
- 47r^
^ ^"' ^^ " r(i-a)ra-y8)r(a+yS) •
* The reader ought to have no difficulty in proving this.
252 THE TRANSCENDENTAL FUNCTIONS [CHAP. XII
12*5. Dirichlet's integral*.
We shall now shew how the repeated integral
I=Jj...Jf(t, + t,+ ...+ tn) «i"'-i «/^-^ . . . *n»"-^ dt, dh... dtn
may be reduced to a simple integral, where /is continuous, a^ > 0 (r = 1, 2, ...n)
and the integration is extended over all positive values of the variables such
that ^1 + ^2+ ... +^„^1.
To simplify f'^ f^"^' f{t+T^\)t'^-' T^-' dtdT
Jo .'0
(where we have written t, T, a, /3 for ti, t^, a^, a^ and A, for ^3 + ^4+ ... +tn),
put t= T{1 — v)/v ; the integral becomes (if X ^t 0)
r~^ r f{X+ T/v) (1 - vy-' v-^-^ T-^+^-i dv dT.
Jo J r/(i-A)
Changing the order of integration (§ 4*5 1), the integral becomes
f r^^^yiX + T/v) (1 - vy-' V-"-! T'^+i^-' dTdv.
J 0 J 0
Putting T ^ VT2, the integral becomes
I I V (\ + T2) (1 - y)«-i v^-' T/+^-i dr. <i-^^
Hence
J. ^ r («l)I>0 [ f . . /'y ( ^. ^ + . . . + t^^^ ^a^a^^ ^ .3-1 . . . tn'^n-^ cIt, dt, . . . dtn ,
the integration being extended over all positive values of the variables such
that T2 + 4 + • • • + ^n ^ 1-
Continually reducing in this way we get
r(7, + a,+ ...+an) Jo-^
which is Dirichlet's result.
Example 1. Reduce
to a simple integral ; the range of integration being extended over all positive valnes
of the variables such that
it being assumed that a, 6, c, a, /3, y, y>, f^, -r are positive. (Dirichlet.)
* Ges. Woke, i. pp. 375, 391.
125] THE GAMMA FUNCTION 253
Example 2. Evaluate j I xt>i/<'dxdy,
m and n being positive and
•^^0, y^O, ji-"»+y»<l. (Pembroke, 1907.)
Example 3. Shew that the moment of inertia of a homogeneous ellipsoid of unit
density, taken about the axis of z, is
- I (a2 + 62) „abc,
where a, b, c are the semi-axes.
Example 4. Shew that the area of the epicycloid x^+y^ = l^ is |jrR
REFERENCES.
N. Nielsen, Handbuch der Gamma-funktion*'.
0. ScHLOMlLCH, Compendium der hoheren Analysis, Bd. ii.
E. L. LiKDELOF, Le Calcul des Re'sidus, Ch. iv.
A. Pringsheim, Math. Ann. xxxi.
Miscellaneous Examples.
1. Shew that
('-'O+O 0-1) H)-'ni^ifvii-i7y
(Trinity, 1897.)
2. Shew that
J^lih iTP ITF- iTl^"^=^^"+'^- ^^""^^^' '^^'-^
3. Prove that
r' (I) v (i)
rfi-)-f^=2log2. (Jesus, 1903.)
4. Shew that
{r(i)}4 32 52-1 72 92-1 112
T6,r2 = 3^:11 • -^2- • 7231 • ^i^ • 11231 •••• (Trinity, 1891.)
5. Shew that
„=o [ (n+/3)(n + y) \ n + iyj tt ^ ^ v'-' /^
(Trinity, 1905.)
8 /r\ 640 / TT \3
6. Shew that n r ( -j = -g^- ( — j . (Peterhouse, 1906.)
7. Shew that, if z = i^ where f is real, then
'''^^^l==\/(c^rhTt)- (Trinity, 1904.)
8. When x is positive, shew thatt
-^- fJ^rZ 22»^! .,h- (^^^^h. Trip, 1897.)
* This work contains a complete biblioKvaphj'.
+ This and some other examples are most easily proved by the result of g 14-11.
254 THE TRANSCENDENTAL FUNCTIONS [CHAP. XII
9. If a is positive, shew that
r(g)r(a + l)_ I ( - )" g (ct - 1 ) (g - 2) . . . (g - ?i) J.
r (2 + g) „=o w ! 2 + ?i'
10. If.^;>Oand
Jo
shew that
''^^ X II x+i.2lx + 2 Slx + S'^'"'
and
.P(.r+l)=.rP(^)-e-i.
11. Shew that if X > 0, .-^ > 0, - ^tt < a < ^tt , then
I i^^-i e-A.<cosa cos (X; sin a) dt=\-=' T {x) cos a.r,
I ^x-ie-A<cosasin(Xi!sina)c/(; = X-^r(a,')sina.r. (Euler.)
12. Prove that, if 5 > 0 then, when 0 < 2 < 2,
r*^ sin 6.*' , , , ,, , , , ,
I — -^ a^ = -|7r6^-i cosec (i7r2)/r (0),
J 0 or
and, when 0 < s < 1,
1 ^^^^^c?.r=i7r6^-isec(|7rs)/r(2). (Euler.)
J a x^
13. If 0<»i<l, prove that
|J(l+.xO"-cos..c^. = r(.)|cos('-J-l)-^^ + ^--...}.
(Peterhoiise, 1895.)
14. By taking as contour of integration a parabola with its vertex at the origin, derive
from the formula
1 /■(0+)
V(a)=---r-. (-zY'-'^e-'dz
2i sm an J ^
the result
Y(a) = —. I e-'^Kv"-'^(l+x'^)^"'[3sm{x + aaYccot(-x)}
^ 2 suigTT jo \ / L I
+ sin {x+{a — 2) arc cot ( — .!•)}] dx,
the arc cot denoting an obtuse angle.
(Bourguet, Acta Mathematica, i. p. 367.)
15. Shew that, if the real part of a„ is positive and ^('g^)-*-oc , then
nr-i:l^^exp(I^^W(g„)n
is convergent when m > 2, where \//(*) (z) = -j-- log r (2). (Math. Trip. 1907.)
16. Prove that
d log r (s) /" " e - « — e - '
jo 1
0^2 / (1 1—e
da-
■y
= f''{(l+«)-'-(l+a)-}'^-y
1 x^-i \
~dx — y. (Legendre.
THE GAMMA FUNCTION 255
1 7. Prove that, when R (z) > 0,
18. Prove that, for all values of z except negative real values,
\ogT{z) = {z-^)logz-z + ^\og{2n)
( 1 - 1 2 " 1 3 =^ 1 ]
"*"* 12 . 3 rli (z + r)'^ "^3.4;.!, {z+rf "'■4.5,.!, (e+r)*"^ -J
19. Prove that, when R {z) > 0,
J^logr(.)=lug.-|^' ^^^^^{i-^+logx}.
20. Prove that, when R (2) > 0,
^-,logr(.) = j^ -^
21. If I logr(«)o?^=M,
log 2,
shew that
du
~dz
and deduce from § 12-33 that, for all values of z except negative real values,
?( = 2log2-2 + ^log27r. (Raabe, Crelle^xxw)
22. Prove that, for all values of z except negative real values,
, ^ , ,,, , . ,^ , "^ C^ dx sin 2?i7rd?
log r {z) = {z-\) log z-z+\ log (2;r) + 2 ;— - •
(Bourguet*.^
23. Prove that
B{p, p)B{p + h, pH) = ^^Y^- (Binet.)
24. Prove that, when —t<i'<t,
„, ,1 f"^ cosh. (2ru)du
25. Prove that, when g' > 1,
B{p, q)+B{p + \, q) + B{p + 2, j) + ... = Z?(p, q-]).
26. Prove that, when p — a > 0,
B{p-a,q)_ aq_ a{a+\)q{q+\)
B{p,q) ^p + q^ l.^.ip + qKp+q + iy-'-
^l. Prove that
B{p, q)B{p + q, r) = B{q, r) B {q + r, p). (Euler.)
28. Shew that
i' a-in- s,-, ^-^-^ ^r(a)r(6) 1__
jo*^ ^' ""^ (^■+p)«-'' r(a + 6) (l+p)«p'''
if rt > 0, 6 > 0, p > 0. (Trinity, 1908.)
* This result is attributed to Bourguet by Stieltjes, Liouville's Journal, (4) v. p. 43-2.
256 THE TRANSCENDENTAL FUNCTIONS [CHAP. XII
29. Shew that, if to>0, n>0, then
/
1 (l + .^^)2»»-l(l -,;)2»-i _, T(m)T{n) _
_1 (I+^2)m + n ' r{m + 7l) '
and deduce that, when a is real and not an integer multiple of ^tt,
'if /cos ^4-sin d\cos2a
/:
/cos ^4- sin d\c
Vcos 6 — sin ^/
c;^=
- in- Vcos ^ — sin 6/ 2 sin (tt cos^ a) '
(St John's, 1904.)
30. Shew that, if a > 0, ^ > 0,
and
' 0 (1 +0 log ^ ^'-^"^ r (ia) r (HP)- (Kummer.)
31 . Shew that, if a > 0, a -f- ^ > 0,
J 0
Deduce that, if in addition a + c>0, a + b4-c>0,
•ix^-i(l-x>')(l-a;') _ ^r(a)r(a + b+c)
(1 - .r) ( - log jcj" ■^- "^g r(a + />)r(a+c)
32. Shew that, if a, b, c be such that the integral converges,
-1(1-^^), 1- rr(a)r(8) V{a+b)T{h)\ , , , ,, ,,,
/:
P (l-.r'')(l-^)(l-.r<') r(6 + c+l)r(c+a4-l)r(a + 6 + l)
Jo (i-.t')(-log.r) ^ ^°r(a+i)r(6+i)r(c+i)r(a+6 + c+i)
33. By the substitution cos^ = l - 2 tan |0, shew that
f y^ mP. (St John',, 18%.)
J 0 (3 - cos ^)* 4 ^TT
34. Evaluate in terms of Gamma- functions the integral / da;, when jo is a
jo -v
fraction greater than unity whose numerator and denominator are both odd integers.
[Shew that the integral is i I sin^ x \--\- 2 (-)"( h I \ dxJ]
'Jo U' )(=! \x + n7r x-nnj]
(Clare, 1898.)
35. Shew that
(l-ism^:^) 2 dx = , 2 -.-7 rr -^r —n ■
36. Prove that
log B (p, ,)==log (P±^) + f ^ (1^(1^^^) rf,. (Euler.)
37. Prove that, if p > 0, p + s > 0, then
B(r ^\A-^(P^P) fii ^(^-1) I ^(^-l)(^-2)(^-3) I
^(P, ^ + ^)- 2« V + 272^1) + 2.4.(2;. + l)(2^ + 3) + -r ^^^"''*-)
38. The curve r'» = 2'"~i a'«-cos ra^ is composed of m equal closed loops. Shew that
the length of the arc of half of one of the loops is
fi^ --1
m ^ a j (^cos.r)™ dx,
J 0
and hence that the total perimeter of the curve is
THE GAMMA FUNCTION 257
39. Draw the straight line joining the points ±t, and the semicircle of | « | = 1 which
lies on the right of this line. Let C be the contour formed by indenting this figure at
-I, 0, t. By considering / «p~9 (2 + 2~^)P + «~2rf2;, shew that, if /) + j> 1, g'<|,
io •
COSP + 9-2^COS(»-o)^C?(9=, ,x^„4.. , D/ ^v •
Prove that the result is true for all values of p and q such that jo + 5' > 1.
(Cauchy.)
40. If s is positive (not necessarily integral), and -^jr^.r^j7r, shew that
1 r(.? + l) f, « „ s(«-2) , . 1
COS*^= r 7 — -4t;; \\-\ ;rCOS 2X'\- , \. , '. COS 4X+...} ,
and draw graphs of the series and of the function cos* x.
41. Obtain the expansion
cos»^-=— -rfa + nf C08a^p_____ cos 3ax "I
2.-1 y^ ^|_r(j,+^« + i)r(is-ja+i)"^r(i«+ia+i)r(i«-fa + i)'^-J'
and find the values of x for which it is applicable. (Cauchy.)
42. Prove that, if jo > ^,
i^p, ^^_ ii^pji !_2p + l r + 2(2p + 3)^2.4.(2p+3)(2jo + 5) + -jJ "
(Binet.)
43. Shew that, if ^<0, a; + 2>0, then
r(-x) (-X ^{-a;)(l-x) J-^)(l-^)(2-^) 1
T{z) \ z ^- z{l+z) ■^•■^ 2(1+2) (2 + 2) ■^•••j
=r(^)/o^-^-'^-^«g(^-^)Hi-0--^^^,
and deduce that, when x+z>0,
d
dz
lo„r(2 + ^)^^ ^x{x-l) x{x-\){x-2)_
■ ^ T{z) z '^ z{z-\-\)'^'^ z{z + \){z-\-2) ■••*
44. Using the result of example 43, prove that
logr(2 + a) = logr(2)+alog2-— ^
a\ t{l-t){2-t)...{n-t)dt- I t (l-t) {2-t) ...(n- t) dt
_ I .' 0 j_o
11=1 {n + l)z{z + l){z + 2) ...{z + nj '
investigating the region of convergence of the series.
(Binet, Journal de I' e'cole poly technique, 1839, cahier 27, p. 256.)
45. Prove that, ii p > 0, q > 0, then
B(p,q)=P'-'-f;~^(2n)^e-^"-'\
W. M. A. 17
258 THE TRANSCENDENTAL FUNCTIONS [CHAP. XII
where
and p^=p^-\-(^-\-pq.
46. If r=2**/r(l-i^), V='L^\V{\-\x\
and if the function Fix) be defined by the equation
shew (1) that F {x) satisfies the equation
F{x->r\) = xF{x)^- ^
r(i-a?)'
(2) that, for all positive integral values of x,
Fix) = T(x),
(3) that F{x) is analytic for all finite values of x,
'1 -X
(4) that i.(..)=-i-^^, log-i-1-
47. Expand
{r(a)}-i
as a series of ascending powers of a.
(Various evaluations of the coefl&cients in this expansion have been given by Bourguet,
Btdl. des Sci. Math. v. (1881), p. 43 ; Bourguet, Acta Math. ii. (1883), p. 261 ; Schlbmilch,
Zeitschrift fur Math. xxv. (1880), pp. 35, 351.)
48. Prove that the G^-function, defined by the equation
(?(.+ l) = (2.r)i^e-*^(^+l)-i^^\n^ iO'^O" ^""^'''^'"
is an integral function which satisfies the relations
G{z+l) = T{z)0{z\ G{\) = \,
(n \)''10 (» + 1) = 11 . 22 . 33 ...»». (Alexeiewsky.)
(The most important properties of the G'-function are discussed in Barnes' memoir
QvAirterly Journal, xxxi.)
49. Shew that
'(2«)l ,
and deduce that
50. Shew that
G'iz->r\) , , ,^ X 1 T'{z)
^l^ = ilog(2.) + i-. + .^^
log ---^ {= I •rrz cot TTzdz-zlog {27r).
riogr(< + l)(^^ = lslog(27r)-|2(z+l)+0logr(2+l)-logG'(s-f-l).
CHAPTER XIII
THE ZETA FUNCTION OF RIEMANN
IZ'l. Definition of the Zeta-function.
Let s = cr + it where a and t are real* ; then if S > 0, the series
is a uniformly convergent series of analytic functions (§§ 2*33, 3*34) in any
domain in which cr'^l + S; and consequently the series is an analytic function
of s in such a domain. The function is called the Zeta-function ; although
it was known to Euler j-, its most remarkable properties were not discovered
before RiemannJ who discussed it in his memoir on prime numbers; it has
since proved to be of fundamental importance, not only in the Theory of
Prime Numbers, but also in the higher theory of the Gamma-function and
allied functions.
13'11. The generalised Zetaf unction^.
Many of the properties possessed by the Zeta-function are particular cases
of properties possessed by a more general function defined, when a'^1 + 8,
by the equation
where a is a constant. For simplicity, we shall suppose || that 0 <a^l, and
then we take arg {a + n) = 0. It is evident that ^(s, 1) = ^(s).
* The letters o-, t will be used in this sense throughout the chapter.
t Commentationes Acad. Sci. Imp. PetropoUtanae, Vol. ix. (1737), pp. 160-188.
:;: Ges. Werke (1876), pp. 136-144.
§ The introduction of this function appears to be due to Hurwitz, Zeitschrift fiir Math, itiid
Phys. XXVII. (1882).
II When a has this range of values, the properties of the function are, in general, much
simpler than the corresponding properties for other values of a. The results of § 13-14 are true
for all values of a (negative integer values excepted); and the results of §§ 13-1'2, 13-13, 13-2 are
true when R (a) > 0.
17—2
260 THE TRANSCENDENTAL FUNCTIONS [CHAP. XIII
13'12. The expression of ^ (s, a) as an infinite integral.
Since (a + w)"* T (s) = I af-'^e-^'^+''^''dx, when arg x = 0 and a >0 (and
JO
a fortiori when a ^1 + S), we have, when <r^l +8,
r (s) ^ (5, a^ = lim t r af-'e-"'+''^='dx
X'-^x, n=OJ 0
= lim ^ I ^^^-- dx - r -r^^ e-(^+i+«)^c^a;[
CO /pS Ip dX
jv-oc U'o 1-e-* Jo l-<?-^ j
Now, when a; ^ 0, e^ ^ 1 + a;, and so the modulus of the second of these
integrals does not exceed
f " af-'e-^^+''^^dx = (N+ ay-^r (o- - 1),
^0
which (when a "^1+8) tends to 0 as JV ^ oo .
Hence, when o- ^ 1 + S and arg x = 0,
this formula corresponds in some respects to Euler's integral for the Gamma-
function.
13*13. The expression* of ^{s, a) as a contour integral.
When o- ^ 1 + S, consider
/ ^ 1 — e~^
l-e-
the contour of integration being of Hankel's type (§ 12'22) and not containing
the points ±2n7ri(n=l, 2, 8, ...) which are poles of the integrand; it is
supposed (as in § 12"22) that | arg (— 2^) | ^ tt.
It is legitimate to modify the contour, precisely as in § 12'22, when^f*
o- ^ 1 + S ; and we get
/•(0+) / ^\s—ip—az foo ^s—ip—ax
^ /^ — d2= {e'^<«-^> - e--<»-i)} ,-- ^ dx.
Therefore
Now this last integral is a one-valued analytic function of s for all values
of s. Hence the only possible singularities of ^(s, a) are at the singularities
of r(l— s), i.e. at the points 1, 2, 3, ... and, with the exception of these
points, the integral affords a representation of ^{s, a) valid over the whole
* Given by Riemann for the ordinary Zeta-function.
t If 0- < 1, the integral taken along any straight line up to the origin does not converge.
1312-1314] THE ZETA FUNCTION OF RIEMANN 261
plane. The result obtained corresponds to Hankel's integral for the Gamma-
function. Also, we have seen that f (s, a) is analytic when cr ^ 1 + S, and
so the only singularity of ^(s, a) is at the point 5 = 1. Writing s= 1 in the
integral, we get
1 r(0+) g-az
which is the residue at £; = 0 of the integrand, and this residue is 1.
Hence lim ^^^ — : = — 1.
^^ir(l-5)
Since F (1 — s) has a single pole at 5 = 1 with residue — 1, it follows that
the only singularity of ^{s, a) is a simple pole with residue + 1 at « = 1.
Example 1. Shew that, when R (s) > 0,
r(s) jo
.8-1
dx.
e^ + l
Example 2. Shew that, when It (s) > 1,
(20 -I) as) = as, i)
2' f^ x»-^e^
T{s) J 0 e^ - 1
Example 3. Shew that
/:
dx.
2i-»r(i-g) /•(0+) (_2)«-i
2iri{2^-'-l) J ^ e' + l '
where the contour does not inchide any of the points ±7ri, ±3Tri, ±5Tri,
13'14. Values of ^{s, a) for special values of s.
In the special case when s is an integer (positive or negative), -^- — ~ —
is a one- valued function of 2'. We may consequently apply Cauchy's theorem,
1 /•(0+) (■_ ^y-ig-az
so that ^ — ; I i"^^ — ^ ^^ ^s ^h® residue of the integrand at 0 = 0, that
/_\«— ig— a^
is to say, it is the coefficient of z~^ in \ .
•^ 1 — e~2
To obtain this coefficient we differentiate the expansion (§ 7*2)
e-az_l ^ (-)«<^„(«)0«
— z ^ ^
e-^-1 „ = i // !
term-by-term with regard to a, where (f)n{a) denotes the Bernoullian poly-
nomial.
(This is obviously legitimate, by § 4"7, when \z\<2Tr, since _^— can be expanded
into a power series in z uniformly convergent with respect to a.)
Then ^-^-= I (-il*!>:)i"
e-^ — 1 « = i n\
262 THE TRANSCENDENTAL FUNCTIONS [CHAP. XIII
Therefore if s is zero or a negative integer (= — m), we have
f (- m,a) = - (/)V2 {a)l[{m + 1) (m + 2)}.
In the special case when a = 1, if 5 = — m, then f (5) is the coefficient
of 2;^~« in the expansion of - L -i •
Hence, by § 7-2,
r(-2m) = 0, r(l-2m) = (-)-5J(2m) (m = 1, 2, 3, ...),
TAese equations give the value of ^(s) when s is a negative integer or zero.
13'15. The formula* of Hurwitz for ^(s, a) when a<0.
Consider — - — ; / — ^^^^ dz taken round a contour G consisting of
a (large) circle of radius (2iV+l)7r, (N an integer), starting at the point
(2iV+ l)7r and encircling the origin in the positive direction, arg (—z) being
zero at ^ = - (2iV + 1) tt.
In the region between C and the contour (2N7r + tt ; 0 +), of which the
contour of §13'13 is the limiting form, (— ^)*~^e~"^(l — e~^)~^ is analytic and
one-valued except at the simple poles + 27ri, ± Atiri, . . . , + 2i\^7ri.
Hence
1 r ( .).-..-.^ _ 1 r<-> (-.)-.- ^^ ^ I ■
where i?„, R^ are the residues of the integrand at 2^t7^^, — Irnri respectively.
Where — z= 27nre~'^^\ the residue is
(2/i7r)'-^e-^''^('-l)e-2«'»'^\
and hence Rn + Rn = {2mrf-'^ 2 sin f ^ -stt + ^iraii) .
Hence
1 /-(o-^) (-4^-ie-«^
27rij(2jv+ij7r 1 — e
_2sin-^S7r E. cos(27ran) 2 cos ^ sir ^ sin(27rcm)
~ 727r)i-^ ,,ri ^^^^^ "^ (27r)i-« " „=i ^^^
; 1 n-zy-^e--^
^27riJa l-e-^ '^'-
Now, since 0<a^l, it is easy to see that we can find a number K
independent of N such that \ e~"^ (1 — e~^')~'^ j < K when 5; is on C.
* Zeitschriftfiir Math, und Phys. xxvii. (1882), p. 95.
1 31 5-13-2] THE ZETA FUNCTION OF RIEMANN 268
Hence
<ir{(2iV+l)7r|'^e-l*"
-^0 as, iV ^ 00 if o- < 0.
Making iV -*- x , we obtain the result of Hurwitz that, if o- < 0,
-., . 2r (!-*)(. (\ \ :?, cos(27ran) l\ \ 5. sin(27raw))
each of these series being convergent.
13*151. Riemamis relation between ^(s) and ^(1 — s).
If we write a = 1 in the formula of Hurwitz given in § 13"15, and employ
§ 12*14, we get the remarkable result, due to Riemann, that
2i-«r(s) ^(s)cos Qstt] = 7r«^(l -s).
Since both sides of this equation are analytic functions of s, save for isolated
values of s at which they have poles, this equation, proved when a < 0,
persists (by § 5*5) for all values of s save those isolated values.
Example 1. If m be a positive integer, shew that
C (2m) = 22'" - 1 TT^*" BJi2m) ! .
Example 2. Shew that r (|«) n' ** f («) is unaltered by replacing s by 1 - s.
(Riemann.)
Example 3. . Deduce from Riemann's relation that the zeros of f (s) at — 2, — 4, — 6, . . .
are zeros of the first order.
13*2. Hermite's* formula for ^{s, a).
Let us apply the result of example 7 of Chapter vii (p. 145) to the
function <f) (z) = (a + z)~^, where arg (a + z) has its principal value.
Define the function q {x, y) by the equation
q {x, y) = 2i '^^^ "^ ^ "^ *2^^~' -{a + x- iy)-']
= - {(a + xy + y-] " *^ sin \s arc tan -— [ .
\ V \ v\
Sincef arc tan "^ does not exceed the smaller of A-tt and - , we
X -\- a • x + a
have
\q(x,y)\^{(a + xy + yf^-^^''\y-'\smhi^^'n-\s\\,
k(^,2/)kl(« + ^)^ + .y1~^" jjsinh^lj.
* Annali di Matematica, ser. in. t. v. (1901).
t If ^ > 0, arc tan ?= I , — - < , . ,; and arc tan ? < | dt.
^ Ji) l + £- jo 1 + t- jo
264 THE TRANSCENDENTAL FUNCTIONS [CHAP. XIII
Using the first result when \y\> a and the second when \y\<a it is
evident that, if o- > 0, 1 q {x, y) {e^^'v — 1)~^ dy is convergent when x'^Q and
J a
tends to 0 as « •-*- 00 ; also I (a + x)~^ dx converges if o- > 1.
Hence, if o- > 1, it is legitimate to make x^-*'qc in the result contained in
the example cited ; and we have
^ {s, a) = 1 a-« + r (a + x)-' dx+2J {a^ + y') - ^^ jsin (s arc tan ^^1 ~.~^— .
So
Us,a) = la-^ + ^^ + 2J\a^ + yr'~'{^^{s^rctanf^^^^^
This is Hermite's formula*; using the results that, ify'^0,
arc tan 2//a ^ 2//a (y<-a7r\, arc tan 2//a < - tt (y > ^air] ,
we see that the integral involved in the formula converges for all values of s.
Further, the integral defines an analytic function of s for all values of s.
To prove this, it is sufficient (§ 5-31) to shew that the integral obtained by diflferentiating
under the sign of integration converges uniformly ; that is to say we have to prove that
j^ ^-^log{a'+f){a'+f) ^«sin(sarctan|)J-5^^
converges uniformly with respect to s in any domain of values of s. Now when | s | :^ A,
where A is any positive number, we have
{a? + y2) - i« arc tan ^ cos is arc tan - ) < (a^ + ?/2)*^ -^ cosh (^ttA) ;
I "^ a \ a) \ '' a ^~ '
suice - I (a^ + iAf ^^ —
a Jo ^ -^ ' p}'y-\
converges, the second integi-al converges uniformly by § 4'431 (I).
By dividing the path of integration of the first integral into two parts (0, \na),
(Jn-(X, 00 ) and using the results
sin ( s arc tan - ) < sinh — ,
sin I s arc tan - ) < sinh in-A
aj \ a ' \ aj \
in the respective parts, we can similarly shew that the first integral converges uniformly.
Consequently Hermite's formula is valid (§ 5"5) for all values of s, and
it is legitimate to differentiate under the sign of integration, and the
differentiated integral is a continuous function of s.
* The corresponding formula when a = l had been previously given by Jensen.
N.
13-21, 13-3] THE ZETA FUNCTION OF RIEMANN 265
13'21. Deductions from Hermite's formula.
Writing s = 0 in Hermite's formula, we see that
f(0,a) = i-a.
Making s-*-l,from the uniformity of convergence of the integral involved
in Hermite's formula we see that
Hence, by the example of § 12*32, we have
1 ) T'(a)
Further, differentiating* the formula for ^{s, a) and then making 5-^0,
we get
+ 2 1 \-^\og(a^ + y'').{a'' + y-)~^^sm(sarctan^\
+ (a^ + v^)~^^ arc tan ^ cos f s arc tan ^H „ ^ ,
/ i\ , , o /*°° 9,rc tan (;?//«) ,
= ^a--jloga-a+2J^ ^^^^ ^^ Uy.
Hence, by § 12-32,
|^^(.,a)|^^^^ = logr(a)-^log(27r).
These results had previously been obtained in a different manner by
Lerchf.
Corollary. lim | ^ (s) - -^ 1 = 7, ^' (0) = - ^ log 27r.
13"3. Euler's product for ^(s).
Let o- ^ 1 + 8 ; and let 2, 3, 5, ... /j ... be the prime numbers in order.
Then, subtracting the series for 2~* ^(s) from the series for ^ (s), we get
^(.).(l-2-0 = ^^ + ^,+J,+ ^, + ...,
* This was justified in § 13-2.
t The formula for f (s, a) from which Lerch derived these results is given in a memoir
published by the Academy of Sciences of Prague. A summary of his memoir is contained in
the Jahrbuch der Math. 1893-1894, p. 484.
266 THE TRANSCENDENTAL FUNCTIONS [CHAP. XIII
all the terms of Xn~^ for which w is a multiple of 2 being omitted ; then in
like manner
r (.).(! -2-) (1-3-)=^+! + ^+...,
all the terms for which w is a multiple of 2 or 3 being omitted ; and so on ;
so that
f (s) . (1 - 2-*) (1 - 3-*) ... (1 -p-')=l + Sn-\
the ' denoting that only those values of n (greater than p) which are prime
to 2, 3, ... p occur in the summation.
CO
Now* |2'n-«j^2'w~^~^< % n-^-^ ^0 as p^oo.
Therefore if a- "^ 1 + B, the product ^(s)II (1 — p~^) converges to 1, where
p
the number p assumes the prime values 2, 3, 5, ... only.
But the product H (1 — p~*) converges when <r ^ 1 + S, for it consists of
p
00
some of the factors of the absolutely convergent product IT (1 — n~^).
M = 2
Consequently we infer that f (s) has no zeros at which cr ^ 1 + 8 ; for if
it had any such zeros, IT (1 - p~^) would not converge at them.
p
Therefore, if o- ^ 1 + S,
This is Euler's result.
IS'Sl. Riemanns hypothesis concerning the zeros of f (s).
It has just been proved that ^{s) has no zeros at which cr > 1.
From the formula (§ 13'] 51)
r(5)=2*-i{r(s)}-^secfj57r)r(l-«) " ^
,2
it is now apparent that the only zeros of ^(s) for which o- < 0 are the zeros
of ■[r(s)|~^sec [- stt) , i.e. the points s = — 2, — 4, —
^2
Hence all the zeros of ^(s) except those at -2,-4, ... lie in that strip of
the domain of the complex variable s which is defined by 0 ^ o- ^ 1.
It was conjectured by Riemann, bub it has not yet been proved, that all
the zeros of ^(s) in this strip lie on the line o" = «; while it has quite recently
been proved by Hardy f that an infinity of zeros of ^(s) actually lie on cr = - .
It is highly probable that Riemarm's conjecture is correct, and the proof of
it would have far-reaching consequences in the theory of Prime Numbers.
* The first term of S' starts with the prime next greater than i).
t Comptes Rendus, t. clviii. (1914), p. 1012.
13'31, 13-4] THE ZETA FUNCTION OF RIEMANN 267
13*4. Riemann's integral for ^(s).
It is easy to see that, if o- > 0,
Hence, when <r> 0,
^{s)r(ls]'7r-^'= lim f f e-'^'^'^x^'-'^ dx.
\^ / JV-^Qo J 0 n = l
00
Now, if «•(«!)= S e""^""^, since, by example 17 of Chapter vi (p. 124),
1 + 2'bt(x) = x~^ {1 + 2-57 (l/x)}, we have lim x^ia (x) = I; and hence
a:-*"0 /:_
/•QO
I cr(a;)a;^*~-'^(Zic converges when o- > 1.
Jo
Consequently/, if a > 2,
^{s)r(ls)7r~^'= lim \r ny(x)x^'-'^ds-r t e' ''"'''' x^'-'^ dxl .
\^ J JV"-*oo L^O Jo n=N+-i J
Now, as in § 13'12, the modulus of the last integral does not exceed
00 ,- ex, ._,■>,. r<x> g-(-N'+l)^TX^^(r-l
0 U=iv+i J Jo i-e (^ + ^)^^
Jo
^ 0 as iV ^ 00 , since o- > 2.
Hence, when cr > 2,
= n_l + la;-^+a;-ii!r(l/a;)U^'-^fZa;+r'sr(a;)a;*^-l(^a;
= -- + -^ + f a;*w(^)a^-*' + ^ (--] dx+r^(x)xi'-\dx.
S S — 1 J oc \ ^'J J 1
Consequently
Now the integral on the right represents an analytic function of s for all
values of s, by § 5'32, since on the path of integration
•GT (x) < e-""" t e-"''^^ < e-""" (1 - e-'^)-^
M = 0
Consequently, by § 5"5, the above equation, proved when a- > 2, persists for
all values of s.
268 THE TRANSCENDENTAL FUNCTIONS [CHAP. XIII
If now we put
we have
no = 2 ~ (^^ + 4) I ^~ ^ ^ («) COS (- ^ log x] dx.
Since I x~ 'i zr {x) i-Xog x y cos f - ^ log a; + - n-n- j cZa?
satisfies the test of § 4*44 corollary, Ave may differentiate any number of times
under the sign of integration, and then put ^ = 0. Hence, by Taylor's
theorem, we have for all values* of t
by considering the last integral a^^ is obviously real.
This result is fundamental in Riemann's researches.
13'5. Inequalities satisfied by ^ {s, a) when a > 0.
We shall now investigate the behaviour of ^ (s, a) as ^ -*- + cx) , for given values of a.
When o- > 1, it is easy to see that, if JV be any integer,
Us, ■')=i„CH-")--(i_,)(],^^).,.-Jy„W
where
. , X j_ r 1 1 ] 1
•^"^^ 1-s t(?i + l +«)«-! (?i + a)»-ij (n + l + ay
/"" + ! u-n
= s I , —-r-.au.
■w- , fn + l ■)/ —Qi
Isow, when a>0, \f„ {s)\^\s\ ,. du
J n (u + a)
(u + a)
'" + 1 du
= \s\{n + a)-^~'^.
CO
Therefore the series 2 /„ (s) is a uniformly convergent series of analytic functions
cc
when cr > 0 ; so that 2 f^ (s) is an analytic function when o- > 0 ; and consequently, by
§ 5-5, the function ^{s, a) may be defined when o- > 0 by the series
C{s,a)= 2 (« + ,,)-- --—-^—— -J- i /„(.).
11=0 [I — s) {jy -t a)" n=N
Now let [t] be the greatest integer in | f | ; and take N'=[t]. Then
[t]
|C(s, «)|^ 2 |(« + n)-«| + !{(l-s)-i(W + «y-1l+ 2 \s\{n + a)-''~'^
«=0 n=[t]
[t] ^
< 2 {a + n)-'' + \t\-^([t] + a)^-'' + \s\ 2 {n + a)-''-\
« = 0 „=[t]
* In this particular piece of analysis it is convenient to regard « as a complex variable,
defined by the equation s = ^ + it; and then f (<) is an integral function of t.
13-5, 13-51] THE ZETA FUNCTION OF RIEMANN 269
Using the Maclaurin-Cauchy sum formula (§ 4*43), we get
\C{s,a)\<a-''+(^'\a+x)-''dx'+\t\-^{[t] + a)^-''+\8\ I (x + a)-"-'^ dx.
Jo J [t]-i
Now when S ^ o- < 1 - 8 where S > 0, we have
Hence ^{s,a) = 0{\t |^~*^), the constant implied in the symbol 0 being independent of s.
But, when 1 — 8<o-^l+S, we have
\C{s,a)\ = 0{\t\}-'')-\-\ {a+xydx
<0{\t\^-'') + {a^-'' + {a + tf-''} \^'\a+x)-'^dx,
J 0
since {a + x)~'^ ^a^'" {a-\-x)-'^ when o-^l, and {a + x)'"^ ■^{a + {t]f~'^ {a+xy^ when
«r^ 1, and so
Wheno-^l + S,
\C{s,a)\^a-''+ 2 (a + n)-i-« = 0(l).
n=l
13*51. Inequalities satisfied by ( {s, a) when tr ^ 0.
We next obtain inequaUties of a similar nature when o-^S. In the case of the
function f (s) we use Riemann's relation
f (s) = 2« ,r»-i r (1 - s) t(l - s)sin (^ stt).
Now when tr < 1 - 8, we have, by § 12 "33,
r (1 - 5) = 0 {e^ -«) log (1 -»)-(! -s)j
and so
f(«) = 0[exp{|7r|«|4-(|-o--z01ogl 1 -s | + i arctan i;/(l -o-)}] f (1 -s).
Since arc tan^/(l -cr)= ±-|7r+0 (^~*), according as t is positive or negative, we see, from
the results already obtained for ({s, a), that
In the case of the function ^(s, a), we have to use the formula of Hurwitz (§ 13'15)
to obtain the generalisation of this result ; we have, when o- < 0,
C {s, a)=-i (2,r)«-i r (1 - .) [e**-" ^ (1 "«) " ^"**"^ C-a (1 - «)]
where f„(l_s)= 2 -j-.
Hence (i _ e^'*") ^^ ( x _ 5) = e27r»a _^ ^ g2«,r;a j-^^g - 1 _ (^^ _ j )8 - 1]
+ {s-\) 2 e^'*^* [" u»-^d^^,
since the series on the right is a uniformly convergent series of analytic functions
whenever a- ^ 1 - S, this equation gives the continuation of ^„ (1 - s) over the range
O^o-^l — 8; so that, whenever o- ^ 1 — 8, we have
!sin7raf„(l-s)|$l+ 2 {/i''~^ + ('/i-l)''~'} + l s-1 I 2 I xi'-'^du.
n='Z n~N+l J n-1
270 THE TRANSCENDENTAL FUNCTIONS [CHAP. XIII
Taking N'=[i], we obtain, as in § 13"5,
= 0(l<riog|i|) (-S^o-<8).
And obviously
Cail-S) = 0{l) (0-<-S).
Consequently, whether a is unity or not, we have the results
C{s,a) = Oi\t\^--'') (<r<8)
= 0(1^!*) (S^o-^l-S)
= 0(|«|*log| ;|) (-8^0-^S).
We may combine these results and those of § 13-5, into the single formula
C{s,a)=0{\tr^''hog\t\)
where*
r(o-) = -^-o-, (o-=$0); r(cr) = |, (O^o-^i); t{(t) = 1-<t, (^<o-^1) ; t(o-) = 0, (<r^I);
and the log 1 1 j may be suppressed except when - S ^ w ^ 8 or when 1— S^cr^l + S.
13'6. The asymptotic expansion of log F (z + a).
From § 12'1 example 3, it follows that
a'»=ilV a + n) j r(z + a)
Now, the principal values of the logarithms being taken,
log
n ifi+_^^,-./4= i rf--- U i (-1^-.-^
00 / \m—i ^m
,„ti m a"""'
If I 2; ] < a, the double series is absolutely convergent since
a\z\ , ( ^ \z\ \ \z
- log 1 - -^-^ +
n=\ \j'^ (ci + n) '^ \ a + nj a + n
converges.
Consequently
, e-y^r(a) z ^. az ^ (-)'""'
_ X r 7r^«
Now consider ^ — . I . - — tl(s, a) ds, the contour of integration beins'
similar to that of § 12'22 enclosing the points s = 2, 3, 4, ... but not the
points 1, 0, —1, —2, ...; the residue of the integrand at s = w (?w ^ 2) is
^m, ^77i^ a^ • and since, as o- — x (where s = a + it), ^{s, a)= 0 (1), the
integral converges if j0;< 1.
* It can be proved that r {<t) may be taken to be i (1 - <r) when 0 -^ cr^ 1. See Landau, Prim-
zahleii, § 237.
13*6] THE ZETA FUNCTION OF BIEMANN 271
Consequently
log j^, — -=^: = - - S —7--^ V - o— . — r - ^{s, a) ds.
Hence
- r(a) r{a) 1 f 7rg«
log _ ,— ^^ = — ^ „ , , — TT — ; I — ^ c (s, a) rt5.
^ r (^^ + a) r (a) 27rt J ^ * sm tt* * ^ ^
Now let Z) be a semi-circle of (large) radius N with centre at s = f , the
semi-circle lying on the right of the line o- = |. On this semi-circle
f(5, a) =0(1), |^*[ = |^|<^e-^'*'"»^ and so the integrand is* 0 {|^ j<'e-''l^i-^"'"K^}.
Hence if | ^ | < 1 and — tt -f- S ^ arg 2^ ^ tt — 8, where h is positive, the integrand
is 0(1 2^ {""e"* 1*^1), and hence
J 2) S sm 7r5
as iV -»- 00 . It follows at once that, if | arg z\-^'k — Z and | 2^ | < 1,
log T^ r^^^\ = — z -^-,- i + ^ — ; — -. t(s, a) ds.
But this integral defines an analytic function of z for all values of | ^^ | if
I arg z\^7r — S.
Hence, by § 5*5, the above equation, proved when I ^ | < 1, persists for all
values of 1 2^ I when i arg ^ j ^ tt — S.
Now consider I — -. f (s, a) ds, where n is a fixed integer and
R is going to tend to infinity. By § 13*51, the integrand is 0 [z'^e~^^'^ R''^'^'>\
where —n --:$cr^-; and hence if the upper signs be taken, or if the lower
signs be taken, the integral tends to zero as i^ -*- 00 .
Therefore, by Cauchy's theorem,
where R^ is the residue of the integrand at s = — m.
Now, on the new path of integration
'^^' ^(5, a)i<ii:^-"-^e-^l"^(-"-i)U|,
.S sm TTS
where K is independent of z and t, and t(o-) is the function defined in
§ 13*51.
* The constants implied in the symbol 0 are independent of s and z throughout.
272 THE TRANSCENDENTAL FUNCTIONS [CHAP. XIII
rcc
Consequently, since I e ~ ^ I ^ ' I ^ | ^ ( " " " 4) cZ^ converges, we have
J — CO
when I ^ I is large.
Now, when m is a positive integer, Rm = ~ ^-^^ ' ^ and so
— m '
by §13-14, R^ = y-l—^^^^±^ where </>,„' (a) denotes the derivate of
Bernoulli's polynomial.
Also Ro is the residue at s = 0 of
1/. . TT^S^
s
l + '^+---)(l+slog^+...)||-a + <(0, a)+...l,
and so R^ = i-^ — a] log z + ^' (0, a)
= (^ - «) log ^ + log r (a) - I log (27r),
by §13-21.
And, using §13-21, R^i is the residue* at >Sf= 0 of
1 /-. c* o. N^ '^''Sf2 N ., ^. /I F(a) N
__(l_S + S=-...)^l + ^ + ..,). (l+Slog.+ ...)(g--piJ + ...).
Hence jR_i = — ^ log 2^ + ^ t-, 7^ + z.
1 (a)
Consequently, finally, if | arg z\ ^tt — B and | ^ | is large,
log r {z + a) = iz + a — ~] log z — z + ~ log(27r)
mti m (77i + 1) (m + 2) ^^« ^ '^ ^•
In the special case when a=l, this reduces to the formula found
previously in §12-33 for a more restricted range of values of arg^.
The asymptotic expansion just obtained is valid when a is not restricted
by the inequality 0 < a ^ 1 ; but the investigation of it involves the rather
more elaborate methods which are necessary for obtaining inequalities satisfied
by ^(s, a) when a does not satisfy the inequality 0< a ^ 1. But if, in the
formula just obtained, we write a = 1 and then put z + a for z, it is easily
seen that, when | arg (z + a,) [ ^^ tt — 8, we have
\ogr{z + a+l) = (z + a+l^ log {2 + a)-z-a + l log (27r) + 0 (1) ;
* Writing s — S + 1.
13-6] THE ZETA FUNCTION OF RIEMANN 273
subtracting log (z + a) from each side, we easily see that when both
I arg (z + a)\^7r — B and | arg ^ | :$ tt — 3,
we have the asymptotic formula
\ogr{z + a)=(^z + a-^\ogz-z + l log (27r) + o (1),
where the expression which is o (1) tends to zero as \z\-* cc .
REFERENCES.
G. F, B. RiEMANN, Oes. Werke, pp. 145-155.
E. G. H. Landau, Handhuch der Primzahlen.
E. L. LiNDELOF, Le Calcul des Residus, Chap. I v.
E. W. Barnes, Messenger of Mathematics, Vol. xxix. (1899), pp. 64-128.
Miscellaneous Examples.
1. Shew that
(2«- 1) C(«) = ^^' + 2 IJ (i+y^)"*' sin (arc tan 2y) ^-2^^^- .
(Jensen.)
2. Shew that
98-1 f" , dy
CW = — T-2» (l+/)-i*sm(arctany) — ^.
s-i. J 0 e"^ + 1
(Jensen.)
3. Discuss the asymptotic expansion of log O {z + a), (Chapter xii, example 48) by
aid of the generalised Zeta-function. (Barnes.)
4. Shew that, if (r> 1,
log C(s) = '2 I — — ,
the summation extending over the prime numbers p = 2, 3, 5, .... (Dirichlet.)
5. Shew that, if a- > 1,
C(«) »=i ^^' '
where A {n)=0 when 71 is not a power of a prime, and A (»)=log^ when w is a power of a
prime p.
6. Prove that
e-^—^^^x^-^dx.
-'0(1 + 4^0
(Lerch, Krakow Itozprawy*, 11.)
* See the Jahrbuch iiber die Fortschritte der Math. 1893-1894, p. 482.
W. M. A. 18
274 THE TRANSCENDENTAL FUNCTIONS [CHAP. XIII
7. If
where | .r | < 1 , and the real part of s is positive, shew that
, , , 1 /"°° xz*~'^dz
(h (s, X) = — T-, I
and, if s < 1,
hm (1 - 07)1 - « 0 (s, ^) = r (1 - s).
(Appell, Coinptes Rendus, lxxxvii.)
8. If .V, a, and s be real, and 0 < a < 1, and s > 1, and if
„2mnx
shew that
<^(^,«.«)=^J^ V
and
^ j- e '(p^-a,x,s) ^
rf) (a\ a, I ~s) = 7~-! i t
(Lerch, Acta Mathematica, xi.)
f ^
CHAPTER XIV
THE HYPERGEOMETEIC FUNCTION
14'1. The hypergeometric series.
We have already (§ 2*38) considered the hypergeometric series*
a.b a(a + l)b(b + l)^ a(a + l)(a+2)b(b+l){b + 2) ^
"^l.c "^ 1.2.c(c+l) 1.2.3.c(c + l)(c + 2) '^ '"
from the point of view of its convergence. It follows from § 2'38 and § 5*3
that the series defines a function which is analytic when | ^ | < 1.
It will appear later (§ 14"53) that this function has a branch point at ^ = 1
and that if a cut f (i.e. an impassable barrier) is made from + 1 to + oo along
the real axis, the function is analytic and one-valued throughout the cut
plane. The function will be denoted by F {a, b; c; z).
Many important functions employed in Analysis can be expressed by
means of hypergeometric functions. Thus +
\og{l+z) = zF{l,\-2;-z),
'e'= lim F{l,B; l;zll3).
' Example. Shew that
d
14"11. The value^ of F(a, b; c; 1) when R{c — a — b) >0.
^ F{a,h; c; 2) = — i^(a + l, 6 + 1 ; c + 1; z).
The reader will easily verify, by considering the coefficients of ;»" in the
* The name was given by Wallis in 1G55.
t The plane of the variable z is said to be cut along a curve when it is convenient to consider
only such variations in z which do not involve a passage across the curve in question ; so that
the cut may be regarded as an impassable barrier.
X It will be a good exercise for the reader to construct a rigorous proof of the third of these
results.
§ This analysis is due to Gauss. A method more easy to remember but more difficult
to justify is given in § 14*6 example 2.
18—2
276 THE TRANSCENDENTAL FUNCTIONS [CHAP. XIV
various series, that if 0 ^^< 1, then
c{c — l—{2c—a — b — l)x]F{a,h;c;x) + (c — a){c — h) xF (a, 6 ; c + 1 ; ^)
= c{c-\){l — x)F{a,h;c-l;x)
= C(C-I)jl+ ^ {Un-Un-^X"-)-,
where Un is the coefficient oi x^^ in F {a,h; c — \; x).
Now make x^l. By § 371, the right-hand side tends to zero if
00
1 4- S {un — it'n-i) converges to zero, i.e. if Un -* 0, which is the case when
»i=i
R{c-a-h)>0.
Also, by the theorems of § 2'38 and § 3"71, the left-hand side tends to
c{a + h— c) F {a,h; c;l) + {c — a){c — h) F {a,h; c i-l;!) under the same
condition.
Therefore
F{a,h-c; l) = ^V^^^'~^V(a, 6; c-M; 1).
^^ c(c — a — 6) ^
Repeating this process, we see that
= -!hm n ) ^-7—-^ , \\ lim F{a,h;c+m;\),
U^oo n=o {c + n)(c-a-b + n)] ,„^„
if these two limits exist.
But (^ 12*13) the former limit is ^, \ „ , j-, if c is not a negative
"^ 1 (c — a) 1 (c - 6) °
integer ; and, if u^ {a, b, c) be the coefficient of x'^ in F(a, b; c; x), and
m>\c\, we have
00
\F {a, b; c + m; 1) — 1\ ^ 2 \un{a, b, c + 7n)\
n = l
00
^ S Un{\a\, \b\, m— \c\)
n = l
\ ab\ °°
< - - — —-, S Un(\a\ + 1, \b\ + l, m + 1 — \c\).
'm-\c\n=o
Now the last series converges, when m>\c\ + \a\ + \b\ — l, and is a positive
decreasing function of m; therefore, since {m — j c j}~^ -* 0, we have
lim F(a, b; c + 7n; 1) = 1 ;
m-*- 00
and therefore, finally,
F(a 5-c-l) = ^^^^^^i^^^^>
14-2, 14-3] THE HYPERGEOMETRIC FUNCTION 277
14"2. The differential equation satisfied by F{a,b;c;z).
The reader will verify without difficulty, by the methods of § 10*3, that
the hypergeometric series is an integral valid near z = 0 of the hyper geometric
equation* ,
from § 10*3, it is apparent that every point is an ' ordinary point ' of this
equation, with the exception of 0, 1, oo , and that these are ' regular points.'
Example. Shew that an integral of the equation
is
z'^Fia + a, b + a; a-^+l; z).
14"3. Solutions of Riemann's P-equation by hypergeometric functions.
In § 10*72 it was observed that Riemann's differential equationf
^ [!-«-«' \-^-^' 1-7-7] du
dz^ \ z — a z — b z — c ) dz
\aoL' {a-b){a-c) ^jS' (b -c){b - a) yj' (c-a)(c-b)
\ z— a z — b z — c
u
{z — a){z — b) {z — c)
= 0,
by a suitable change of variable, could be reduced to a hypergeometric
equation; and, carrying out the change, we see that a solution of Riemann's
equation is
/z - aY /2-cy j^( _ _, ^ / (2- - a) (c - b))
provided that a — a' is not a negative integer ; for simplicity, we shall,
throughout this section, suppose that no one of the exponent differences
a — a',/3 — /3',y~ 7' is zero or an integer, as (§ 10'32) in this exceptional
case the general solution of the differential equation may involve logarithmic
terms ; the formulae in the exceptional case will be found in a memoir J by
Lindelof, to which the reader is referred.
Now if a be interchanged with a, or 7 with 7', in this expression, it must
still satisfy Riemann's equation, since the latter is unaffected by this change.
* This equation was given by Gauss.
t The constants are subject to the condition a + a' + ^ + ^' + y + y' = 1.
X Acta Soc. Sclent. Fennicae, xix. (1893). See also Klein's Lectures, Ueher die hijpfr-
geometrische Funktion.
278 THE TRANSCENDENTAL FUNCTIONS [CHAP. XIV
We thus obtain altogether four expressions, namely,
fZ - ay' f2 - C\y ri { ^ n > a' -i / {c-h)(z- Ci)]
(z-aY(z-c\y' J, ( „ , , n/ . / 1 . / (c-&)(^-a)l
which are all solutions of the differential equation.
Moreover, the differential equation is unaltered if the triads (a, a', a),
{13, ^', h), (7, 7', c) are interchanged in any manner. If therefore we make
such changes in the above solutions, they will still be solutions of the
differential equation.
There are five such changes possible, for we may write
\b, c, a], [c, a, h], [a, c, h], [c, h, a], [h, a, c]
in turn in place of [a, h, c], with corresponding changes of a, a', /8, /3', 7, 7'.
We thus obtain 4 x 5 = 20 new expressions, which with the original four
make altogether twenty-four particular solutions of Riemann's equation, in
terms of hypergeometric series.
The twenty new solutions may be written down as follows :
/z — b\^/z — aY „ f _ ^ , , _ ^, (a — c) (z - b)]
fz -cy fz-by ^{ ^ , n -, > (b - a) (z - c)]
/^ — CX^' /■S' — &\^ rr f / /I / / /I -. / (b — a) (z — c)
c)(^-tOj'
z - cy (z - by „ ( _, / ^/ 1 , (6 - a) (^ - c)l
\Z — a) \z — al [' ' {b — c){z — a)]
cy fZ - Z>\^' r, f , , ^ , 0/ / , ^/ , /O' . 1 , ' . (^ - a) (^ - C)]
144] THE HYPERGEOMETRIC FUNCTION 279
f fz-ay- /z-h\^ ^{ _ / o , , {b-c){z- a)\
z-c) \z-cJ - |--v • ^^' - ' / ■ A-' - ■ - -' (6_a)(z-c)l'
/z - ay (z - by „ f ^, / ^/ -, / (& - c) (^^ - a)]
/^ - c\y fz-ay -^{ ^ _, ^ , (a -h){z- c)]
fz — c\y fz — ay' { „ , „, , , , (a—b)(z — c)]
fz — cy' /z~ay' { , „ / / 0/ /I ' (a-b)(z — c)]
\2r — a/ \z — a/ [ {c — b){z — a))
^.,___Y/^y^( ,^ (c-«)(.-i))
V^ — a/ V^-a/ i (c — 6)(2^— a)j
= (i:i^yf-^)V|0 + „ + y, ^ + a' + y; l+;3-;3-;<^-">(^-''n,
V^ — a/ V^ - aj { {c~b){z — a)]
fZ — by'fZ—cy' ri{n> , , ' a' , / , ' ^ , a> /^ (c - «) f^' - 6)]
I, V^^ - a) \z-aj { (c - 6) {z - a))
By writing 0, 1 - G, A, B, 0, G - A - B, x for a, a', yg, /S', 7, 7',
^ ^-^ ^ respectively, we obtain 24 solutions of the hypergeometric
equation satisfied by F {A, B,G; ai).
The existence of these 24 solutions was first shewn by Kummer*.
14"4. Relations betiueen 'particular solutions of the hypergeometric
equation.
It has just been shewn that 24 expressions involving hypergeometric
series are solutions of the hypergeometric equation; and, from the general
theory of linear differential equations of the second order, it follows that
if any three have a common domain of existence, there must be a linear
relation with constant coefficients connecting those three solutions.
* Crelle's Journal, xv. They are obtained in a different manner in Forsyth's Treatise
on Differential Equations, Chap. vi.
280 THE TRANSCENDENTAL FUNCTIONS [CHAP. XIV
If we simplify Wj, Wg, ^s, W4 ; u^^, Ui^', "21. ^22 in the manner indicated at
the end of § 14"3, we obtain the following solutions of the hypergeometric
equation with elements A, B, C, x\
y, =F(A,B;C;x),
y^ =(^- xf-c F{A-G +\, B - C +1; 'i.- G ; x),
2/3 ={l-xf-^-^F{G-B,G-A-G;x),
y^ ={-xf-''{l-xf-^-''F{l-B,l-A;2-G;x),
y,r=^FiA, B;A+B-G+l;l-x),
y,, = {l-xy-^-''F(G-B,G-A;C-A-B + l;l-x),
y,, = (-x)-''F{A, A-G+1;A-B + 1; x-'),
y^ = (-x)-^F(B,B-G+l;B-A + l;x-').
If I arg (1 — a;) I < TT, it is easy to see from § 2'53 that, when | a; | < 1, the
relations connecting 2/1, 2/2? ^a, Vi must be y^ = y.^, y.^ = y^, by considering the
form of the expansions near a; = 0 of the series involved.
In this manner we can group the functions u^, ... ii^i into six sets of four,
VIZ. Ui, 'W3, Mis, Mj5 , U2, M4, Wi4, Wig, W5, Uj , U'Zli ^23 3 ^6> ^8) ^22} '^24 5 ^9 > ^11) ^17 > '^19 3
Uio, 1*12, Wi8> U20, such that members of the same set are constant multiples of
one another throughout a suitably chosen domain.
In particular, we observe that Ui, u^, Wjg, u^^ are constant multiples of
a function which (by §§ 5*4, 2"53) can be expanded in the form
{z - ay |l + i en {z - ay\
when \z — a\ is sufficiently small ; when arg {z — a) is so restricted that
{z — ay is one-valued, this solution of Riemann's equation is usually written
P'"'. And P^"'^ ; P'^', P<^'> ; P<>', P'y* are defined in a similar manner when
\z — a\, \z — h\, \z — c\ respectively are sufficiently small.
To obtain the relations which connect three members of separate sets
of solutions is much more difficult. The relations have been obtained by
elaborate transformations of the double circuit integrals which will be obtained
later in 1 14"61 ; but a more simple and singularly elegant method has recently
been discovered by Barnes ; of his investigation we shall give a brief account.
14'5. Barnes contour integrals for the hypergeornetric function*.
Consider ^^j_^^ -^ -^.^ (- zyds,
where j arg (— z)\< tt, and the path of integration is curved (if necessary) to
ensure that the polesof r(a + 6') r(6 + s), viz. s = — a — 7i,~b-n(n = 0,l,2, ...),
* Proc. London Matli. Soc. Ser. 2, Vol. vi. pp. 141-177. References to previous work on
similar topics by Barnes and Mellin are there given.
14-5]
THE HYPERGEOMETRIC FUNCTION
281
lie on the left of the path and the poles of r(—s), viz. s = 0, 1, 2, ..., lie on
the right of the path*.
From § 136 it follows that the integrand is
0[\s |«+&-«-i exp {- arg {- z) . I (s) -7r\I(s)\]]
as 5 -^ 00 on the contour, and hence it is easily seen (§ 5 "32) that the inte-
grand is an analytic function of z throughout the domain defined by the
inequality | arg 2; | ^ tt — 8, where 8 is any positive number.
Now, taking note of the relation F (— s) F (1 + s) = — tt cosec stt, consider
^iri J c^{c + s)V{l+s) sin sir
where C is the semicircle of radius i\^ + ^ on the right of the imaginary axis
with centre at the origin, and N is an integer.
Now, by § 13*6, we have
T{a + s)V{h + s) irj-zr ^ ^ .^«^,_,_,x {-^
F (c + 5) F (1 + s) sin sir ' ' sin sir
as JSf^oo , the constant implied in the symbol 0 being independent of args
when s is on the semicircle ; and, if s = (N + - j e^^ and | 2^ | < 1, we have
(— zy cosec sir — 0
exp \lN+^jcos6log\z\-(N + ^j sin 0 arg (- z)
- N +
sin 6
exp \(N + ljcosd\og\z\-(N+^jB\ sin 6
Orexp|2-4('i\^+^')log|
0 ^ I ^ k 7 TT,
0
exp
-2-^^8(n + 1
1
4^^
1
Hence if \og\z\ is negative (i.e. \z\<l), the integrand tends to zero
sufficiently rapidly (for all values of 6 under consideration) to ensure that
/.
0 as iV^ ^ X .
Now
J.-.y,i (7-Oc/ JC .(.V+i)ij
by Cauchy's theorem, is equal to minus 27ri times the sum of the residues
of the integrand at the points s = 0, 1, 2, ... N. Make N ^ cc , and the last
* It is assumed that a and b are such that the contour can be drawn, i.e. that a and b
are not negative integers (in which case the hypergeometric series is merely a polynomial).
282 THE TRANSCENDENTAL FUNCTIONS [CHAP. XIV
three integrals tend to zero when \ arg {— 2)\^7r — 8, and | ^ | < 1, and so, in
these circumstances,
^ — -■ TTT^ T — ^ (— ^y ds = hm X — ^-Y^, — ' — ^ — ; — - z^,
2771 J. ^i r(c + s) N^oo n = 0 F (c + tl) . H \
the general term in this summation being the residue of the integrand
at s = n.
Thus an analytic function {namely the integral under consideration) exists
throughout the domain defined by the inequality \ arg z\< it, and, when | ^ j < 1,
tills analytic function may be represented by the series
„to r (c + n) . n !
The symbol F(a,b;c;z) will, in future, be used to denote this function
divided by T (a) T {b)/r (c).
14'51. The continuation of the hypergeometric series.
To obtain a representation of the function F{a, b; c; z) in the form of
series convergent when | ^ | > 1, we shall employ the integral obtained in
§ 14'5. If D be the semicircle of radius p on the left of the imaginary axis
with centre at the origin, it may be shewn* by the methods of § 14"5 that
1 r i>+^)r(i+^rj-.) ^ ^^. ^^ , ^
^injj) V{c + s) ^
as p -* CO , provided that j arg (— z)\ <7r, \z\ >1 and /a -* oo in such a way
that the lower bound of the distance of D from poles of the integrand
is a positive number (not zero).
Hence it can be proved (as in the corresponding work of § 14"5) that,
when j arg (— 2;) | < tt and ! ^ | > 1,
2-77* J r (c + s) ^
_ ^ r (g + n) r (1 - c + a+ n) sin {c- a- n) ir , s ^_,j
„=o r (1 + n) r (1 - 6 + ft + n) cos nir sin (6 — a — n) tt ^
I r (6 + n) V {l-c + b +ji) sin (c - 6 - n^Tr __ ^^_^^
„=o r (1 + ?i) r (1 — ft + 6 + n) cos niT sin {a — b — n)7r
the expressions in these summations being the residues of the integrand at
the points s = — a — n, s = — b — n respectively.
It then follows at once on simplifying these series that the analytic
* In considering the asymptotic expansion of the integrand when [ s | is large on the contour
or on D, it is simplest to transform V {a + s), T (b + s), V (c + s) by the relation of § 12-14.
14-51, 14-52] THE HYPERGEOMETRIC FUNCTION 283
continuation of the series, by which the hypergeometric function was
originally defined, is given by the equation
r(a)r(6)„, , - r(a)r{a-b), ,_„„, , ,17, . _u
Tic) ^^^'^' ^' ^) = ~-YW~-'^^~^^ ^^' ' ' ^
^r(b)r{b-a) 1-c + b; 1 -a + b;z-^),
r(6 — c)
where | arg (— ^) | < tt.
It is readily seen that each of the three terms in this equation is a solution
of the hypergeometric equation (see § 14-4).
This result has to be modified when a — 6 is an integer or zero, as some of the poles of
r(a + s)r{b + s) are double poles, and the right-hand side then may involve logarithmic
terms, in accordance with § 14'3.
Corollary. Putting b—c, we see that if | arg {- z)\ < tt,
r(a)(l-2)-« = -i-. /"' T{a + s)r{-s){-zyds,
Znl J -xii
where (1 — 2)~"-^l as z-^-O, and so the value of | arg(l -z) \ which is less than tt always
has to be taken in this equation, in virtue of the cut (see § 14'1) from 0 to +oo caused
by the inequality | arg {—z)\ < tt.
14 '52. Barnes' lemma.
If the path of integration is curved so that the poles of Y {y — s)T {b — s) lie on the right
of the path and the poles of r (a + s) r O + s) lie on the left*, then
Write / for the expression on the left.
If C be defined to be the semicircle of radius p on the right of the imaginary axis with
centre at the origin, and if p-»-cc in such a way that the lower bound of the distance of
C from the poles of r (y — s) r (8 - s) is positive (not zero), it is readily seen that
T(a + s)ri^ + s)T{y~s)T{8-s) = ^^^^^^^-^7r'^coseoiy-s)neOBecid-s)^
= 0[s<^+^+y+«-2exp{-27r|/(s)|}],
as I s I -»• X on the imaginary axis or on C.
Hence the original integral converges ; and I -*.0 as p -*. oo , when R(a + l3 + y + 8-l)<0.
Thus, as in § 14-5, the integral involved in /is ivi times the sum of the residues of the
integrand at the poles of r (y — s) r (S — s) ; evaluating these residues we gett
rt=or(?i+l)r(l+y-S + 7i) sin(S-y)7r "*' «=o T ()i+ 1) T (1 + 8-y + n) sin(y-8)7r"
* It is supposed that a, /3, y, d are such that no pole of the first set coincides with any pole
of the second set.
+ These two series converge (§ 2-38).
284 THE TRANSCENDENTAL FUNCTIONS [CHAP. XIV
And so, using the result of § 12-14 freely, by § 14*11 :
_7rr(l-a-3-y-S) f T{a + 8)r{^ + 8) F (g + y) T (g + y) ]
sin(y-8)7r lr(l-a-y)r(l-^-y) T (1 - a-S) T (1 - ^-8)J
- sin (a + 8) TT sin O + 8) rr}.
But 2sin (a + y)7rsin 0 + y) 7r-2 sin(a + S) tt sin(/3 + 8)7r
= COs(a — 0) TT -COs(a + ^ + 2y) tt — COS (a-/3) tt+COS (a + 3 + 28) tt
= 2sin(y-S)7rsin(a + i3+y + S)n-.
Therefore ^^r(a + y) r(^+y) r (a + 8) r (/3 + 8)
r(a + i3 + y + S) '
which is the required result ; it has, however, only been proved when
7E(a + /3 + y+8-l)<0;
but by the theory of analytic continuation, it is true throughout the domain through
which both sides of the equation are analytic functions of, say, a ; and hence it is true for
all values of a, /3, y, 8 for which none of the poles of r (a + s) r 0+s), qua function of s,
coincide with any of the poles of r (y — s) r (8 — s).
Corollary. Writing s+^, a — k, ^-k, y+k, 8 + ^' in place of s, a, ^, y, 8, we see that
the result is still true when the limits of integration are —k±cci, where k is any real
constant.
14 "SS. The connexion between hypergeometric fimctions of z and of\—z.
We have seen that if | arg ( - 2) | < tt,
V{a)V{b) „, , , 1 /■*'■ T(a+s)T(b + s)T(-s) , , -
^^ T{-s){-zY ,„
r {c -a)V {c- b) '
by Barnes' lemma.
If ^ be so chosen that the lower bound of the distance between the s contour and the
t contour is jjositive (not zero), it may be shewn that the order of integration * may be
interchanged.
Carrying out the interchange, we see that if arg (1 -2) be given its principal value,
V {c-a)V {c-b)V {d)V {b) F{a, b; c; z)jT {c)
'r{a + t)T{b + t)r{c-a-b-t) i~ I "' T {s-t)r (-s) (-zydsl dt
t-ooj {.^TTl J _a=i )
T {a + t) T {b + t) V {c - a-b -t)T {-t) {\ - zY dt.
* Methods similar to those of § 4-51 may be used, or it may be proved without much difficulty
that conditions established by Bromwich, Infinite Series, § 177, are satisfied.
277J j
14-53, 14'6] THE HYPERGKOMETRIC FUNCTION 285
Now, when | arg (1 - a) | < 27r and \l—z\<l, this last integral may be evaluated by the
methods of Barnes' lemma (§ 14"52) ; and so we deduce that
r {c- a) r (c -b) r (a) T {b) F {a, b ; c ; z)
= r{c)T{a)T{b)r(c-a-b)F{a,b; a+b-c + 1; 1-z)
+ r{c)r{c-a)T{c-b)r{a + b-c){l-zy-<*-^F{c-a,c-b; c-a-b + 1; l-z\
a result which shews the nature of the singularity of F{a, b ; c ; z) at 2=1.
This result has to be modified if c — a — 6 is an integer or zero, as then
T ia + t)T {b + t)T {c-a-b -t) r {- t)
has double poles, and logarithmic terms may appear. With this exception, the result is
valid when | arg (-z)\<7r, \ arg (1 - 2) | < tt.
Taking | 2 | < 1, we may make 2 tend to a real value, and we see that the result still
holds for real values of 2 such that 0 < 2 < 1 .
14*6. Solution of Rierrumns equation hy a contour integral.
We next proceed to establish a result relating to the expression of the
hypergeometric function by means of contour integrals.
Let the dependent variable u in Riemann's equation (§ 107) be replaced
by a new dependent variable 7, defined by the relation
u = {z-aY{z-hY {z-c)y I.
The differential equation satisfied by / is easily found to be
d^I (1 + a-a' H-/3-/S' 1+7-7']^/
dz^ [ z — a z — b z — c ] dz
^ (a + ^ + 7) {(« + yg + 7 + 1)^ + ^0^ (« + /3^+7- 1)1 j_Q
{z — a){z — h) {z — c)
which can be written in the form
+ {I (A. - 2) (A, - 1) Q" {z) + (X - 1) R' {z)] 1=0,
where ( A, = 1 — a — /3 — 7 = a' + /3' + 7',
Q{z) = (z-a)(z-b){z-c),
R{z) = 'E(a.' + ^ + y){z-b){z-c).
It must be observed that the function / is not analytic at qo , and consequently the
above differential equation in / is not a case of the generalised hypergeometric equation.
We shall now shew that this differential equation can be satisfied by an
integral of the form
1= [ (t- ay+P+y-' {t - by+^'+y-' {t - cY+^+y'-' (z - i)-»-^-r dt,
■' c
provided that C, the contour of integration, is suitably chosen.
286 THE TRANSCENDENTAL FUNCTIONS [CHAP. XIV
For on substituting this value of / in the differential equation, the
condition* that the equation should be satisfied becomes
( (t- aY+P^y-' (t - hy+^'+y-^ {t - cY+p-^y'-' {z - ty^-^-y-^ Kdt = 0,
J c
where
K = (X-2)\q (z) + (t-z) Q {z) + \{t- zf Q" (z)^
+ (t-z){R{z) + (t-z)R'(z)]
==(X-2){Q{t)-(t-zy]+{t-z){R(t)-{t-zyi(a' + ^ + y)}
= {\-2)Q (t) + {t-z)R (t)
= -(l + a + ^ + ry)(t-a){t-b)(t-c)
+ t(a' + ^ + ry){t-b)(t-c){t-z).
f dV
It follows that the condition to be satisfied reduces to —5- dt = 0, where
J c at
V=(t- ay+^+y (t - bY+^+y (t - cY+^+y' (t - 2)-a+-+p+y),
The integral / will therefore be a solution of the differential equation,
provided C is such that V resumes its initial value after t has described C.
Now
F= {t - aY+^+y-' (t - bY+^'-^y-' {t - cY+p+y'-' {z - ty'^-^-y u,
where U=- (t -a)(t- b) (t -c)(z- t)-\
Now ?7 is a one-valued function of t ; hence if C be a closed contour, it
must be such that the integrand in the integral / resumes its original value
after describing the contour.
Hence finally any integral of tJie type
(z - aY {z - by (z - c)y ! {t- ay+y+''-'{t-b)y+''-^P'-'{t-cY--p+y'~'{z-t)---p-ydt,
where 0 is either a closed contour in the t- plane such that the integrand
resumes its initial value after describing it, or else is a siinp)le curve such that
the quantity V has the same value at its ter^mini, is a solution of the differential
equation of the general hyper geometric function.
The reader is referred to the memoirs of Pochhammer {Math. Ann. xxxv.) and Hobson
{Phil. Trans. 187 a) for an account of the methods by which integrals of this type
are transformed so as to give rise to tlie relations of i^i^ 14"51, 14*53.
Example 1. We can now deduce a real definite integral which, in certain circumstances,
represents the hypergeometric series.
* The differentiations under the sign of integi-ation are legitimate (§ 4*2) if the path C does
not depend on z and does not j^ass through the points a, h, c, z ; if C be an iutinite contour or if
C passes through the points a, b, c or z, further conditions are necessary.
14-61] THE HYPERGEOMETRIC FUNCTION 287
The hyi-Kjrgeometric series F{a, b; c; z) is, as already shewn, a solution of the differential
equation defined by the scheme
* r 0 X 1 ^
{, l-c b c-a-b ]
If in the integral
which is a constant multiple of that just obtained, we make b-^-cc (without paying
attention to the validity of this process), we are led to consider
/.
c
Now the limiting form of V in question is
and this tends to zero at ^ = 1 and t = cc , provided R{c) > R {b) > 0.
We accordingly consider I t'^'" {t-lY~''~''- {t -z)~"'dt, where z is not* positive and
greater than 1.
In this integral, write t = u~'^ ; the integral becomes
/:
u'>-^ {I - uy-^-'^ (l -uzy du.
We are therefore led to expect that this integral may be a solution of the differential equation
for the hypergeometric series.
The reader will easily see that if R{c)> R {b) > 0, and if arg « = arg (1 — «) = 0, while the
branch of l-uz is specified by tlie fact that {l—7iz)~'^-*'l as u-*-0, the integral just
found is
This can be proved by expandingt {l — uz)~°' in ascending powers of z when |2| < 1 and
using § 12-41.
Example 2. Deduce the result of i^ 14'11 from the preceding example.
14"61. Determination of an integral u-hich represents P^'^'.
We shall now shew how an integral which represents the particular solution P^'^^
(§ 14-3) of the hypergeometric differential equation can be found.
We have seen (§ 14-6) that the integral
] = {z-af{z-h)\z-cy\{t-af+y^^'-\t-b)y^^^^'-\t-cT^^+'''-\t-z)-''-^--^dt
satisfies the differential equation of the hypergeometric function, jjrovided C is a closed
contour such that the integrand resumes its initial value after describing C. Now the
singularities of this integrand in the ^plane are the points a, b, c, z; and after describing
the double circuit contour (i^ 12'43) symbolised hy {b + ,c + ,b-, c - ) the integrand returns
to its original value.
* This ensures that the point t—ljz is not on the path of integration,
t The justification of this process by § 4-7 is left to the reader.
288 THE TRANSCENDENTAL FUNCTIONS [CHAP. XIV
Now, if z lie in a circle whose centre is a circle not containing either of the points
h and c, we can choose the path of integration so that t is outside this circle, and so
1 0 - a i < I ^ - a I for all points t on the path.
Now choose arg (2 - a) to be numerically less than ir and arg {z — h\ arg (2 — c) so that
they reduce to* arg (a -6), arg(a — c)when 2-*- a; fix arg(<-a), arg(< — 5), arg(^ — c) at
the point N at which the path of integration starts and ends ; also choose arg (f — 2) to
reduce to arg(^-a) when z-^a.
Then (2-j/ = (a-6/|l + fl(^j + ...l,
and since we can expand {t-z)""-'^"^ into an absolutely and uniformly convergent series
we may expand the integral into a series which converges absolutely.
Multiplying up the absolutely convergent series, we get a series of integer powers of
z-a multiplied by {z-af. Consequently we must have
(6+,c+,6-,c-)
J N
We can define p("'\ p(^\ P^^'\ p(v), pW) by double circuit integrals in a similar
manner.
14'7. Relations between contiguous hypergeonietric functions.
Let P{z) be a solution of Riemann's equation with argument 2^, singularities
a, b, c, and exponents a, a', y8, jS', 7, 7'. Further let P{z) be a constant
multiple of one of the six functions P<"), P*"^', P'^», P'^', PO', P<>''. Let
P^+i^jft-i (^) denote the function which is obtained by replacing two of the
exponents, I and m, in P(^) by ^ + 1 and m - 1 respectively. Such functions
P;+i,m.-i (^) are said to be contiguous to P {z). There are clearly 6 x 5 = 30 con-
tiguous functions, since I and m may be any two of the six exponents.
It was first shewn by Riemann*f- that the function P(z) and any two of
its contiguous functions are connected by a linear relation, the coejjicients in
which are polynomials in z.
There will clearly be - x 30 x 29 = 435 of these relations. To shew how
to obtain them, we shall take P {z) in the form
p (^) = {z- ay {z - by {z - c)y \ {t - ay^y+'^'-' (t - 6)y+-+^'-i
{t-cy+P+y-'{z-t)-^-^-ydt,
where C is a double circuit contour of the type considered in § 14"61.
* The values of arg {a - h), arg (a - c) being fixed.
t Abhandlurifien der Kon. Ges. der Wins, zu Gottingen, 1857; Gauss had previously obtained
15 relations between contiguous hypergeometric functions.
14*7] THE HYPERGEOMETRIC FUNCTION 289
First, since the integral round G of the differential of any function which
resumes its initial value after describing C is zero, we have
0 = f i-Aii- ay-^^'-y {t - by+p'+y-' (t - cy-^^+r-^ (t - zy-^-p-y] dt
J CCI'i
On performing the differentiation by differentiating each factor in turn,
we get
(a' + /3 + 7) P + (a + yS' + 7 - 1) Pa'+^,^'-r + (« + ^ + 7' - 1) i'a'+i.y-i
_(a + ^ + 7)p
Considerations of symmetry shew that the right-hand side of this
equation can be replaced by
(a + ^ + y) J,
These, together with the analogous formulae obtained by cyclical inter-
change* of {a, a, a) with (b, ^, yS') and (c, 7, 7'), are six linear relations
connecting the hypergeometric function P with the twelve contiguous
functions
■^a+l,P'—l> -^^+1,-)''— 1) -^-y+l.a'-l, -Ta+l.v'-lj ■Pfi+l,a'—l, X"^y+i,^'— i.
Next, writing t - a = {t — b) ■{■ {b - a), and usingf Pa'_i to denote the result
of writing a' — 1 for a! in P, we have
Similarly P = P„-_i_ y+i + {c-a) P^'_^ .
Eliminating Pa'_i from these equations, we have
(c-b)P + (a-c) Pa'-i,§'+i + (b-a) Pa'-i,y+i = 0.
This and the analogous formulae are three more linear relations con-
necting P with the last six of the twelve contiguous functions written above.
Next, writing (t — 2!) = (t — a) — (z — a) we readily find the relation
P = ^ P^-M.y-i - (^ - ay^' (^ -hnz- c)y
X
' c
I (t- af+y+^'-' {z - a)>+»+^'-i {z - by+^+y'-' {t - ^)-«-^-y-l dt,
which gives the equations
(^ _ a)-i [p-{z- by- p,+,,y-x] = (^ - by- [P-{z- cy- P,+,,a--i}
= {z- cy- [P-{z- ay- p„
+1,0-1 )
* The interchange is to be maile only in the integrands ; the contour C is to remain
unaltered.
t -Pa'-i is not a function of Riemann's type since the sum of its exponents at a, h, c is not
unity.
W. M. A. 19
290 THE TRANSCENDENTAL FUNCTIONS [CHAP. XIV
These are two more linear equations between P and the above twelve
contiguous functions.
We have therefore now altogether found eleven linear relations between
P and these twelve functions, the coefficients in these relations being rational
functions of z. Hence each of these functions can be expressed linearly in
terms of P and some selected one of them ; that is, between P and any two of
the above functions there exists a linear relation. The coefficients in this
relation will be rational functions of z, and therefore will become polynomials
in z when the relation is multiplied throughout by the least common multiple
of their denominators.
The theorem is therefore proved, so far as the above twelve contiguous
functions are concerned. It can, without difficulty, be extended so as to be
established for the rest of the thirty contiguous functions.
Corollary. If functions be derived from P by replacing the exponents a, a', /3, /3', y, y'
by a+p, a' + 5, i3 + ?', i3' + «, y + ^, 7+^) where jtj, q., r, s, t, u are integers satisfying the
relation
p + q + r + s + t + u = 0,
then between P and any two such functions there exists a linear relation, the coeflBcients
in which are polynomials in z.
This result can be obtained by connecting P with the two functions by a chain of
intermediate contiguous functions, writing down the linear relations which connect them
with P and the two functions, and from these relations eliminating the intermediate
contiguous functions.
Many theorems which will be established subsequently, e.g. the recurrence-formulae
for the Legendre functions (§ 15*21 ), are really cases of the theorem of "this article.
REFERENCES.
C. F. Gauss, Oes. Werke, Bd. lii. pp. 123-163, 207-229.
E. E. KuMMER, Crelle, Bd. xv. pp. 39-83.
G. F. B. RiEMANN, Ges. Math. Werke, pp. 67-84.
E. Papperitz, Math. Ann. Bd. xxv. pp. 212-221.
E. W. Barnes, Proc. London Math. Soc. Ser. 2, Vol. vi. pp. 141-177.
H. J. Mellin, Acta Soc. Fennicae, Vol. xx. No. 12.
Miscellaneous Examples.
1. Shew that
F{a, h+l;c; z)-F{a, b; c; z) = '!^ F{a + l, 6+1 ; c + 1 ; z).
2. Shew that if a is a negative integer while id and y are not integers, then the ratio
F{a, j3; a + ^ + l-y; l-.v)-^F{a, /3; y; x) is independent of .v, and find its value.
THE HYPERGEOMETRIC FUNCTION 291
dP cPP
3. li P(z) be a hypergeoraetric function, express its derivates -r- and -t„ linearly in
terms of P and contiguous functions, and hence find the linear relation between P, -t- ,
dz
, d^P .
and -ry , I.e. verify that P satisfies the hypergeometric differential equation.
4. Shew that F {^, I; 1 ; 42 (1 -2)} satisfies the hypergeometric equation satisfied by
^(h i ; 1 ; ■^)- Shew that in the left-hand half of the lemniscate | z (1 —z) | = ^, these two
functions are equal ; and in the right-hand half of the lemniscate the former function is
equal to F{^, | ; 1 ; 1 -2).
5. If Fa+ =F{a + l, b; c; x), Fa-=F{a — l, b; c; x), determine the 15 linear relations
with polynomial coefficients which connect F{a, b;c; x) with pairs of the six functions
-^o + j Fa-, Fb + , Fi,_, Fc + , F^_. (Gauss.)
6. Shew that the hypergeometric equation
is satisfied by the two integrals (supposed convergent)
\^-^ {\-z)-i-^-^ {\- xzy dz
and
/:
[" z^-^{\-zf-^\l~{^\-x)z)-''dz.
7. Shew that, for values of x between 0 and 1, the solution of the equation
is ^F{|a, i^;|;(l-2^)2}-fi5(l-2^)i^{|(a+l), H/3 + l);|;(l-2^n,
where .4, B are arbitrary constants and F {a, i^;y,x) represents the hypergeometric series.
(Math. Trip. 1896.)
8. Shew that
x^l-oL ^''^'^' ^ nj ' 7i!r(y-a)r(y-^)r(«)r(^) ^' ^> J
^r^(y-„_/3)r(y)
r(y-a)r(y-/a)
where k is the integer such that ^ ^ 7^ (a + ^ - y) < ^ -f 1.
(This specifies the manner in which the hypergeometric function becomes infinite when
•*-■-► 1 - 0 provided a -f /3 - y is not an integer.) (Hai-dy.)
9. Shew that, when R{y-a-li) <0, then
r(y)^°+^-v
''"•(«4-^-y)r(a)r(^)
as n-^ CO ; where Sn denotes the sum of the first n terms of the series for F{a, ^■,y;\).
(M. J. 1\I. Hill, Proc. London Math. Soc. Ser. 2, Vol. v.)
19—2
292 THE TRANSCENDENTAL FUNCTIONS [CHAP. XIV
10. Shew that if yi, y^ be independent solutions of •
the general solution of
d^z . „ ^d^z
d^
is z = Ay-^ + Byxy.^+ Cy^^ where A, B^ Care constants.
(Appell, Comptes Rendus, xci.)
11. Deduce from example 10 that if a + 6 + | = c,
/w,, .-.-..v. r(c)r(2c-l) - r(2a + n)r(a + 6 + n)r(26 + r0 ■
^^*-"' ''''''•^^■' ~r(2a)r(26)r(a + 6)n=o /i ! r(c + «)r(2c- l+«)
(Clausen, Crelle, ill.)
12. Shew that if | :r | < 1 and | ;r (1 -.r) | < |,
F{2a, 2^;a + ^+^;x} = F{a, ^; a + ^ + i ; 4^(1 -«)}. (Kummer.)
13. Deduce from example 12 that
i^{2a, 2,3, - + 0 + hh-^j^^jr^j:^jy
14. Shew that, if w = e*'^' and /i (a) < 1,
F(a,3a-l;2a; - o,^) = 3^'^ - ^ exp {^.rt (3a - 1)} |; Igf,^)^^ '
F(a,3a-1; 2a; - co) = 3^° ' ^ exp {^^^ (1 - 3a)} ^ |3°LV)7(| '
(Watson.)
15. Shew that
r(i)T(n + S)
F{-^n,-in + Ji;n + ^; - J) = (|)« ^,A|L,|_±|j .
(Heymann, Zeitschrift f. Math. XLiv.)
16. If {l-x)"^^"^ F{2a,2^;2y;x)^l-\-Bx + Cx'^ + Da^ + ...,
shew that
F{a,(i;y + \;x)F{y-a,y-^;y + ^;x)
7+i (y+i)(y+'l) (y+i)(y+^)(7+l)
(Cayley, PA«7. il/agr. Ser. 4, Vol. xvi.)
17. If the function F{a, ^, ^', y, x, y) be defined by the equation
F{a,^,^',y;x,y)=_-~^ ^ f u''-' {I -u)y-'^-' {I - uxy^ {l -uy)-^' du,
I (a) r (y - a) Jo
then shew that between F and any three of its eight contiguous functions
F(a±l), >(/3±l), i^(/3'±l), i^(y±l),
there exists a homogeneous linear equation, whose coefficients are polynomials in x and y.
(Le Vavasseur.)
THE HYPERGEOMETRIC FUNCTION 293
18. Tf-y-a-j8<0, shew that, as 07-^1-0,
and that if y — a — ^=0, the corresponding approximate formula is
(Math. Trip. 1893.)
19. Shew that when | .r | < 1,
f{x+,0+,x-,0-) ,
where c denotes a point on the straight line joining the points 0, x, the initial arguments
of V -.r and of v are the same as that of a;, and arg (1 - v) ^-0 as v -*0.
(Pochhammer.)
20. If, when | arg (1 - a?) | < 2n-,
^ ^^^ == 2^- /^l ^^ ^ - ') ^ ^H «)}' (1 - ^)* ^«,
and, when | arga? | < 27r,
by changing the variable s in the integral or otherwise, obtain the following relations :
K{a;) = K'{l-x), if | arg(l-^) | < tt.
K(l-x) = E'{x), if |arg:y|<7r.
K{a;) = (l-a;)-^K(^-^y if | arg(l-^)| < ,r.
Kil-x)=x-^K(^^y if |arg.ri<7r.
K'{x)=x-^K'il/x), if |arg^!<7r.
^'(1 -x) = (l-x)-i K' (]4^.) , if I arg(l -^) | < tt. (Barnes.)
21. With the notation of the preceding example, obtain the following results :
^^'<^'=- J, fit T" h^-^'-'^'+^G - ^+--2»)}'
when I a; I < 1, I arg^ I < TT ; and
K{x)= i;.i{-x)- ^K{\lx)^{-x) -^K' (l/.r),
when I arg ( -^) | < tt, the ambiguous sign being the same as the sign of I{x).
(Barnes.)
294 THE TRANSCENDENTAL FUNCTIONS [CHAP. XIV
22. Hypergeometric series in two variables are defined by the equations
i^i (a ; /3, ^' ; y ; ^, j/)= 2 ^^^^^ ^-y",
m, n 11'' • •I' • 7m + n
i^'a (a, a', ^, ^' ; y ; ^, 3/)= 2 ^^^f"^. x^y\
m,n mini ym + n
»i, Ji '"' • "' • /m yn
where a,„ = a (a+ l)...(a + wi~- 1), and 2 means 2 2.
m, n m=0 Ji=0
Obtain the diflferential equations
and four similar equations, derived from these by interchanging x with 3/ and a, fi, y with
a', /3', y' when a', /S', y' occur in the corresponding series.
(Appell, Comptes Rendus, xc.)
23. If a is negative, and if
a= —v\-a,
where v is an integer and a is positive, shew that
V{x)T{a)_ % \ Rn
, o (-)»(a-l)(a-2)...(a-»)^, ^
where Rn = ^ — -^ ^-^^-1 — ^ Cf ( - n).
^wK'-^.-^OO^-^.) 0^^)'
O^^ (^) = ^i^^)__^l_5) . (Hermite, Cre^^e, xcii.
24. When a < 1, shew that
Tix)Tia-x)^ 2 ^'^- 2 ^
T (a) n,=i J7+% M=i .3?-a-«'
where /£„ = — ^^ ; — ^^ .
n I
25. When a > 1, and v and a are respectively the integral and [fractional parts of
a, shew that
T{x)T(a-x)^ - 0{.v)_pj,_ I G'(^)p^+^
T (a) n=l x+n n=l x-a-n
\_X-a X-a-\ X- a — v\-\j
THE HYPERGEOMETRIC FUNCTION 295
where «(,) =(i.?) (: ._i_) ... (i ._^^^)
(-)»«(a + l)...(a + »-l)
and pn —
n\
(Hermite, Crelle, xcii.)
26. If
/ f^ » ,,^-^ _ r -^Cy + ^ + ^-l) . ^ x{x + l)(j/ + v+n-l)(2/ + v + n)
where «. is a positive integer and „(7i, nC^, ... are binomial coefficients, shew that
f (r ,/ ..s_'^ij/)'^i^-x + n)T{x + v)T{v + n)
JnK^,y,i^) Y{^-x)T(jf + n)T{v)V{x + v + n)'
(Saalschiitz, ZeitschHft f. Math, xxxv.)
[See also Dougall, Proc. Edinburgh Math. Soc. xxv., for a large number of similar
results.]
27. If Fia,^,y; S, . ; ^)=l + _.,+ S (§+]) ^(^ + 1) l72" * +-'
shew that, when Ti (S + e — f a - 1) > 0, then
i^(a,a-S+l,a-. + l,S,., l)-2 r (8-|a) T (e-Ja) T (i + i«) T (S + f -a-1) '
(A. C. Dixon, Proc. London Math. Soc. xxxv.)
28. Shew that, if i? (a) < g, then
, =° fa(a + l)...(a + %-l)P r(l-fa)
(Morley, Proc. London Math. Soc. xxxiv.)
CHAPTER XV
LEGENDRE FUNCTIONS
■ . _. I
15'1. Definition of Legendre polynomials.
Consider the expression (1 — 2zh + A^) ~ ^ ; when i '2zh — h?\<l, it can be
expanded in a series of ascending powers of 2zh — ¥. If, in addition,
I 2zh\ + \hf<l, these powers can be multiplied out and the resulting series
rearranged in any manner (§ 2*52) since the expansion of [1 — {| 2zh \ + \h p}]~^
in powers of | 2zh \ + \h\^ then converges absolutely. In particular, if we
rearrange in powers of h, we get
(1 -2zh + h')-i = P,{z) + hP,{z) + h-'P.iz) + h'P,(z) + ...,
where
Po(^) = l, P^{z) = z, P,(z) = l(Sz^'-l), P,{z) = l{5z'-Sz),
P, (z) = I {S5z' - SOz^ + 3), P, (z) = I (6Sz' - 70z' + 15z),
and generally
p (,^ = (2>0' L« _ ^ (^Lull^n-. . nS^v-l)(n- 2)(n-S) _ 1
"^^ 2'\{niy\ 2(2/1-1) ^ 2.4.(2n-l)(2w-3) ■"]
= I (_Y (2n-2r)l
r=o 2''.rl{n-r)l{n-2r)l
where m = ^ n or ^{n — 1), whichever is an integer.
If a, b and 8 be positive constants, b being so small that 2ab + b^ ^ 1 — S, the expansion
of (1 — 2zh+h'^) " 2 converges uniformly with respect to z and h when | 2 1 ^a, | A | ^ 6.
The expressions P^ (z), P^ (z), . . . , which are clearly all polynomials in z,
are known as Legendre polynomials*, Pn{z) being called the Legendre
polynomial of degree n.
It will appear later (§ 15-2) that these polynomials are particular cases of a more
extensive class of functions known as Legendre functions.
* Other names are Legendre coefficients and Zonal Harmonics. They were introduced into
analysis in 1784 by Legendre, Memoires par divers savans, x. (1785).
15'1-1512] LEOENDRE FUNCTIONS 297
Example 1. By giving z special values in the expression (1 - 2«A+ A*) ~ *, shew that
/'„(1) = 1, P„(-l) = (-l)»
Example 2. From the expansion
shew that
1.3.(2»)(2«-2) „ , .s>,, 1
+ 2:4.(2Lli(2.-3)^^"^(^-'^^ + -r
Deduce that, if ^ be a real angle,
,„, ^,, 1.3...2n-l L l.(2«) „ 1.3.(2«)(2?i-2) „, ]
so that I P„ (cos e)\^l. (Legendre.)
Example 3. Shew that, when 2= — ^,
Pn=PoPu-PlP^n-l + P2P2n-2-.:+P2nPo' (Clare, 1905.)
15*11. Rodrigues'* formula for the Legendre polynomials.
It is evident that, when n is an integer,
i^«^^^^ ~rf^M.=o^ ^ r\{n-r)l' ]
dz
r=o r!(w-7-)! (ri-2r)!
where 7/1 = 571 or ^(ti— 1), the coefficients of negative powers of z vanishing.
From the general formula for Pn (z) it follows at once that
this result is known as Rodrigues' formula.
Example. Shew that P„ (2) = 0 has n real roots, all lying between ±1.
15*12. Schldjiisf integral for Pn (z).
From the result of § 1511 combined with § 5"22, it follows at once that
^"^^^ 27riJc2-{t-zr-^^'^^'
where (7 is a contour which encircles the point z once counter-clockwise ; this
result is called Schldfli's integral formula for the Legendre polynomials.
* Correnp. sur Vecole poly technique, iii. (1814-1816), pp. 361-385.
t Schlatli, Ueber die zwei Heine'schen Kugelfunctionen, Bern, 1881.
298 THE TRANSCENDENTAL FUNCTIONS [CHAP. XV
15*13. Legendre's differential equation.
We shall now prove that the function u = P^ {z) is a solution *of the
differential equation
which is called Legendre's differential equation for functions of degree n.
For, substituting Schlafli's integral in the left-hand side, we have,
by § 5-22,
(w + 1) r d [(f^-if+'i
1'jri.2'^Jcdt\{t-zY-^A '
and this integral is zero, since (t^ — 1)'*+' (t - ^•)~'*-2 resumes its original value
after describing G when n is an integer. The Legendre polynomial therefore
satisfies the differential equation.
The result just obtained can be written in the form
A (/-, .,x dPn (z)]
dz
i^(l-,^^^\ + n(n^l)P,iz) = 0.
It will be observed that Legendre's equation is a particular case of Riemann's equation,
defined by the scheme
drp M
Example 1. Shew that the equation satisfied by — ," is defined by the scheme
Example 2. If z^=r], shew that Legendre's differential equation takes the form
dr}'^ [2rj l-i;J dr] 4rj (l - rj)
Shew that this is a hypergeometric equation.
Example 3. Deduce Schlafli's integral for the Legendre functions, as a limiting case of
the general hypergeometric integral of § 14-6.
[Since Legendre's equation is given by the scheme
^-1 . 1 I
P< 0 n + l 0 z\,
0 -n 0
15-13, 1514] LEGENDRE FUNCTIONS 299
the integral suggested is
Urn (l-0"*' ( («+l)"(<-l)" lim 6-0 "^ {t-z)-^-''cU
= I (i2-l)"(«-2)-«-l0^^
taken round a contour C such that the integrand resumes its initial value after describing
it ; and this gives Schlafli's integral.]
15*14. The integral properties of the Legendre polynomials.
We shall now shew that*
1, 2w -I- 1 ^ ^
Let {u]r denote ^ ; then if r < n, {(z' - 1)%. is divisible by (z' - l)"-*- ;
and so, if r < n, {(z^ — lY]r vanishes when z =1 and when z = — 1.
Now, of the two numbers m, n, let m be that one which is equal to or
greater than the other.
Then, integrating by parts continually,
' {{z'-ir]m{(z'-ir]ndz
{{Z^ - irU-^ {(Z' - ir}« \^ - W {{^' - l)'"}m-a {{Z' - If
dz
= {-T f (^' - 1)"" K^' - l)"}n+m dz,
since {(^- - l)"»}m-i, K^' " '^T]m-2, ••• vanish at both limits.
Now, when m > n, {{z^- l)"}w,+n = 0, since differential coefficients of (z^-1)''
of order higher than 2n vanish ; and so, when m is greater than n, it follows
from Rodrigues' formula that
[' Pm{z)Pn(z)dz = 0.
When m = n, we have, by the transformation just obtained,
j ^ {(z^ - iy% {(z^ - ir]n dz = (-)" j_^(z^ - ir ^, (z' - 1)" dz
= (2/0! [ (1-^T^^
= 2 . (2n) \[ (1 - z^y dz
Jo
rhtr
= 2.(2/?)! sm^^'+'-ede
Jo
2.4... 2/7
_2.(2w).^^^ --2,7+-^)'
* These two results were given by Legendre in 1784 and 1789.
300 THE TRANSCENDENTAL FUNCTIONS [CHAP. XV
/:
where cos 6 has been written for z in the integral ; hence, by Rodrigues'
formula,
j_i^ "^ ^^ * (2".w!)Hiiw+l)! 271 + 1*
We have therefore obtained both the required results.
It follows that, in the language of Chapter xi, the functions (w + D^ P„ (z) are normal
orthogonal functions for the range ( — 1, 1).
Example 1. Shew that, \i x > 0,
(cosh2^-z)-ip„(3)c?2 = 2~*(w + i)-ie-(2» + i)*.
(Clare, 1908.)
Example 2. If /= f * P,„ (2) P„ {z) dz, then
(i) /=l/(2?i + l) {m = n\
(ii) /=0 (m- 71 even),
i-T'^*' n^ m^
(ni) /= -—- — -—^ — ^-— :., , .,., , .,„ (7i = 2v + l, m = 2u).
(Clare, 1902.)
15'2. Legendre functions.
Hitherto we have supposed that the degree n of Pn{z) is a positive
integer ; in fact, P„ {z) has not been defined except when n is a positive
integer. We shall now see how P„ {z) can be defined for values of n which
are not necessarily integers.
An analogy can be drawn from the theory of the Gamma-function. The expression
z\ as ordinarily defined (viz. as 0 (0- 1) (2-2) ... 2 . 1) has a meaning only for positive
integral values of z ; but when the Gamma-function has been introduced, 2 ! can be defined
to be r (2 + 1), and so a function 2 I will exist for values of 2 which are not integers.
Referring to § 15*13, we see that the differential equation
is satisfied by the expression
n _^2) ^ _ 2^ ^ + n (71 + l)7t = 0
dz'^ dz
u^j-.( j':-^>:.di.
27ri j c 2" {t - ^)"+i
even when n is not a positive integer, provided that C is a contour such that
(^2 — 1)^*+! {t — z)~^~^ resumes its original value after describing C.
Suppose then that n is no longer taken to be a positive integer.
The function (t^ — 1)"+^ {t — z)~^''~'^ has three singularities, namely the
points ^=1, ^ = — 1, t = z; and it is clear that after describing a circuit round
the point ^ = 1 counter-clockwise, the function resumes its original value
multiplied by e^'^* '"+^* ; while after describing a circuit round the point t = z
counter-clockwise, the function resumes its original value multiplied by
15'2, 15*21] LEGENDRE FUNCTIONS 301
g!hri(-n-2) jf therefore G he n contour enclosing the points t = l and t = z,
but not enclosing the point t = —l, then the function (t^ — 1)^+^ (t — z)~^*~^
will, after describing C, resume its original value. Hence Legendre's differential
equation for functions of degree n,
,, „.d'^u a du , ,.
is satisfied hy the expression _
27rtj.
dt,
for all values of n ; the many-valued functions will be specified precisely
by taking A on the real axis on the right of the point t = l (and on the
right of 2 if z be real), and by taking arg(^— 1) = arg(^ + 1) = 0 and
|arg(^-2:)| < TT at ^.
This expression will be denoted by Pn (z), and will be termed the Legendre
function of degree n of the first kind.
We have thus defined a function Pn {z), the definition being valid whether
n is an integer or not.
The function P„ {z) thus defined is not a one-valued function of z ; for we might take
two contours as shewn in the figure, and the integrals along them would not be the same ;
to make the contour integral unique, make a cut in the t plane from — 1 to — qo along the
real axis ; this involves making a similar cut in the z plane, for if the cut were not made,
then, as z varied continuously across the negative part of the real axis, the contour would
not vary continuously.
It follows, by § 5-31, that /*„(2) is analytic throughout the cut plane.
15*21. The Recurrence Formulae.
We proceed to establish a group of formulae (which are really particular
cases of the relations between contiguous Riemann P-functions which were
shewn to exist in § 14'7) connecting Legendre functions of different degrees.
If C be the contour of § 15'2, we have*
^n {Z) 2U+ Vi j ^ (^ _ ^y.4-1 ^^' ^ n {2) - 2«+i^^- I ^ ^^ _-^^^, dt.
* We write P„' (2) for ~ P„ {z).
302 THE TRANSCENDENTAL FUNCTIONS [CHAP. XV
^°" dt w
and so integrating
°^ dt (t - ^)«+i ~ {t- ^)"+i (t - ^)«+2
t(f'-ir^^ f (t^-l)n+^^^^
0 = 2 f *^^^::^dt-[ ^^^
Therefore
2'*+i7ri J c (« - ^)" 2"+^7rz j c (i^ - ^)'*+' 2'^+i7rt J c (t - zf+'
Consequently
P„«W-.P„W = 2J,„.J;|^* (A).
Differentiating*, we get
P'n+, (Z) - zP'n (Z) - Pn (Z) = nP^ (z),
and SO P'n+i{z)- zP'r,(z) = {n+ 1) P.„{z) (I).
This is the first of the required formulae.
Next, expanding the equation
f ditii-iy)
Jcdt\(t-zf\
we find that
h ^ dt + 2>i -^ ^ dt-n ,-f -^ dt = 0.
Writing (^- — 1) + 1 for t- and {t — z) + z for t in this equation, we get
Using (A), we have at once
{n + 1) [Pn+, {Z) - zPn {Z)] + nPn-^ {z) - UzP^ {z) = 0.
That is to say
{n + 1) P„+, {z) - (2n + 1) zP,, {z) + nP,_, {z) = 0 (II),
a relation f connecting three Legendre functions of consecutive degrees. This
is the second of the required formulae.
We can deduce the remaining formulae from (I) and (II) thus :
Differentiating (II), we have
(n + 1) [Fn^, {z) - zP\ (z)] - n [zP'n (z) - P^i (z)] - {2n + 1) P,, {z) = 0.
Using (I) to eliminate P n+i {z), and then dividing by j n, we get
zP\,{z)-P'n_,(z) = nPn{z) (III).
* The process of differentiating under the sign of integration is readily justified by § i-2.
t Heine states that this relation is due to Gauss.
J If 71 = 0, we have Fq(z) = 1, P_^ (2) = !, and tlae result (III) is true but trivial.
15-21] LEGENDRE FUNCTIONS 303
Adding (I) and (HI) we get
P'n+,(z)-P'n_,(z) = (2n4-l)Pn(z) (IV).
Lastly, writing n- 1 for n in (I) and eliminating P'n-i (z) between the
equation so obtained and (III), we have
(z'-l)P'r,(z) = mPr,(z)-nPn-^(z) (V).
The formulae (I) — (V) are called the recurrence formulae.
The above proof holds whether n is an integer or not, i.e. it is applicable to the general
Legendre functions. Another proof which, however, only applies to the case when n is
a positive integer (i.e. is only appUcable to the Legendre polynomials) is as follows :
Write F=(l-2/i3 + A2)-i.
Then, equating coefficients* of powers of h in the expansions on each side of the
equation
{\-^hz+h^)^^={z-h)V,
we have nP„ (z) - {2n-\) zP^.^ {z)^{n-\) P„_2 (3) = 0,
which is the formula (II).
Similarly, equating coefficients* of powers of h in the expansions on each side of the
equation
we have z-£J - -«^:=nP, (.),
which is the formula (III). The others can be deduced from these.
Example 1. Shew that, for all values of n,
j^{z{PJ + P'n^i)-2PnPn^i}=-{^n + 3) P^„^,-{2n+l) P,,K
(Hargreaves.)
Example 2. If Mn{x) = \ (-j-j (ze*^ eosech 2) ,
dM Lv) P
shew that — J' =nMn^i{x) and I Mn{x)dx = 0. (Trinity,
ax J _i
Example 3. Prove that if ??i and n are integers such that ?«^«, both being even
or both odd,
fl f/p ri p
/ ^x ^ '^^' ^ '" ^''* ■*■ ^^* ^^^*^^' ^^^^-^
Example 4. Prove that, if m, n are integers and m ^ n,
x{l+(-)"-^"'}.
(Math. Trip. 1897.)
* The reader is recommended to justify these processes.
1900.)
304 THE TRANSCENDENTAL FUNCTIONS [CHAP. XV
15'211. The expression of any polynomial as a series of Legendre
polynomials.
hetfn (z) be a polynomial of degree n in z.
Then it is always possible to choose ao, a^, ... a„ so that
fn (z) = aoPo (z) + a^Pi (z) + ... + anPn {z),
for, on equating coefficients of z^, z'^~^, ... on each side, we obtain equations
which determine «„, a^-i, ••• uniquely in turn, in terms of the coefficients of
powers of z in /„ (z).
To determine a^, a^, ... an in the most simple manner, multiply the
identity by Pr{z), and integrate. Then, by § 1514,
when r = 0, 1, 2, ... w ; when r> n, the integral on the left vanishes.
Example 1. Given 2" = aoA (2) + «i A {z) + ... + anPn (^X to determine ao> ^\^ ••• ««•
[Equate coefl&cients of 2" on both sides ; this gives
2».(n!)2
"^"^ {2n)\ '
Let In,m= \ z^Pm {^) dz, SO that, by the result just given,
"•'" (2w + l)! •
Now when n — m'm odd, /„, ^ is the integral of an odd function with limits + 1, and so
vanishes ; and /„, ^ ^l^o vanishes when n — m is negative and even.
To evaluate /„, »» when n — m is a positive even integer, we have from Legendre's
equation
= -[2"(l-^^)i^,„'(2)]^^ +^i j'^Z--^{l-Z^)P,r:iz)dz
= n[z--^{l-z-')P,n{z)J ^~nj' ^{in-l)zr^-^-{n-\-l)z^}P,^{z)dz,
on integrating by parts twice ; and so
m (m + 1) /„, ,„ = 7Z (« + 1) /„, ,n-n {n-\) /„_2, w
Therefore
J ^L^z2) r
^"•'"~(n-m)(rt + m + l) "-'^•"'
% («— 1) ... (m + 1) -
{7i-m){n-2-m) ...2.{n+m + l)in + m-l) ...{2m + 3) *"• ""
by carrying on the process of reduction.
15-21 1, 1522] LEGENDRE FUNCTIONS 305
Consequently /,. ^= (^^_a^) ; (^^ J+i), '
and so «to=0, when n- m is odd or negative,
a„. = Vi f — s . / ^TTT- when n- w is even and positive.]
(^n-^m): (ti + w + I)! ^ ■•
Example 2. Express cos «^ as a series of Legendre polynomials of cos 6 when n is an
integer.
Example 3. Evaluate the integrals
(St John's, 1899.)
Example 4. Shew that
J Jl -^2) {P„' (2))2^,=?!|^^) . (Trinity, 1894.)
Example 5. Shew that
nP„(cos(9)= 2 cosr^i'„_r(cos^). (St John's, 1898.)
r=l
Example 6. If «„ = I (1 -22)»P2m (2) c?2, where m<n, shew that
(%-m)(2?i + 2m+l)w„ = 2«-«„_i. (Trinity, 1895.)
15*22. Murphy's expression* of Pn{z) as a hypergeometric function.
Since (§ 15"13) Legendre's equation is a particular case of Riemann's
equation, it is to be expected that a formula can be obtained giving P„ {z) in
terms of hypergeometric functions. To determine this formula, take the
integral of § 15"2 for the Legendre function and suppose that 1 1 — 2: j < 2 ; to
fix the contour C, let h be any constant such that 0 < 8 < 1, and suppose that
z is such that 1 1 — ^ j ^ 2 (1 — 8) ; and then take G to be the circle f
i 1 - « I = 2 - S.
Since ^j I :$ ^ — k- < 1, v^e may expand {t — zy^'^ into the uniformly
convergent series J
Substituting this result in Schlafli's integral, and integrating term-by-
term (§ 4"7), we get
P (z\= S (^-1X0^ + 1)0^ + 2). ..(n+r) p + .-H (^2_i)n
""^ ^ r%'l''+'iri r\ J A {t-lf+'-
^n+^+rdt
^ I (z-iy.{n+l)(n + 2)...(n+ r)
r=o 2-.(r!)-^
* Electricity (1883). Murphy's result was obtained only for the Legendre polynomials.
t This circle contains the points t = l, t = z.
X The series terminates if ji be a negative integer.
W. M. A. 20
306 THE TRANSCENDENTAL FUNCTIONS [CHAP. XV
by § 5-22. Since arg (^ + 1) = 0 when t = l, we get
(t + l)n =2''-^n(n-l)...{n-r + l),
at jt=i
and so, when 1 1 — ^^ | ^ 2 (1 — 8) < 2, we have
p .._^ {n + l)(n + 2) ... (n + r) .(-n) {1 -n) ...(r-1 -n) (i i V
This is the required expression ; it supplies a reason (§ 14'53) why the cut
from — 1 to — 00 could not be avoided in § 15'2.
Corollary. From this result, it is obvious that, for all values of n,
Note. When « is a positive integer, the result gives the Legendre polynomial as
a polynomial in 1 — s with simple coefficients.
Example 1. Shew that, if m be a positive integer,
[d^-^'Pm.n{t^)\ _ r(2m + n + 2) fTrinitv 1907 )
1 ^"- + 1 7^=i-2'» + i(m + l)!r(«)- (irmity, 1907.)
Example 2. Shew that the Legendre polynomial P„ (cos 6) is equal to
{-YF{n->r\, -n; I ; coa'^ ^6),
and to cos" \e F{-n, -n;\; tan^ \e). (Murphy.)
15*23. Laplace s integrals'^ for Pn{z)-
We shall next shew that, for all values of n and for certain values of z,
the Legendre function Pn {z) can be represented by the integral (called
Laplace s first integral)
- \ [z-\- (^2 - 1)* cos <f)|'' d<b.
(A) Proof applicable only to the Legendre polynomials.
When n is a positive integer, we have, by § 15'12,
"V ; 2^+Wi J c {t - .^^''+'
■zf
where C is any contour which encircles the point z counter-clockwise.
Take G to be the circle with centre z and radius |^^^ — Ij^. so that, on C,
t — z + {z^ — l)^e^'^, where <^ may be taken to increase from — tt to tt.
* Mecanique Celeste, Livre xi. Chap. '2. For the contour employed in this section, and
for some others introduced later in the chapter, we are indebted to Mr J. Hodgkinson.
15-23] LEGENDRE FUNCTIONS 307
Making the substitution, we have, for all values of z,
d<f>
= ^r- f {z -{■ (z- - 1)^ COS <j>Y' d<b
= -\ [Z^ {Z"" - l)i COS <^}» C?<^,
since the integrand is an even function of ^. The choice of the branch of
the two- valued function {z^ — 1)^ is obviously a matter of indifference.
(B) Proof applicable to the Leg endre functions, where n is unrestricted.
Make the same substitution as in (A) in Schlafli's integral defining
P„(2^); it is, however, necessary in addition to verify that ^ = 1 is inside the
contour and ^ = — 1 outside it, and it is also necessary that we should specify
the branch of {z + (z^ - 1)^ cos <f)]'^, which is now a many-valued function of cf).
The conditions that t=l, t = — l should be inside and outside C re-
spectively are that the distances of z from these points should be less and
greater than | ^^^ _ l |i. These conditions are both satisfied ifj^ — 1|< |0 + 1|,
which gives R (z) > 0, and so (giving arg z its principal value) we must have
{argzlK-Tr.
Therefore P„ (z) = ^ f " [z + (z' - 1)^ cos <f)}« d6,
where the value of arg [z + {f- — 1)^ cos ^| is specified by the fact that it
[being equal to arg (^^— 1) — arg (^ — 2^)] is numerically less than tt when t is
on the real axis and on the right of z (see § 15*2).
Now as <^ increases from - tt to tt, 2 + (32 — 1)^ cos0 describes a straight line in the
Argand diagram going from 2 — (z^ — 1)^ to z-\-{^^ — \Y and back again; and since this line
does not pass through the origin*, arg {z + (0^- 1)^ cosc^} does not change by so much as
TT on the range of integration.
Now suppose that the branch of {2 + (0^ - 1) cos (^}" which has to be taken is such that
it reduces to z^e^''^^'^ (where k is an integer) when ^ = \ir.
tnkni r^
Then p,^(^) = ___ {2 + (22-i)4cos</)}«(;0,
^tr J -TT
where now that branch of the many-valued function is taken which is equal to 2" when
Now make 2^-1 by a path which avoids the zeros of /*„ (z) ; since P,j {z) and the
integral are analytic functions of z when I args j < ^tt, k does not change as z describes the
path. And so we get e^"''"' = l.
* It only does so if 2 is a pure imagiuary; and such values of z have been excluded.
20—2
308 THE TRANSCENDENTAL FUNCTIONS [CHAP. XV
Therefore, when | arg ^ | < - tt and ?i is unrestricted,
where arg [z + {z^ — 1)^ cos </>} is to be taken equal to arg z when <f) = ^7r.
This expression for P„ (z), which may, again, obviously be written
- r {z + (z'- - 1)^ cos (^}« d<b,
T Jo
is known as Laplace's first integral for P,j (z).
Corollary. From § 16'22 corollary, it is evident that, when | arg 2 | < ^tt,
a result, due to Jacobi {Crelle, xxvi.), known as Laplace^ s second integral for P„ (s).
Example 1. Obtain Laplace's tirst integral for the Legendre polynomials by con-
sidering
2 A" f " {«+ (22 - l)i cos (/)}» (^0,
»=o j 0
and using § 6'21 example 1.
Example 2. Shew, by direct differentiation, that Laplace's integral is a solution of
Legendre's equation.
Example 3. If s < 1, ] A | < 1 and
(1 - 2/i cos e-\-h?)-*= 2 6„ cos n6,
n=0
shew that ^'-^^ j„ (1"^^)' (T^A2)3«^^- (^^"^^-^
Example 4. When 2 > 1, deduce Laplace's second integral from his first integral by
the substitution
{z-{z'-l)^cos6}{z + {z^-l)icos(f)}==l.
Example 5. By expanding in powers of cos (p, shew that for a certain range of
values of z,
- [''{5 + (22_l)^C0S<i}"rf(A = S"i^(-i?i, i-i%; 1; 1-2-2).
TT J 0 - - -
Example 6. Shew that Legendre's equation is defined by the scheme
( 0 X 1 1
pj -|h i+hi 0 M,
[^ + hi -in 0 J
where 2=i(|^ + ^~^).
15-231. The Mehler-Dirichlet integral* for P„ (2).
Another expression for the Legendre function as a definite integral may be obtained in
the following way :
* Dirichlet, Crelle, Bd. xvii. (1837), p. 35; Mehler, Math. Ann. Bd. v. (1872), p. 141.
15-231] LEGENDRE FUNCTIONS 309
For all values of n, we have by the preceding theorem
F„ (3) = i [ '' {2 + cos (^ (?2 - l)i}" rf</).
In this integral, replace the variable (^ by a new variable A, defined by the equatitm
and we get ' P„(2) = - f'^'^^''"^' h^{l-2hz+k^)-idh;
the path of integration is a straight line, argA is determined by the fact that h = z when
<^=-in-, and (1 -2Az + A''^) - i= -1(22- 1)* sin (^.
Now let a=cos 6 ; then
P„(cos^)=- / k'{\-2hz+h^)-^dh.
■f J -id
e
Now {0 being restricted so that — ^tt < ^ < I^tt when n is not a positive integer) the
path of integration may be deformed* into that arc of the circle |A| = 1 which passes
through A=l, and joins the points A = e~**, h = e^, since the integrand is analytic throughout
the region between this arc and its chord t.
Writing h=e^'^ we get
1 re e(" + i)*>
P„(cosd) = - -dci>,
■^ J ~e (2 cos (^-2 cos ^)2
and so
D / /IN 2 f C0S(/l + l)<^ ,,
P„(cos^) = - ^ -^ ,#;
^ J^ {2(cos(^-cos^)P
it is easy to see that the positive value of the square root is to be taken.
This is known as Mehler's simplified form of Dirichlet^s integral. The result is valid for
all values of n.
Example 1. Prove that, when w is a positive integer,
P„(cos^) = - / ^ -^ ^;l-
^ •/« {2(cos^-cos(^)p
(Write TT - ^ for ^ and ir — (p for (ft in the result just obtained.)
Example 2. Prove that
If A"
P„(cos^) = -^. ^d/i,
27^^j (A2_2Acos^+l)^
the integral being taken along a closed path which encircles the two points /i~e ' , and
a suitable meaning being assigned to the radical.
* If ^ be complex and R (cos ^) > 0 the deformation of the contour presents slightly greater
difficulties. The reader will easily modify the analysis given to cover this case.
+ The integrand is not analytic at the ends of tlie arc but behaves like (/t-c*'^)"^ near
them ; but if the region be indented (§ G-23) at e*** and the radii of the indentations be made to
tend to zero, we see that the deformation is legitimate.
310 THE TRANSCENDENTAL FUNCTIONS [CHAP. XV
Hence (or otherwise) prove that, if 6 lie between Jtt and f tt,
P /pn^m-^ 2.4...2?^ , co.s(^g + (^) , P cos(n^ + 3«^) >
jr d.5...(2%+l) (2sin^)* 2(2n + 3) (2 sin ^)^ |
1 1^32 cos(«^ + 5(^) I".
2.4.(27i + 3)(2» + 5) (2sin^)^ |
where 0 denotes |^ - \ir.
Shew also that the first few terms of the series give an approximate value of P„ (cos Q')
for all values of 6 between 0 and tt which are not nearly equal to either 0 or tt. And explain
how this theorem may be used to approximate to the roots of the equation P„ (cos ^) = 0.
(Math. Trip. 1895.)
15'3. Legendre functions of the second kind.
We have hitherto considered only one solution of Legendre 's equation^
namely P>i(^). We proceed to find a second solution.
We have seen (§ 15 "2) that Legendre's equation is satisfied by
I {t^-iy(t-zy-'dt,
taken round any contour such that the integrand returns to its initial value
after describing it. Let i) be a figure-of-eight contour formed in the following
way : let z be not a real number between ± 1 ; draw an ellipse in the ^-plane
with the points + 1 as foci, the ellipse being so small that the point t = z is
outside. Let A be the end of the major axis of the ellipse on the right
of « = 1.
Let the contour D start from A and describe the circuits (1 — , — 1 -i-),,
returning to A (cf. § 12'43), and lying wholly inside the ellipse.
Let I arg z\^7r and let ] arg (z — t)\^ arg z as t-*0 on the contour. Let
arg(^ + l) = arg(i5— 1) = 0 at A.
Then a solution of Legendre's equation valid in the plane (cut along the
real axis from 1 to — x ) is
if n is not an integer.
When R{n+ 1) >0, we may deform the path of integration as in § 12*43,
and get
Qn(z)=^,l\i-tr(^-t)-^'-'dt
(where arg(l — ^) = arg(l -f-^) = 0); this will be taken as the definition of
Qn (^) when 71 is a positive integer or zero. When n is a negative integer
(= — TO— 1) Legendre's differential equation for functions of degree n is
identical with that for functions of degree in, and accordingly we shall take
the two fundamental solutions to be Pm (z), Qm (z).
Qn {z) is called the Legendre function of degree n of the second kind.
15'3, 15-31] LEGENDRE FUNCTIONS 311
15*31. Expansion of Qn (z) as a power-series.
We now proceed to express the Legendre function of the second kind as
a power-series in z~^.
We have, when the real part of w + 1 is positive,
Suppose that 1 2; | > 1. Then the integrand can be expanded in a series
uniformly convergent with regard to t, so that
where r = 2s, the integrals arising from odd values of r vanishing.
Writing t^ = 11, we get, without difficulty, from § 12*41,
Qn (^) = 2^+1 rln+i)^^ F(^ln+l,ln+l;n + l; ^-^) .
The proof given above applies only when the real part of {71 + 1) is positive
(see § 4*5) ; but a similar process can be applied to the integral
the coefficients being evaluated by writing I (t^— ly^t^ dt in the form
J D
/•(1-) /•(-! + )
Jo Jo
and then, writing t^ = u and using § 12*43, the same result is reached, so
that the formula
is true for unrestricted values of w (negative integer values excepted) and for
all values* of z, such that \z\ > 1, | arg z\< tt.
Example 1. Shew that, when n is a positive integer,
«.(»=<^f ,^ I-: f- D" /; (»- 1)-- *■} ■
* When n is a positive integer it is unnecessary to restrict the value of arg z.
312 THE TRANSCENDENTAL FUNCTIONS [CHAP. XV
[It is easily verified that Legeiidre's equation can be derived from the equation
d^v)
by diflferentiating n times and writing w = -j-^ .
Two independent sohitions of this equation are found to be
/•OO
(22-1)™ and (22-l)» I (v2_l)-"-irfj;.
It follows that ~ 1(32 -\Y i ( y2 _ 1 ) - ~ - 1 ^i.j
is a solution of Legendre's equation. As this expression, when expanded in ascending
powers of 2~i, commences with a term in 2~"~\ it must be a constant multiple* of Qn (z) ;
and on comparing the coefficient of 0~."~i in this expression with the coefficient of z~'^~^ in
the expansion of §„ {z), as found above, we obtain the required result.]
Example 2. Shew that, when n is a positive integer, the Legendre function of the
second kind can be expressed by the formula
C OO r ao '"00 /"qo
$„(0) = 2»n! j j^ j^ ...j^_ (i;2-i)-«-i(f;^;)» + i.
Example 3. Shew that, when n is a positive integer,
" 9" ■»? ' /* ""
Q,{z)= 2 — -^(-2)-t i;'C.2-l)-"-irf,.
[This result can be obtained by applying the general integration-theorem
r r /," - /," ^<"' <*>■"' - ,L4f?9i.r "-^f"' ^'
to the preceding result.]
15*32. The reciirrence-formulae for Qn{z)-
The functions P„ (2) and Q^ {z) have been defined by means of integrals of precisely the
same form, namely
\ {fi-lY{t-zy-^dt,
taken round different contours.
It follows that the general proof of the recurrence-formulae for Pn{z), given in ^ 15"21,
is equally applicable to the function Qn (z) ; and hence that the Legendre function of the
second kind satisfies the recurrence-formulae
{n + l) Qn^,{z)~{^n + \) zQn{z) + nQ,,_^{z) = 0,
^V. + 1 (--)-<?'. -1 (2) = (2H-fl)(>„ (4
{z'-\)Q'n{z)-^nzQ„{z)-nQ,_,{z).
Example 1. Shew that
and deduce that ^2 (2) = ^-^''2 (•') log ^—- - |2,
z — \
Q,iz) = lP,iz)\i>g'^-iz^+-^.
* P„ [z) coni&ina poititive powers of 2 when n is an integer.
1532, 15-33] LEGENDBE FUNCTIONS 313
Example 2. Shew by the recurrence-formulae that, when n is a positive integer*,
jP„(2)logJ-+-i-Q„(2)=/„_,(2),
where fn-i{z) consists of the positive (and zero) powers of z in the expansion of
2 + 1
^P^ (z) log — — in descending powers of z.
[This example shews the nature of the singularities of Qn (z) at ± 1, when n is an integer,
which make the cut from — 1 to +1 necessary. For the connexion of the result with
the theory of continued fractions, see Gauss, Werke, iir. pp. 165-206, and Frobenius,
Crelle, lxxiii. p. 16.]
15*33. The Laplacian integral^ for Legendre functions of the second kind.
It will now be proved that, when i2 (?i + 1) > 0,
Q„ {z) =r [z+ {z^ - 1)^ cosh 6'}-"-^ de,
Jo
where arg [z + (^■^ — 1 y cosh 6] has its principal value when ^ = 0, if n be not
an integer.
First suppose that z >\. In the integral of § 15"3, viz.
^ e'{z+l)^-{z-lf
write t= - , — ^- \,
e%z + \Y + {z-l)^
so that the range (—1, 1) of real values of t corresponds to the range (— oo , oo )
of real values of 6. It then follows (as in § 15*23 A) by straightforward
substitution that
/•oo
Qn {Z) ^\\ [Z + (Z' - 1)2 cosh ^}-'»-i de
/•oo
= {^ -f (0^ - 1 )* cosh 6]-^-^ dO,
Jo
since the integrand is an even function of 6.
To prove the result for values of z not comprised in the range of real values greater
than 1, we observe that the branch points of the integrand, qua function of s, are at the
points +1 and at points where z + {z^ — l)^ conh 6 vanishes; the latter are the points at
which z= ±tanh^.
Hence Q,, (z) and / {z + {z'^- 1)^ cosh 0}-"'-'^dd are both analytic J at all points of the
plane when cut along the line joining the points z= ±1.
* If - 1 < 2 < 1, it is apparent from these formulae that Q^ (^ + 0() - Q„ (^ - 0/) = - 7r/P„ (:).
It is convenient to define Q^iz) for such values of z to be iQ„{z + Oi) + hQ,^(z-Oi). The
reader will observe that this function satisfies Legendre's equation for real values of r.
t This formula was first given by Heine ; see his Kiigelfunktionen, p. 147.
X It is easy to shew that the integral has a unique derivate in the cut plane.
314 ' THE TRANSCENDENTAL FUNCTIONS [CHAP. XV
By the theory of analytic continuation the equation proved for positive values of 2 — 1
persists for all values of z in the cut plane, provided that arg{2 + (22 — 1)2 cosh 6\ is given
a suitable value, namely that one which reduces to zero when 2 — T is positive.
The integrand is one-valued in the cut plane [and so is $„(2)] when n is a positive
integer; but arg {2 + (2^ — 1)^ cosh &\ increases by %ir as arg2 does so, and therefore if % be
not a positive integer, a further cut has to be made from 2= — 1 to2=— oc.
These cuts give the necessary limitations on the value of 2 ; and the cut when n is not
an integer ensures that arg {2 + (22 — 1)2} =2arg{(2 + l)2 + (2— l)2},has its principal value.
Example 1. Obtain this result for complex values of 2 by taking the path of
integration to be a certain circular arc before making the substitution
^^6^2 + 1)^ -(2-1)^
where 6 is real.
Example 2. Shew that, if 2 > 1 and coth a=z,
Qn (2) = f " {z - (22 - 1 )i cosh ^» die,
where arg {2 - (2^ - l)i cosh u] = 0. (Trinity, 1893.)
15*34. Neumamtis* formula for $„ (2), when n is an integer.
When n is a positive integer, and 2 is not a real number between 1 and — 1, the
function Qn{z) is expressed in terms of the Legendre function of the first kind by the
relation
which we shall now establish.
When 1 2 j > 1 we can expand the integrand in the iniiformly convergent series
Pniy) 2 ;^K
ni=0 *
Consequently
^ J -\ ^-y ^ ni=0 J -1
The integrals for which m — ?i is odd or negative vanish (§ 15-211) ; and so
■^ J -I ^~y ^ m=0 J -1
1 * , ,2» + ^(n + 2m)\ (n + m)\
2,„=o m\{2n + 2m + l)l
2" (il n2
by § 15'31. The theorem is thus established for the case in which | 2] > 1. Since each
side of the equation
I'epresents an analytic function, even when | 2 | is not greater than unity, provided that 2 is
not a real number between —1 and +1, it follows that, with this exception, the result is
true (§ 5"5) for all values of 2.
* F. Neumann, Crelle, Bd. xxxvii. p. 24.
15'34, 15-4] LEGENDBE FUNCTIONS 315
Example 1, Shew that, when -\^R{z)^\, | $„(a)| ^[/(z) |-i; and that for other
values of z, |§n(«)l do^s not exceed the larger of |2-1|~', |3+1|"^
Example 2. Shew that, when n is a positive integer, (?„ (2) is the coefficient of A" in
the expansion of (1 - 2hz + A'^) ~ 2 arc cosh J r L .
1(22- l)ij
[For, when | A | is sufficiently small.
= (1 - 2hz + A2) - i arc cosh | A^'^ ]
This result has been investigated by Heine, Kugelfunktionen, i. p. 134, and Laurent,
LiouvUle^s Journal, (3) I. p. 373.]
15'4. Helms* development of {t — z)"^ as a series of Legendre poly-
nomials in z.
We shall now obtain an expansion which will serve as the basis of
a general class of expansions involving Legendre polynomials.
The reader will readily prove by induction from the recurrence-formulae
(2m+ 1) tQ,,{t)-(m + 1) Q,n+^ (t) -mQ,n-^ (0 = 0,
{2m + 1) zPm (z) - (m + 1) Pm+i (z) - mPm-, {z) = 0,
that
I Z OT=o f — ^
Using Laplace's integrals, we have
Pn+A^)Qn(t)-Pn(^)Qn+dt)
n 1"^ f
1
TTJo Jo
z + (z'- 1)^C0S<^}'
Now consider
x[z + (z- - 1)^ cos (^ - {t + (t- - 1)- cosh u]-^] d(J3du
j z + {z^- 1)2 cos<^ i
\t + {f-- l)"*coshM
Let cosh a, cosh a be the scnii-major axes of the ellipses with foci ±1 which pass
through z and t respectively. Let 6 be the eccentric angle of z ; then
s=cosh {a + i6),
\z±{z^-l)^cos(})\ = \ cosh (a + id) ± sinh (a + id) cos cji |
= {cosh2 a - sin- d + (cosh"^ a - cos- 6) cos^ 0 ± 4 sinh a cosh a cos <^}2.
This is a maximum for real Vcilues of 0 when eos(f)= + l; and hence
] s± (s2 - l)i cos 4)\^%2 cosh2 a - 1 + 2 cosh a (cosh- a - l)i = exp (2a).
Similarly \t + {t--Vy-i cosh ic \ ^ exp a.
* Crdle, Bd. xlii. p. 72.
316 THE TRANSCENDENTAL FUNCTIONS [CHAP. XV
Therefore
P~^i (z) Qn (0 - Pn (z) Qn+i (t) \ $ ^"^ exp {n(a - a)] f / Vd<f>du,
n+i '
where I ^| = ! —^-^^ 1 ~
t + {P-l)^coshu
Therefore | Pn+i (z) Qn (0 - ^« (^) Qn+i (t)\^0, as 7i^ cc , provided a < a.
And further, if t varies, a remaining constant, it is easy to see that
the upper bound of Vd(f)du is independent of t, and so
Jo Jo
Pn+i{z)Qn(t)-Pn(z)Qn+Ut)
tends to zero uniformly with regard to t.
Hence if the point z is in the interior of the ellipse which passes through
the point t and has the points ± 1 for its foci, then the expansion
J-= 2 {%l+l)Pn{z)Qn{t)
t — Z n=0
is valid ; and if t be a variable point on an ellipse with foci ± 1 such that z is
a fixed point inside it, the expansion converges uniformly ivith regard to t.
15 '41. Neumann's* expansion of an arbitrary function in a series of
Legendre polynomials.
We proceed now to discuss the expansion of a function in a series of
Legendre polynomials. The expansion is of special interest, as it stands next
in simplicity to Taylor's series, among expansions in series of polynomials.
Let f{z) be any function which is analytic inside and on an ellipse C,
whose foci are the points e = ± 1. We shall shew that
f{z) = ttoPo {z) +a,P, (z) + a,P^ (z) + a,P, {z) + ...,
where a^, a^, a^... are independent of z, this expansion being valid for all
points z in the interior of the ellipse C.
Let t be any point on the circumference of the ellipse.
00
Then, since S (2n + 1) Pn{z) Qn{t) converges uniformly with regard to t
71=0
/<^> - 2S- l/t"^ = 2S J. /. (^" ^ 1) ^» (^> «•■ <')/<*> '"'
00
X anPniz),
n = 0
2n + 1 f
where Un = ^ . j f{t) Qn (t) dt.
* K. Neumann, Ueher die Enticickelung einer Funktion nach den Kugelfunktionen (Halle,
1862). See also Thome, Crclle, Bd. lxvi. p. 337. Neumann also gives an expansion, in Legendre
functions of both kinds, valid in the annulus bounded by two ellipses.
15-41, 15o] LEGENDRE FUNCTIONS 317
This is the required expansion ; since S (2n + 1) P„ (z) Q^ (t) may be proved*
w = 0
to converge uniformly with regard to z when z lies in any domain C lying
wholly inside G, the expansion converges uniformly throughout C.
Another form for a„ can therefore be obtained by integrating, as in
§ 15-211, so that
an = (n +1) j /(^) ^n (a?) dx-
A form of this equation which is frequently useful is
""^I^J!/"'^^^^^-^^"'^'^"^^'
which is obtained by substituting for P„ (w) from Rodrigues' formula and
integrating by parts.
When f{x) is a function of a real variable x in the range ( - 1 ^ « ^ 1), f(x) can he
expanded in a series of Legendre polynomials v«,lid in this range and has limited total
fluctuation in the range (-1, 1).
This theorem bears the same relation to Neumann's expansion as Fourier's theorem
bears to the expansion of § 9'11.
For a proof of a more general form of the theorem, the reader is referred to memoirs
by Hobsont and BurkhardtJ.
Example 1. Shew that, if p (> 1) be the radius of convergence of the series 2c„z", then
2c„P„(2) converges inside an ellipse whose semiaxes are ^(/> + p"*), i(p-p~^).
Example 2. If z=(^), k^= \'^--]\^l±^ ^„heve y > x> 1,
provethat [' ^=.{(^+i)(y- i)}i i P,{x)Q,{,j).
[Substitute Laplace's integrals on the right, integrate with regard to (f) and then
take a new variable t defined by the equation
(y + l)^ + (i/-l)*cosh(9
(y-l)^ + (?/ + l)i cosh^
Example 3. Shew that
= t.]
15-5. Ferrers associated Legendre functions P,i'" {z) and Q,j"* {z).
We shall now introduce a more extended class of Legendre functions.
If m be a positive integer and -\ <z <1, n being unrestricted §, the
functions
P, - (.) = (1 - .^)i- ^J^^"> , Q, - (.) = (1 - .•^)*- ^'"^^^
* The proof is similar to the proof in § 15'4 that that convergence is uniform with regard
to t.
t Proc. London Math. Soc. Vol. vi. (1908), pp. 388-395; Vol. vii. (1909), pp. 24-39.
:;; Munchcn. Sitzungsbericltte, Bd. xxxix. (1909), No. 10.
g See p. 311, footnote. Ferrers writes T„"'(z) for F,^"^{z). ,
318 THE TRANSCENDENTAL FUNCTIONS [CHAP. XV
will be called Ferrers' associated Legendre functions of degree n and order m
of the first and second kinds respectively.
It may be shewn that these functions satisfy a differential equation
analogous to Legendre 's equation.
For, differentiate Legendre's equation
m times and write v for -r-^ . We obtain the equation
{l-z^)-r-^-22(7n + l)^ +{n-m){7i + m + l)v = 0.
Write w = {1 — 0^)^"' v, and we get
This is the differential equation satisfied by P^^ (z) and Qn^ (z).
From the definitions given above, several expressions for the associated Legendre
functions may be obtained.
Thus, from Schlafli's formula we have
where the contour does not enclose the point ^= — 1.
Further, when n is a positive integer, we have, by Rodrigues' formula,
p m (,\ _ \ i ^ i_
Example. Shew that Legendre's associated equation is defined by the scheme
r 0 00 1 \
p}. ^m n + 1 ^m ^-^zi. (Olbricht.)
\ —\m —n —^m J
15"51. The integral properties of the associated Legendre functions.
The generalisation of the theorem of § 15*14 is the following:
When n, r, m are positive integers and n > in, r > m, then
[ Fn'"" {z) Pr (z) dz=0 (r :jfe n) ]
2 (71 + m) !
2n + l {n — m.y.
(r = n) I
To obtain the first result, multiply the differential equations for P,/"' (z),
P/" (z) by Pr^ (z), P,i'^ (z) respectively and subtract ; this gives
+ {n -r)(n + r + l) P,'" (z) P,,»" (z) = 0.
15-51, 15'6] LEOENDRE FUNCTIONS 319
On integrating between the limits — 1, +1, the result follows when n
and r are unequal, since the expression in square brackets vanishes at each
limit. "^
To obtain the second result, we observe that
, fjp m{g\ .
Pn"*+' (Z) = (1 - Z')i ^Y^ + WIZ (1 - Z^) - 4 PrT {z) \
squaring and integrating, we get
+ ^lAPn^\z)Y]dz
= -J[^Prr(z) ^ j(l-^^)^-^«^>|ci.-mJ'^ {Pn"^(z)Ydz
+ j_^^^APn^{z)}^dz,
on integrating the first two terms in the first line on the right by parts.
If now we use the differential equation for P^^{z) to simplify the first
integral in the second line, we at once get
[ {P«"*+i {z)Y dz = (n-m){n + m + l)j {P^^ {z)Y dz.
By repeated applications of this result we get
/
1
{P„''*(z)}2 dz = {n-m + l){n-m+2)...7i
x(n + m){n + m-l)...{n-\- 1)1 {Pn{z)Ydz,
and so f {P,r(z)Ydz=^^ ^:^'^.
15'6. Hohson's definition of the associated Legendre functions.
So far it has been taken for granted that the function (1 — z'^)^'^ which
occurs in Ferrers' definition of the associated functions is purely real ; and
since, in the more elementary physical applications of Legendre functions, it
usually happens that — 1 < 2; < 1, no complications arise. But as we wish
to consider the associated functions as functions of a complex variable, it is
undesirable to introduce an additional cut in the ^-plane by giving arg (1 — 2^)
its principal value.
Accordingly, in future, ivhen z is not a real number such that — 1 < z < 1,
we shall follow Hobson in defining the associated functions by the equations
p- (z) = {z^ - 1)4- ^";^/"> , Q, - (.) ^ (z^ - i)i- ^"2:i"^ ,
where m is a positive integer, 71 is unrestricted and arg z, arg (z + 1), arg (2— I)
have their principal values.
320 THE TRANSCENDENTAL FUNCTIONS [CHAP. XV
When 7)1 is unrestricted, Pn"* (z) is defined by Hobson to be
and Barnes has given a definition of $„'" {z) from which the formula
_sin(?2H-m)n- r(% + m + l)r(^) {f-\)^'^
^»'"(2) = -
may be obtained.
Throughout this work we shall take m to be a positive integer.
15'61. Expression of Pn^ {z) as an integral of Laplace's type.
If we make the necessary modification in the Schlafli integral of § 15"5,
in accordance with the definition of 1 15"6, we have
P, » (,) = (»» + l)(« +^2).. ■(» + «!) (,, _ i)4,„ J^-' '+>(,, _!)„(,_ ,^_,._.._, ^,
Write t = z + {z'- 1)^6'*, as in § 15-23 ; then
P m /,x _ (ri+l)(n+2)...(7i + 7n) j,„ pT+a |^ + (^2_i)lcos<^p^
^^ ^^^~ 2^r ~^'-^^ j. |(,._i)V*}- "^^^
where a is the value of ^ when Hs at J., so that
I arg {z^ — 1)^ + a I < TT.
Now, as in § 15*23, the integrand is a one- valued periodic function of the
real variable <p with period 27r, and so
Since [z + iz^ — l)2cos0}'* is an even function of ^, we get, on dividing
the range of integration into the parts (— tt, 0) and (0, tt),
p_ .„^^^ ^ (.. + !)(« + 2)... (,. + m) r j^ ^ ^^, _ j^j _^^^ ^ „ ^^^ ^^
The ranges of validity of this formula, which is due to Heine, (according as
n is or is not an integer) are precisely those of the formula of § 15'23.
Example. Shew that, if | arg 2 | < Jtt,
p^ m (;;) _ / Nm '<^ {n-\)...(n-m->r\) f^ cos mcjidcf)
^ ^0 {2+(s''^-l)*COS0}" + l'
where the many-valued functions are specified as in § 15-23.
15*7. The addition theorem for the Legendre polynomials*.
Let z=xx' — {x'^—\y^ {x"^ — 1)^ cos co, where x, x\ a are unrestricted complex numbers.
* Legendre, Calc. Int. t. ii. pp. 262-2G9. An investigation of the theorem based ou physical
reasoning will be given subsequently (§ 18-4).
LEGENDRE FUNCTIONS
15-61, 15-7]
Then we shall shew that
x + ^x^ - 1)^ cos {<>>-<f>)
321
First let R (a') > 0, so that
is a bounded function of (f> in the
range 0 < 0 < 27r. If 31 be its upper bound and if | A | < M~^, then
« ^J^ + (:g--i-l)^C0s(a>-(^)}"
«=" {a-' + (:p'2-1)^C08<^}" + i
converges uniformly with regard to cf), and so (§ 4'7)
* /■ T {^+(^2_i)i cos (« - <;())}» , . _ r " 5 A™J^(^2 - iji cos (co - (^)}»
n=0 j -T {^ + (,r'2-l)ic0S^}'' + l j -1)1=0 {ji;' + (;(7'2-l)i cos (/)}" + !
/:
rf(^
-:r ^' + (.??'^- 1)^ COS (jy-h {x + {x^- 1)^ cos (w - <^)}
Now, by a slight modification of example 1 of § 6 "21, it follows that
dcf) 27r
/:
A + Bcoscfy + Csincl) (A^-B^-C^)^'
where that value of the radical is taken which makes
\A-{A^- 2?2 _ (;2)i I < I (^2 + (72)^ I ,
Therefore
/
d(f)
-n ^ + (^'2-l)icos(^-A{.r + (.r2-l)4cos(a)-<^)}
2rr
[(X' - hxf - {(^'2 _ 1 )4 _ /j (^2 _ 1 )i COS ^}2 _ |/i (^2 _ 1 )4 gin a>ff
2tt
(1-2A2+A2)i'
and when /i-».0, this expression has to tend to 27rPo C'^') by § 15-23. Expanding in powers
of h and equating coefficients, we get
Pn{^ = l
{x + {x"- - l)i cos (q) - 0)}»
d4>.
2 "■ ./ _ ^ {.f' + (a,-'2 - 1 )4 cos (^}™ + 1
Now P„ (s) is a polynomial of degree n in cos w, and can consequently be expressed in
n
the form ^^o+ 2 J,„cosOTa), where the coefficients Jq? -^i ■■■ -Ui ai"e indei)endent of w;
»i=i
to determine them, we use Fourier's rule (§ 9'12), and we get
1
Am —
1
9^
Pn (2) COS w^a) dco
^ {x 4- (,i;2 - 1 )^ cos (w — c^)}" COS wiw
l''2
2- 1)- COS (w — d))}"C0SHia) ,,1 ,
^ -T — ^-^ dcf) dco
;' + (.x''2-l)4cos0}»-'i J
1 fn r /"t {.*; + (.*-2- 1)2 cos (co -<i)}"COS J?la) , 1 , ,
^-^, I ! !^ J — dco \d(fi
^■^ J -n\_j -n {^' + (x'2-l)icOS(^}"+l J
1 [■^ \ i"^ {a' + (^^— 1)^ cos\//'}"cosm (0 + a//-)
27^- J -TT L j '^ (.,/ + (.t;'2 _ 1)1 COS 0;» - 1 ' -
c/0,
on changing the order of integration, writing co = (^ + \/^ and changing the limits for y\f
from + TT — 0 to ± TT.
w. M. A. 21
322 THE TRANSCENDENTAL FUNCTIONS [CHAP. XV
Now I {^ + (072-1)2 cos !!/■}" sin m^//c?^//• = 0, since the integrand is an odd function;
and so, by i:^ 15"61,
n\ [-^ cos m(p . Pji'" (x) ,
TT (% + m) ! 7 _^ |_^.' + (^2 _ 1)4 COS <^}» + 1
Therefore, when | arg z' | < ^tt,
n f'Y) — qi-)\ t
P^{z)=.P,,{x)P,{x') + 2 2 (-)™^V-^P„-(^)P,-(.r')cosm«.
m=I (^/(.-(-//Ij .
But this is a mere algebraical identity in x, x' and cos u> (since w is a positive integer)
and so is true independently of the sign of R [x').
The result stated has therefore been proved.
The corresponding theoi-em with Ferrers' definition is
P„ {xx' + ( 1 _ .r2)4 ( 1 - .r'2)2 cos a,} = P„ {x) P„ {x') + 22 ^ 3-^^ ^n"^ .(*■) /^J" (-r') cos mco.
m=i {n + inj .
15*71. The addition theorem for the Legendre functions.
Let X, x' be two constants, real or complex, whose arguments are numerically less than
\if ; and let {x±l)^^ (-^'±1)^ be given their principal values; let u) be real and let
Z = xx — (^2 — 1 ) 2 (^ 2 _ 1 )2 cos O). •
Then toe shall shew that, if j argi | < Itr for all vahies of the real variable ta, and n he
not a positive integer,
P,,{z)=P,„{x)P,{x') + 2 2 ( - )'» ^ L , „: 1 1 1^:"' (^) Pn' (^') ^O^ »^"-
)H = i J^ (^71 -t- /M -f- i ;
Let cosh a, cosh a be the semi-major axes of the ellipses with foci ±1 passing through
.r, x' respectively. Let ^, ^' be the eccentric angles of x, x' on these ellipses so that
-Itt </3< .^TT, -\n <fi' <hn.
Let a + i0 = I, a' + 2/3' = $', so that ,r = cosh |, :r' = cosh |'.
Now as CO passes through all real values, R (z) oscillates between
R {xx') ± R (,f2 - 1 )i (.*-'2 - 1 ) i = cosh (a ± a') COS (3 ± fi'),
so that ?Y is necessary/ that 13 + 13' he acute angles positive or negative.
Now take Schlafli's integral
and write
_ e"^ {e"'"'^ sinh g cosh -^-f - cotsh g sinh 1 g'j + cosh ^ cosh Af - e''" sinh g sinh jg'
cosh If + e^*sinhig'
The path of t, as <^ increases from — tt to tt, may l)e shewn to be a circle; and the
reader will verify that
_ ^ _ 2je^^*2!i^^shJ|-i- sinh \^) {sinh i^ cosh ||' - e''" cosh 1 g sinh ^f |
cosh If + </* sinh ig'
■). U ("^ - '-) sinh |g + cosh ig] {cosh |^ cosh \^ - e'"' sinh h^ sinh if J
i + 1^
C(wh|f + /*sinhif
{e'* cosh Jf + sinh ;}f } {e'" sinh g sinh- h^' + e " '" sinh g cosh- |-f - cosh g sinh f }
cosh if + t''* sinh ^f
15-71, 158] LEGENDRE FUNCTIONS 323
Since* | cosh J^' | > | siuh ^^' |, the argument of the denominators does not change when
<f) increases by 27r ; for similar reasons, the arguments of the first and third numerators
increase by 27r, and the argument of the second does not change; therefore the circle
contains the points t=l, t=z, and not ^=—1, so it is a possible contour.
Making these substitutions it is readily found that
P,,{z) = — f" {^' + (^^-l)^cos(<o-<^)}"
d(f),
and the rest of the work follows the course of § 15--7 except that the general form of
Fourier's theorem has to be employed.
Example. Shew that, if n be a positive integer,
Qn {XX' + (^2 _ 1 )i (^2 _ 1 )i cos O)} = $„ {x) P„ {x') + 22 ^^^ {x) 1\ " '" {x') COS ?««,
TO = 1
when o) is real, R{x') ^0, and \{x' — \) {x ■\-\)\ <\{x —\) {od ->r\)\.
(Heine, Kugelfunktionen ; K. Neumann, Leipzig. Ahh. 1886.)
15-8. The functioni C^ {z).
A function connected with the associated Legendre function P„™ {z) is the function
€n (z), which for integral values of 7i is defined to be the coeflBcient of A" in the expansion
of (1 -2hz + h'^)~'' in ascending powers of h. ,
It is easily seen that Cn {z) satisfies the differential equation
^ (2v+^ di _ n{n + 2v)
dz'^ z'-l dz z'-l ~^~^-
For all values of n and v, it may be shewn that we can define a function, satisfying
this equation, by a contour integral of the form
^^ ') jp («-2)»+l '^^'
where C is the contour of § 15-2 ; this corresponds to Schlafii's integral.
The reader will easily prove the following results :
(I) When n is an integer
Clz)- {-2rv{u + l)...{u + n-l) _ I _ , d- n + v - h .
^'' ^^'^7ili2n + 2v-l){27i + 2v-2)...{n + 2p)^^ '' dz^^^^ "> '*'
since P^ (2) = C,r (2), Rodrigues' formula is a particular case of this result.
(II) When r is an integer,
^n-r (^) = (2r~iy"(27- ^X^.ZT\ dz' ^'' ^^^'
The last equation gives the counexiun between the functions C„"(i) and P,,' {z).
"" This follows from the fact that cos /3' > 0.
t This function has been studied by Gegenbauer, Wicn. Sitzuiujbberichtc, 13de. lxx, lxxv,
xcvii, cii.
21—2
324 THE TRANSCENDENTAL FUNCTIONS [CHAP. XV
(III) Modifications of the recurrence-formulae for Py^ (z) are the following :
dCr," (2) _ o„/f +1 M
nC:'{z) = in-\ + 2v)zCl_^{z)-2v{\-z'^)C]-\{^).
REFERENCES.
A. M. Legendre, Calcul Integral^ t. 11.
H. E. Heine* Handbuch der Kugdfunktionen (Berlin, 1878).
N. M. Ferrers, Spherical Harmonics (1877).
I. Todhunter, Functions of Laplace, Lame and Bessel (1875).
L. ScHLAFLi, Ueber die zwei ■Heine' schen Kugelfunktionen (Bern, 1881).
E. AV. HoBSON, Phil. Trans, of the Royal Society, Vol. 187 a, pp. 443-531.
E. W. Barnes, Quarterly Journal, Vol. xxxix. pp. 97-204.
R. Olbricht, Studien ueber die Kugel- und Cylinder -funktionen (Halle, 1887).
Miscellaneous Examples!.
1. Prove that when n is a positive integer,
(Math. Trip. 1898.)
/"I dP dP
2. Prove tlmt zil-z^^^^dz
J —I az ctz
is zero unless m — n= ±1, and determine its value in these eases.
(Math. Trip. 1896.)
3. Shew (by induction or otherwise) that when n is a positive integer, •
(2h + 1) I' Pa' (S) dz=l-zP,;^-2z (PjHp.,2+ ... + p2^_j) + 2 {P,P., + P.,Ps + ... + P,,_iP,,).
(Math. Trip. 1899.)
4. Show that
zP,: {z) = nP, (z) + (2h - 3) P„ _ , (.-) + (2/i - 7) P, -,{z) + ....
(Clare, 1906.)
5. Shew that
z^ P,:' (z) = n {n - 1 ) P„ (z) + 2 (2/. - 4/- + 1 ) {r (2/i - 2/- + 1 ) - 2} P,, _ 2,- {^),
r=l
where p = In or I (n - 1). (Math. Trip. 1904.)
* Before studying the Legendre function P„(2) in this treatise, the reader should consult
Hobson's memoir, as some of Heine's work is incorrect.
t The functions involved in examples 1-30 are Legendre 'polynomials.
LEGENDRE FUNCTIONS 325
6. Shew that the Legendre polynomial satisfies the relation
(a2-l)!i"^' = n(«-l)(w + l)(H + 2) / dz I Pn{z)dz.
(Trin. Coll. Dublin.)
7. Shew that
/
0 ^ ^-» (') ^"-> ^'^ ^^= (27.-1) (2^ + 1) (2.^ + 3) •
(Peterhou.se, 1905.)
8. Shew that the values of I (1 —z^yPm" (z) Pn i^) dz are as follows :
(i) 8n (n + 1) when m — n'vi positive and even,
(ii) — 2w (Ti^ — 1) (7i_2)/(2?i+l) when ?H = n,
(iii) 0 for other values of m and n. (Peterhouse, 1907.)
9. Shew that
iihV'dP,„(sme)= 2 (-)'"-7-; — ^^^ cos'" ^P^ (cos ^).
r=o ^! (n — r) ! r\ /
(Math. Trip. 1907.)
10. Shew, by evaluating I P^ (cos ^) dd (§ 15-1 example 2), and then integrating by
r
parts, that I P„ (/x) arc sin fi . dfj, is zero when n is even and is equal to n- j^ '/■'"/ 1 \f
when % is odd. (Clare, 1903.)
11. If m and n be positive integers, and m^n, shew by induction that
P (Z)P (Z)- S An-r^yAn-r f27l + 2m - 4r+l\
1.3.5...(2to-1)
where -^«i= ; •
7n !
(Adams, Proc. Royal Soc. xxvfi.)
12. By expanding in ascending powers of ?i shew that
( — V r/" 1
where m^ is to be replaced by (1 -z-) after the differentiation has been performed.
13. Shew that P„ (s) can be expressed as a constant multiple of a determinant in
which all elements parallel to the auxiliary diagonal are equal (i.e. all elements are equal
for which the sum of the row-index and column-index is the same) ; the determinant
containing n rows, and its elements being
_1 1 _1 1 1
^' ~3' 3^' 5' 5''' -a^T^^-
(Heun, Gott. Nach. 1881.)
14. Shew that, if the path of integration passes above <=1,
„,, 2 r {z{\-f-)-2t(\-z'^fY , ,.,., ,
15. By writing cot ^' = cot 6 — h cosec 6 and expanding sin Q' in powers of h by Taylor's
theorem, shew that
P„ (cos^) = ^ cosec" + i ^ t'r^-^} ■ (^lath. Trip. 1893.)
«1 rt(cot^)" "■
326 THE TRANSCENDENTAL FUNCTIONS [CHAP. XV
16. By considering 2 A" P,i (2), shew that
« ! \/7r J -00 V dzj
17. The equation of a nearly spherical surface of revolution is
r = 1 + a {Pi (cos 6) + P3 (cos d) + ... + P2n-i (cos d)},
where a is small ; shew that if a^ be neglected the radius of curvature of the meridian is
n-l
1 + a 2 {%(4?rt + 3)-(m + l)(8m-|-3)} P2»j + i(cos^).
TO = 0
(Math. Trip. 1894.)
18. The equation of a nearly spherical surface of revolution is
r = a {1 + fPn (cos 6)},
where e is small.
Shew that if e^ be neglected, its area is
ina^ |l +i.2 '-'^t^l . (Trinity, 1894.)
19. Shew that, if k is an integer and
(\-2kz + h^)-^^= 2 a„P^(z),
A»_ 2^(^-3) (2M + 1) (,^,l^l\^^^-^^^^,-U2n-k + i) i(2n + k-2)
then
«« = (T^A2p2 1.3. 5. ..'(>(;- 2) V'^ dx^dy) ^ ^' ' " ''^'
where x and y are to be replaced by unity after the difterentiations have been performed,
(Routh, Proc. London Math. Soc. xxvi.)
20. Shew that
j[^~~{Pn{:v)P,,_^iz)-P,,_,{x)P,,i^)}dx=-^^,
J,2,^l l[^"(^) G^"-(^) + nil^"-(^))]=-l- (Catalan.)
21. Let A'2 + ?/^ + 2- = /2, z = fxr, the numbers involved being real, so that — 1 </i< 1.
Shew that
(-)"r»^i 8" /1\
where r is to be treated as a function of the independent variables .v, 1/, z in performing
the difterentiations.
22. With the notation of the preceding example (cf p. 313, footnote *), shew that
(,. + 1)P„(m) + M^,/(m) =
7i ! Si" V'"''
23. Shew that, if j A | and ] z \ are sufficiently small,
-^-^-, = 2 (27i+ \)h"P, (z).
(l-2/iz + h^)'^ "=«
LEGENDRE FUNCTIONS
327
24. Prove that
Pn.x{z)Qn.x{z)-P.-X {z) Qn.X i') = ^^^^Y)'-
(Math. Trip. 1894.)
25. If the arbitrary function /(.r) can be expanded in the series
f{x)= 2 a„P„(^),
converging uniformly in a domain which includes the point x=\, shew that the expansion
of the integral of this function is
/;/Wofc=-«.-^<.,+ J_ G:"f\- |^)i'.W. (Bauer.)
26. Determine the coefficients in Neumann's expansion of e"^ in a series of Legendre
polynomiala (Bauer.)
27. Deduce from example 25 that
77-/1.3. 5. ..(2/1- 1)^2
arc sm 2 = - 2 ,
2 0 ( 2. 4. 6. ..2/1
{P2.^l{z)-P2n-l{z)].
(Catalan.)
28. Shew that
Qn {Z) = \ log (J^) . P„ (.) - |P„_, (3) Po (^) + \ Pn-2(Z) A C^)
+ lPn-3iz)P2{z) + ...+lPoiz)Pn-Az)\-
(Schlafli, Hermite.)
29. Shew that
«"<^)=2^|J.f-'-')"'»8;-^}-l^"WH':4{
Prove also that
Qniz) = lPAz)log'--^-f.,,,{z),
u * j: / \ 2/1 - 1 „ , ^ 2/t - 5 „ / V 2% - 9 „ , X
where* f,i-i(z) = F,, ^ (z) -\ P„ ■t(z)+ P„ .-, (s) + ,..
, - J,..,(,. ^ 1) '11-1^:1) (irl) ^ (,. _ I _ I) " (..- 1) (..y) (;i±j) (eri)-
^/. . 1 rwi(n-l)(vi-2)(» + l)(7i + 2)(« + 3) /2-l\3^ I'
( +(^^'.-1-2-3; 122232 (^-^j +-^
where >{-„=l+^ + ^ + ...+-. (Math. Trip. 1898.)
30. Shew that the complete solution of Legendre's difterential equation is
dt
y = APn{z) + BP,.{z)
(?^-l){/^„(OP'
the path of integration being the straight lino which when produced backwards pas.se.s
through the point ^ = 0.
The tirst of these expressions for f^-x {^) was given by Christoffel, Grelle, lv.
328 THE TRANSCENDENTAL FUNCTIONS
31. Shew that {2 + (22- l)i}«= i 5,„fen-a-i (s),
?n=0
where B a{a-^m + ^) T{m-\)T{m-a-\)
™ 2iT m\T{m-a+l)
[chap. XV
(Schliifli.
A-»-i
32. Shew that, when R{n + l)>0,
J Z+{Z^-])^ (l-2/«2 + A2)5
and
Qn(z) = j'^'
■(z'-l)^
A"
<^/i,
c^A.
33. Shew that
(l-2A2 + /j2)i
cosh ?»?<
r(?i-m + l)7o {s + (52_i)icoshtt}"+i
c??^,
where the real part of (n + l) is greater than m.
(Hobson.)
34. Obtain the expansion of P„ (z) when | arg 3 1 < tt as a series of powers of l/z, when
n is not an integer, namely
Fu{z) = ~~'^{QAz)-Q-n-l{z)}
TV
2"J>^jJ)_ /l-_9i _'_i i_^ 1
+ r(-?or(i)"'
[This is most easily obtained by the method of § 14-51.]
2 ' ^' 2 ' '^■*"'^"' 02
35. Shew that the differential equation for the associated Legendre function P,/" {z)
is defined by the schemes*
0 00
0
X 1
P\ -in on -hi —^ \[, p\ -hi hn 0 -~
\;\« + i -m h'^ + l ) \hi + l -hn J
(Olbricht.)
36. Shew that the differential equation for (7/ (z) is defined by the scheme
f -■' " ' 1
P-l h-v 7i + 2v i-p z y
[ 0 -n 0 J
37. Prove that, if
^ (27^+ 1) {2n + 3U. (2n + 2s-l) _ d>P,
^'' ni^n^-\){^n^-\) ...{n^-{s-\f}{n + sy^ ' dz' '
u n 2(2« + l) 2n + 3 „
then ^. = P,.,- 2„rr ^" + 2-]rrT^^-2,
_3(2n + 3) 3(2n + 5) (27^ + 3) (2» + 5)
" + 3 2h-1 '»->^ 2;i-3 '"' (2n-l)(2n-3) "-'''
and find the [general formula.
(Math. Trip. 189G.)
See also § 1.5-5 example.
LEGENDBE FUNCTIONS 329
38. Shew that
p^m (cos 6) = — ^(^ + "^+^) r<iO^{{n+\)e-^n + lmn} ^ V' - 4m^ C08{(7^ + §)^-J7r + |m7r}
^ir r{n + ^) L (2sin^)i 2(2« + 3) (2sin^)*
(12 - 4m2) (32^- 4m2)^ cos {(n + f) ^-f7r + ^W7r} "1
2.4.(2/H-3)(2ri + 5) (2 sin ^)^ "'J'
obtaining the ranges of values of m, n and ^ for which it is valid.
(Math. Trip. 1901.)
39. Shew that the values of n, for which /*„~ '" (cos 6) vanishes, decrease as 6 increases
from 0 to TT when vi is positive; and that the number of real zeros of P„~"*(cos5) for
values of B between —n and tt is the greatest integer less than n — m+l.
(Macdonald, Froc. London Math. Soc. xxxi, xxxiv.)
40. Obtain the formula
-— I " [1 - 2A {cos 0) cos (^ + sin co sin 0 cos {6' - 6)] + A^] ~^dd= 2 k^Pn (cos a>) 1\ (cos <^).
277 j 0 «=0
(Legendre.)
41. li f{x)=x'- (^^0) and /(^) =—572 (^^<;0), shew that, if/(^) can be expanded
into a uniformly convergent series of Legendre polynomials in the range ( — 1, 1), the
expansion is
^/ N -i rw X % , .,. 1.3...(2r-3) 4r + 3 „ . ,
f^.y^lP^x)- ^2^ (-)' 4.6.8...2.W4^^-^^(-^)-
(Trinity, 1893.)
42. If ^- = 1 /i»(7/(3),
(1-2A2 + A2)'' n=0
shew that
Cn {ooxx — {x^ — 1)^ (^r — 1 )- cos 0}
_r(2i/-i)A=n ■^4^r(n-x + i){r(v+X)}2
{r(v)P ;,io r(«+2v+x)
X (.^^ - 1)^' (.^.^ - 1)*' ^r: (.r) C^:!: (^0 Cf ^"-^^ (COS c/.).
(Gegenbauer.)
43. If fTn\z)=\ {fi-Ztz^-l)-^e'dt,
where ej is the least root of ^^ — 3i;2 4- 1 = 0, shew that
(2?H-l)(r„ + i-3(2?i-l)2o-,,_i + 2(»-l)o-„_2=0,
and
4 (4^3 - 1) o-,/" + 144sV,/' - z (12rt2 - 24?i - 291) o-,,' - (n - 3) {2n - 7) (2« + 5) o-„ = 0,
where o-«"'= -J'z^ i etc. (Pincherle.)
44. If (A3 -3^3 + 1)-^= 2 Rn{z)li'\
shew that 2 (« + !) /i'„ + i -3(2h + ]) s^„ + (2?i- 1) /4_2 = 0,
nR„ + R\,_2-zB,: = 0,
and
4(423-l)i2,/" + 96^2/?,/'-2(l2tt2 + 24m-91)/?;-?i(2ji + 3)(2« + 9)/i:„ = 0,
where R,'" =—,-.-, etc. (Pineherlo.)
330 THE TRANSCENDENTAL FUNCTIONS [CHAP. XV
obtain the recurrence formula
(7i + l){2n-l)A,ix)-{{4n^-l).v-\-l}Ar,.i{x-) + {n-l){2n + l)A„_^{x)=0.
(Schendel.)
46. If n is not negative and m is a positive integei', shew that the equation
(^2_l)g + (2n + 2).^^g=m(m + 2«+l)y
has the two solutions
K„,{x) = (.v^-l)- ^ (.^^2_ ])„. + „^ x^^ (a;) = (a^2-l)-» ^\^^^^ K,,,{t) dt,
when X is not a real number such that — 1 ^.^' ^ 1.
47. Prove that
, X In + m l /-/"■'""* /r2_l\»
^ ^ T ^ i V ^ „=,„, (?i + to)! ?i d(;" + '» V 2 /
(Clare, 1901.)
48. If F,,M= i ^^r^-^-,
shew that i^„, „ (.*--) = |^. (e«« + ^«')|^_^^ = e^P„ (.i-, a),
where P^ {.i; a) is a polynomial of degree n in a; ; and deduce that
)•
(Trinity, 1905.)
d
Pn + 1 {X, a) = {.V + a) P,, (x, a) + x -^ P,, (x, a).
49. If Fn (x) be the coefficient of z"- in the expansion of
2hz
(>XZ
phz p — hz
in ascending powers of z, so that
Zx- - h^
F(, {x) = 1, F^ (x) = X, F. (x) = — - — , etc.,
shew that
(1) F„ {.v) is a homogeneous polynomial of degree a in x and k,
^2^ ^-^=/^„_,(..) coi),
(3) f" FJx)dx = 0 (n^l),
(4) If _y = a,)/^Q {x)-\-a-^^F^ (.r) +a2-^2 (■^) + --m where «(,, aj, a2i ••• ^^<^ ''^al constants
then the mean value of — ^ in the interval from x= - h to x= +/i is a,.. (Leautd.)
50. If F„{x) be defined as in the preceding example, shew that, when —h<x<h,
,;.COS-^+..
^ 2.. + 1 GO = ( - )"' 2 ^^,,^ ^ J ( sm -^- - ^^^ sni - ^^ + g,„^ ^-, sm -^ + • • ■
(Appell.)
CHAPTER XVI
THE CONFLUENT HYPERGEOMETRIC FUNCTION
16'1. The confluence of two singularities of Riemann's equation.
We have seen (§ 10"8) that the linear differential equation with two
regular singularities only can be integrated in terms of elementary functions ;
while the solution of the linear differential equation with three regular
singularities is substantially the topic of Chapter xiv. As the next type
in order of complexity, we shall consider a modified form of the differential
equation which is obtained from Riemann's equation by the confluence of
two of the singularities. This confluence gives an equation with an irregular
singularity (corresponding to the confluent singularities of Riemann's equation)
and a regular singularity corresponding to the third singularity of Riemann's
equation.
The confluent equation is obtained by making c -* oo in the equation
defined by the scheme
[0 00 c
1 7
p I ;^ + 7?l — C C — K
\t. — m 0 k
The equation in question is readily found to be
(}?ii du (k \ — m-\ ^ ,..
We modify this equation by writing u = e~ -~Wk,m{2) and obtain as the
equation* for Wjc^y^iz)
*Z+|_J + ^%t-^'l,r = o (B).
dz- \ ^ z z- )
The reader will verify that the singularities of this equation arc at
0 and X, the former being regular and the latter irregular; and when "2m
* This equation was given by Whittaker, Bulletin American M(Uh. Soc. x.
332 THE TRANSCENDENTAL FUNCTIONS [CHAP. XVI
is not an integer, two integrals of equation (B) which are regular near 0 and
valid for all finite values of z are given by the series
M^^,,\z}-z e |^ + i,(2m+l)^ 2\{%n + l){2m + 2) '^■"]'
^r /x 4-j» -i? (-. ^ — m — k (l — m — k)(% — m — k) „
*,_..(.) = .4 "e " |l + 1^ (132^) ^ + 2 ! (1 - 2iV(2 - 2m) ' +
These series obviously form a fundamental system of solutions.
[Note. Series of the type in { } were first considered by Kummer* and more recently
by Jacobsthalt and Barnes J. In the notation of Kummer, modified by Barnes, they would
be written ^Fi {\±m-k; ±2m + l ; z); the reason for discussing solutions of equation (B)
(j 7/ (ill
rather than those of the equation z-j^~{z- p) ~'--ai/ = 0, of which iFi{a;p;z) is a
solution, is the greater appearance of symmetry in the formulae, together with a simplicity
in the equations giving various functions of Applied Mathematics (see § 16-2) in terms of
solutions of equation (B).]
16'11. Kummer s formulae.
(I) We shall now shew that, if 2«i is not a negative integer, then
0 - 4 - '«7lf,,,, {z) = (- ^) - i - "M-it,,. (- z),
that is to say,
^ + m — k {\-\- m — k){^-{-m — k) „
^ 1!(27H + 1)^''" 2!(2m + l)(2m + 2) ^"^■"
_ l + m^ {\ + mfk){^rn^k) „_
~ 1 ! (2w +1) ^ "^ 2 ! (2m + 1) (2m + 2) ^' * ' "
For, replacing e"^ by its expansion in powers of z, the coefficient of z"^ in
the product of absolutely convergent series on the left is
11 \ V I n\ V{iin-{-\^-k)V (2m + 1 + n)
by § 14'11, and this is the coefficient of ^" on the right§; we have thus
obtained the required result.
This will be called Kummer s first formula.
(II) The equation
°'"'^ ^"'" i'^,ri2^^.^!(m + l)(m + 2)...(m + j9)['
valid when 2m is not a negative integer, will be called Kummer s second
formula.
* Crelle, xv. p. 139. t 3Iath. Ann. lvi. pp. 129-154.
% Trans. Cmnh. Phil. Sor. xx. pp. 253-279.
§ The result is still true when m + h + k is a negative integer, by a slight modification of the
analysis of § 14-11.
16*11, 1612] THE CONFLUENT HYPERGEOMETRIC FUNCTION 333
To prove it we observe that the coefficient of 2"+"' + 7 j^ the product
of which the second and third factors possess absolutely convergent expansions, is (§ 3'73)
w! (2»i + l)(2m + 2)...(2w+n) ^ ' ' 2 > ./
(it + m)(^ + m) ...(n-m + h) i,/ 1 1 ,1 i\
= —,-,^ \\ ,r.- ?i^ — /., — \ F( - hi, -m-hi; -n + -k-m; 1),
n\ {2m + l){2m + 2)...{2m + n) ^2) 2 » 2 > /.
by Rummer's relation*
F{2a, 2^; a-\-$ + ^', x) = F{a, fi; a+fi+^; 4x (l-.v)},
valid when 0^.»^^; and so the coefficient of /^ + "* + ^ (by § 14-11) is
{^ + m){^+m) ...{n-m+^) _r {-n + ^-r-m) r^|)
»! (2m + l)(2>n + 2) ... (2m + n) r(^-m-|n) r('|-^»)
^ r(i-m)r(i)
/^ ! (2?tt + 1 ) (2m + 2) . . . (2»i + /i) r (^ - m - hi) T{h- \n) '
and when n is odd this vanishes; for even values of n { = 2p) it is
r(i-m)(-i)(-f)...(|-p)
- 2p\ 2:^P{m-irh){in+f) ...{m+p-h) (m + 1) (m+2) ... (wi+^) r (i-m-^)
1.3...(2/)-l) 1
2j9! 23p(//i + l)(«i + 2) ... {m+p) 2^P.^!(//i + l)(m + 2)...(m+jo)'
16*12. Definition^ of the function Wk,m{z)-
The solutions ii;fc,±„i,(^) of equation (B) of § 16'1 are not, however, the
most convenient to take as the standard solutions, on account of the
disappearance of one of them when ^.m is an integer.
The integral obtained by confluence from that of § 14"6, when multiplied
by a constant multiple of e'-^ , is;|;
It is supposed that arg z has its principal value and that the contour is so
chosen that the point t= — z is outside it. The integrand is rendered one-
valued by taking j arg (— ^) [ $ tt and taking that value of arg (1 -f- tjz) which
tends to zero as ^ -* 0 by a path lying inside the contour.
Under these circumstances it follows from § 5'32 that the integral is an
analytic function of z. To shew that it satisfies equation (B), write
CO
* See Chapter xiv, examples 12 and 13, p. 292.
t The function W,^^„^{z) was defined by means of an integral in this manner by Wliittaker,
loc. cit., p. 125.
X A suitable contour has been chosen and the variable f of § ll-G replaced by - t.
334 THE TRANSCENDENTAL FUNCTIONS [CHAP. XVI
and we have without difficulty*
^ /2^'_ \c?iJ l-m'' + k{k-l)
dz^ \ z ) dz z'^
= 0,
since the expression in | } tends to zero as ^ -»► + oo ; and this is the condition
that e~^^z^v should satisfy (B).
Accordingly the function Wk^mi^) defined by the integral
-_L.r(i + l-,„).-»^//"'-^'(-0-*-4+"'(i + J)'"*^'".-'rf«
is a solution of the differential equation (B).
The formula for Wk,m{z) becomes nugatory when k — ^ — m is a negative
integer. To overcome this difficulty, we observe that luhenever
R(k-l-m]<0
2
and k— V)— m is not an integer, we may transform the contour integral into
an infinite integral, after the manner of § 12'22 ; and so, when
R [k — ^ — m] ^0,
This formula suffices to define Tfjfc,,„(^) in the critical cases when
m-\--^—k is a positive integer, and so Ti^A;,w('2') is defined for all values of
k and m and all values of z except negative real values f.
Example. Solve the equation
<F-u , ( , b , c^
in tet-ms of functions of the type TF^.^ ^,^ (z), where a, b, c are any constants.
16'2. Expressio7i of various functions hy functions of the type W]c,m{z)-
It has been shewn;]: that various functions employed in Applied Mathe-
matics are expressible by means of the function TF;t,;» {z)\ the following are a
few examples :
* The differentiations under the sign of integration are legitimate by § 4-44 corollary.
t When z is real and negative, Wj,,^{z) may be defined to be either JFj.,„j(2 + 0(:) or
TFj. „j {z - Oi), whichever is more convenient.
X Whittaker, Bulletin American Math. Soc. x. ; this paper contains a more complete account
than is given here.
16*2] THE CONFLUENT HYPERGEOMETRIC FUNCTION 335
(I) The Error function* which occurs in connexion with the theories of
Probability, Errors of Observation, Refraction and Conduction of Heat is
defined by the equation
Erfc (a?) = r e-''dt,
where x is real.
Writing <=a;-(w'^ — 1) and then w = s/a; in the integral for W_i i(x^),
we get
and so the error function is given by the formula
Erfc (cc) = ^ a;-*e-*-^'F_^, ^ (x').
Other integrals which occur in connexion with the theory of conduction
of heat, e.g. I e~^'~^''*''dt, can be expressed in terms of error functions, and
so in terms of Wk,m functions.
Example. Shew that the formula for the error function is true for complex values of x.
(II) The Incomplete Gamma function, studied by Legendre and others +
is defined by the equation
r^{n, x)= jP'-'e-'dt.
By writing t = s — x in the integral for 14'^/,j_ix i,j(«), the reader will
verity that
7(n,^-) = rOO-^*(«-i).--^Tf,(„_i),,,(^.).
(III) The Logarithmic-integral function, which has been discussed by
Euler and others j, is defined, when | arg {— log 2]\ < tt, by the equation
,. , , [' dt
* This name is also applied to the function
Erf (x) = I "" <--«" dt = Tri - Erfc (x).
t Legendre, Exercicen, i. p. 289; Hocevar, Zcitxc.hrift fijr Math. xxi. p. 449; Schluniilcli,
Zeittichrift filr Matli. xvi. yy. 261 ; Prym, Crelle, lxxxii. p. 1G5.
X Euler, Iimt. Cede. Int. i. ; Soldner, Moiiatliclie Corre.'ipojidenz, von Zach (ISll), p. 182;
Briefwechsel zwischen Gauss und Bessel (1880), pp. 114-12(J; Bessel, Krinuinberfjt'r Archir, 1812;
Laguerre, Bulletin de la Soc. .Math, de France, vii. (1879); Stieltjes, Ann. de I'Kc. Svrin. Sup.
ser. 3, t. III. The logarithmic-integral function is of considerable importance in the highe)
parts of the Theory of Prime Numbers. See Landau, Prnnzahlen, p. 11.
336 THE TRANSCENDENTAL FUNCTIONS [CHAP. XVI
On writing s — log z = u and then u = — log t in the integral for
^-J,o(-log^),
it may be verified that
li (2) = -{- log z)'hiW_.^Q(- log z).
It will appear later that Weber's Parabolic Cylinder functions (§ 16'5) and
Bessel's Circular Cylinder functions (Chapter xvii) are particular cases of the
Wjc,m function. Other functions of like nature are given in the Miscellaneous
Examples at the end of this chapter.
[Note. The error function has been tabulated by Encke, Berliner ast. Jahrhuch, 1834,
and Burgess, Trans. Roy. Soc. Edin. xxxix. The logarithmic-integral function has been
tabulated by Bessel and by Soldner. Jahnke und Emde, Funktiontafeln (Leipzig, 1909),
and Glaisher, Factor Tables., should also be consulted.]
16'3. The asymptotic expansion of Wk, m (2), when \z\ is large.
From the contour integral by which Tf i, ,„ (z) was defined, it is possible
to obtain an asymptotic expansion for Wk^ ^ (z) valid when | arg z\< ir.
For this purpose, we employ the result given in Chap. V, example 6, that
V'^'z)^^'^ \z + ••• + n\ J- + ^'^^^' ^)'
where
Rn{t, Z) = ^ ^--^ ^^ (1 + -) j^ U- (1 + u)-'-^du.
Substituting this in the formula of §16*12, and integrating term-by-term,
it follows from the result of § 12-22 that
r,„.(.)= .-*'/ ji + ^!i_(^y^f.»'-(^-CTN^--(^-»)i ^ ... .
{m' -{k- If] \m' - (k - ^y] . . . {m^ -(k- n + m
+
n ! z'-
k-h + m
provided that n be taken so large that R yn — k — .,-(- /h] > 0.
Now if I arg z\^'7r — a and U j > 1, then
1^1(14-^/^)1^1+^ R(z)^0
1(1 + ^/5)1^ sin a R{z)^0
and so*
X(X-l)...(X-n):,, , ,,„,, ,„, /-K^/--)!
Rn (t, Z) i ^
(1 -1-O'^'(coseca)l^l iO'{l-¥u)\^\du.
J {)
* It is supposed that \ is real ; tlie inequality has to be slightly moJified for coniiDlex values
of X.
16-3-16'4] THE CONFLUENT HYPERGEOMETRIC FUNCTION 337
Therefore
< I H^-'^)^-^-(^-n) I ^j _^ ^)1A| (cosec a)l^l | (t/z) \^+' (1 + 0'^' (^ + 1)"S
since I +u<l + t.
Therefore, when 1 2^ | > 1,
r(-A; + ^ + w;Jo
= 0 (z—^),
since the integral converges. The constant implied in the symbol 0 is
independent of arg z, but depends on a, and tends to infinity as a -* 0.
That is to say, the asymptotic expansion of W]c^rn{z) is given by the formula
( M=i nl 2 )
for large values of \z\ when \ arg 2 [ ^ tt — a < tt.
16*31. The second solution of the equation for Wj^^^ni^)-
The differential equation (B) of §16'1 satisfied by W]c,m{z) is unaltered if
the signs of z and k are changed throughout.
Hence, if | arg {— z)\< ir, PF_a;,„i (— z) is a solution of the equation.
Since, when | arg z\< ir,
Wj,,,„(z) = e-^'z^{l + 0(z-'^)],
whereas, when | arg ( - 2^) j < tt,
W_,,,,(-z) = ei^-zy^{l + 0{z-%
the ratio Wic^,,^{z)/W-k,m(— 2) cannot be a constant, and so Wi;^),^(z) and
^V^-k,m(—z) form a fundamental system of solutions of the ditferential
equation.
16"4. Contour integrals of Barnes type for Wk, ,,> (^)-
Consider now
. e^}^ /•-■ r{s)ri-s-k-m+^)r{-s-k + m + ^J
where | arg z\< \^tt, and neither of the numbers k ± m + ., is a positive integer
338 THE TRANSCENDENTAL FUNCTIONS [CHAP. XVI
or zero*; the contour has loops if necessary so that the poles of r(s) and
those of r( — s — k — m + 2) r[ — s — A^ + m + g) are on opposite sides of it.
It is easily verified, by § IS 6, that, as s^ qo on the contour,
r (s) r (- s - k-m + l^ r (- s - k + m + 1) = 0 (e-^''^'^ \s\-^^-^),
and so the integral represents a function of z which is analytic at all pointsf
3 3
in the domain | arg 2;\%2'^~^'^2''^-
Now choose N so that the poles of T (— s — k — m + ^] T (- s — k + m + ;^]
are on the right of the line R (s) =^ — N — 2', and consider the integral taken
round the rectangle whose corners are ± ^i, - i\^ — 2 ± ^h where ^ is positive:!:
and large.
3
The reader will verify that, when ] arg^^ | ^2'^ ~ ^' ^^® integrals
N-i-^i r-N-i + ^i
tend to zero as | ^ go ; and so, by Cauchy's theorem,
g - i^ ^'^ /• 00 i r{s)r(-s-k-m +^)r(-s-k + m + ^)
^TTi J -ooi ' r{-k-m + i)r{-k + m + ^)
z^ds
^ Ar .. ^ + cc i r {s)V{-s-k-m + i,)V{-s-k + m + i) )
+ 27^^■j_;,_^_^, r(-A;-m+4)T(-^ + m + ^) '' '}'
where i^„ is the residue of the integrand at 5 = — «.
Write s = — iV — ^ + lY, and the modulus of the last integrand is
\z\-''-^0[e--\'\\t\'''''\
where the constant implied in the symbol 0 is independent of z.
Since I g-aiti \t\^'~-^dt converges, we find that
in=0 ' }
* In these cases the series of § 16-3 terminates and Wic,„^[z) is a combination of elementary
functions.
t The integral is rendered one-valued when R(z) <(i by si)ecif}ing arg z.
% The line joining ±t' niay have loops to avoid poles of the integrand as ex])lained above.
16'4] THE CONFLUENT HYPERGEOMETRIC FUNCTION 339
But, on calculating the residue Rn, we get
_r(n-k-m+i!)T(7i-k + m + ^)
^ {m" - (k - I)'} {m' - (k - f )"} . . . {m=' - (A; - 71 + hf]
nlz"
and so 7 has the same asymptotic expansion as Wk^^ni'^)-
Further / satisfies the differential equation for Wk^mi^)', for> on
substituting I T (s) V (— s — k — m + 2) ^ [ — (>' — k + m + A z^ds for v in
the expression (given in § 16*12)
we . get
p' T{s)V(-s-k-m + ^^ vl-s-k + m^^^^ds
-T' T{s-vl)v{-s- k-m + 1] r [-6-- ^ + m + g) 0*+ic;s
= (["'.- ['^"1 r (s) r (- s - ^^ - m + I) r f- s - A; + m + ^j 2*d6'.
Since there are no poles of the last integrand between the contours, and
since the integrand tends to zero as { 5 | ^ 00 , s being between the contours,
the expression under consideration vanishes, by Cauchy's theorem ; and so
I satisfies the equation for Wk,mi^)-
Therefore I = AWk,,n (^) + BW^k,^,^ (- 2),
where A and B are constants. Making | ^ j ^ x when R(2)>0 we see, from
the asymptotic expansions obtained for / and W±k,m{+ ^)> that
^ = 1, B = 0.
Accordingly, by the theory of analytic continuation, the equality
i=WkM^)
persists for all values of z such that j arg 2r j < tt ; and, for values* of arg ^
such that TT ^ j arg z\ < ,^ tt, Tf^■,„( (z) may be defined to be the expression /.
Example 1. Shew that
IV fA-"!^ r"' r(^-/(-)r(-.s-;;^ + i)r(-^+y».+A)., ,
"'""^ '~~27ri J _-,,i r(-y(--/« + i)r(-/(- + m + |)
taken along a suitable contour.
* It would have been possible, by modifying the path of integration in § 16-3, to have shewn
that that integral could be made to define an analytic function when | arg z | < jtt. But the
reader will see that it is unnecessary to do so, as Barnes' integral affords a simpler definition
of the function.
o o •)
340 THE TRANSCENDENTAL FUNCTIONS [CHAP. XVI
Example 2. Obtain Barnes' integral for IFj., „j (2) by writing
_ i^'
2iTi J -^i T{-k-m + l)
for (1 +t/z) '^'^'"^ in the integral of § 16'12 and changing the order of integration.
1 j'-' ris)T{-s-k-vi + ^)^^^_^^^
16"41. Relations between Wjc, m (2) f^"c^ ^h, ±m (^)-
If we take the expression
F(s) = V(s)v(-s-k-m + f\ r (-s-k + m + l)
which occurs in Barnes' integral for W^., ,„ (z), and write it in the form
ir'-Tis)
r {s + k + m + ^)T (s + k — m + ^) cos {s + k + m) tt cos (s + A; — m) tt '
we see, by § 13"6, that, when R {s) ^ 0, we have, as j s j ^ 00 ,
F{.) = 0
exp ■] ( — s - ^ — 2^• ) log s + s
sec (.§ + h + m) tt sec {s + k — m) tt.
Hence, if |arg^'j<27r, jF{s)z^ds, taken round a semicircle on the
right of the imaginary axis, tends to zero as the radius of the semicircle
tends to infinity, provided the lower bound of the distance of the semi-
circle from the poles of the integrand is positive (not zero).
Therefore Wh ,„ (z)= - .^ y-j , x U 7 7^ ,s.
where SJ?' denotes the sum of the residues of F(s) at its poles on the
right of the contour (cf. § 14'5) which occurs in equation (C) of § IG'4.
Evaluating these residues we find without difficulty that, when
iarg^|<.^7r,
and 2m is not an integer*,
-iir / N r(— 2//^) -, , ^ T (2l}l)
w,. „ (.) = J, ^^'- ^^ _'^.^ ih, „ (.) + p ^^ y^^^ >_ ^^^^ M,, _„. (4
Example 1. Shew that, when j arg { — z) \ < Stt and 2»i i.s not an integer,
W , (-r\~ r(-2«0 r(2m)
(Earnest.)
Example 2. When - .W < arg 2 < f tt and - 'in < arg ( - i) < Itt, .shew that
* When 2m is an integer some of the poles are generally double poles, and their residues
involve logarithms of z. The result has not been proved when h - i±m is a positive integer or
zero, but may be obtained for such values of h and m by comparing the terminating series for
t Barnes' results are given in the notation explained in § IG'l.
16'41-16"51] THE CONFLUENT HYPERGEOMETRIC FUNCTION 341
Example 3. Obtain Kummer's first formula (§ 16-11) from the result
1 /"*■
z^e~* — - — . I T(n~8) z'ds. (Barnes.)
16'5. The parabolic cylinder functions. Weber's equation.
it IS
Consider the differential equation satisfied by w = z * W^ _ , (
this reduces to -j-^ + \2k — jz'^l w = 0.
Therefore the function
satisfies the differential equation
Accordingly Dn{z) is one of the functions associated with the parabolic
cylinder in harmonic analysis*; the equation satisfied by it will be called
Weber's equation.
From § 16'41, it follows that
when I arg z\<-^'ir.
and these are one-valued analytic functions of z throughout the ^•-plane.
Accordingly Dn {z) is a one-valued function of z throughout the 2^-plane ; and,
3
by § 16'4, its asymptotic expansion when | arg^^ j < j tt is
^ - 12- " (i 'U'i - 1 ) , n {n - 1) {n - 2) (n - 8)
( ^z- 2Az*
16"51. The second solution of Weber's equation.
Since Weber's equation is unaltered if we simultaneously replace n
and z by — ?i — 1 and ± iz respectively, it follows that D_u~iii^) -^^id
D^n-i{~iz) are solutions of Weber's equation, as is also D,^{—z).
* Weber, Math. Ann. i.; Wbittaker, Proc. London Math. Soc. xxxv.
342 THE TRANSCENDENTAL FUNCTIONS [CHAP. XVI
It is obvious from the asymptotic expansions of Dn(z) and Z)_,i_i (^e^'^*),
valid in the range — ^ vr < arg z< tt, that the ratio of these two solutions is
not a constant.
16'511. The relation between the functions Dn{z), D_n-i{± iz).
From the theory of linear differential equations, a relation of the form
Dn {z) = aD_n-i (iz) + bD_n-i (- iz)
must hold when the ratio of the functions on the right is not a constant.
To obtain this relation, we observe that if the functions involved be
expanded in ascending powers of z, the expansions are
jr(i)2;:^^_r(-i)22l;^. )
+ 'l T{l+in) r{^ + ^n) ''+■■']■
Comparing the first two terms we get
a = (27r)-* r 0^+1) e•^"'^^ b = (27r)-* r(n+ 1) e"^'
and so
Anwi
i).(.)=^^^^^^
e^'"^'i)_„_l(^■^) + e-■^"'^'i)_,_l(-^■^)
V(27r) L
16'52. The general asymptotic expansion of Dn(z).
So far the asymptotic expansion of i)„ (z) for large values of z has only
been given (§ 16"5) in the sector |arg2:| < /Tt. To obtain its form for values
of arg z not comprised in this range we write — iz for z and —n—1 for n in
the formula of the preceding section, and get
A. (.) = ."'^^' A.(- z) + f^^^ ^ ("+^) '^^ i)_._, (- iz).
5 1
Now, if I TT > arg z > -^'jt, we can assign to - ^ and — iz arguments between
3 , 1
± ^ TT ; and arg (— z) = arg z —ir, arg (— iz) = arg ^ — g tt ; and then, applying
the asymptotic expansion of § 16"5 to Dn(— z) and l)_n-i{—iz), we see that,
...5 1
if I TT > arg ^ > ^ TT,
r, i.\ o-i~-^''U n{n-l) nin-l)in-1){n-Z) \
V(27r) nTTi } z-^ -«-![-, (n + 1) (n + 2)
1 i-n) [ 2z"
(n + l){n + 2)(n + S){n-h4<)
16-51 1-16-6] THE CONFLUENT HYPERGEOMETRIC FUNCTION 343
This formula is not inconsistent with that of § 16'5 since in their common range of
validity, viz. Jrr <arg3 < |»r, e^ 2-2h-i is o(z~"^) for all i)Ositivc values of m.
1 3
To obtain a formula valid in the range — ^ tt > arg z > ^ir, we use the
formula
Dn{z) = e ^"^~^^ + r(-wV D-n-ii^Z),
and we get an asymptotic expansion which differs from that which has just
been obtained only in containing e ~ "'^^ in place of e"'^*.
Since Dn (z) is one- valued and one or other of the expansions obtained
is valid for all values of arg z in the range —tt^ arg z ^tt, the complete
asymptotic expansion of D„ (z) has been obtained.
16*6. A contour integral for Dn (z).
Consider I e'^^~^^' {-t)-'^-'^dt, where |arg(-0l<7r; it represents a one-valued
analytic function of z throughout the s-plane (§ 5-32) and further
the. differentiations under the sign of integration being easily justified; accordingly the
integral satisfies the differential equation satisfied by e"^^ Dn {z) ; and therefore
e-^'- r^\-~^--^^^\-t)'^-'dt = aDn{.z) + hD_r,.x{iz),
where a and h are constants.
Now, if the expression on the right be called E^ (2), we have
^,^(0)= e--~U-t)-^'-'dt, En'{0)= e-i^ {-t)-dt.
J cc ./go
To evaluate these integrals, which are analytic functions of /i, we suppose first that
^(«)<0; then, deforming the paths of integration, we get
En(0)= -2inin{n + l)7j- I e"^^' t-''-'^dt
/GO
= 2'"4"isin(?i7r)r(-|H).
Similarly En' (0) = - 2^ ~ i" i sin (mr) r (J - hi).
Both sides of these equations being analytic functions of n, the equations are true for
all values of n ; and therefore
b^O, a = ^^ ^ - " -/'^ 2 ^ ^" i sin («7r) r ( - hi)
r(,i.)2-"
= 2ir ( - ii) sin /in.
Therefore D,.{z)^ -"^^ e'^^ j'^^'e' ^^~^^ (-t)-'^'^ dt.
344 THE TRANSCENDENTAL FUNCTIONS [CHAP. XVI
16*61. Recurrence formulae for D-n{z).
From the equation
= 1 ^. — z{—t) +{-t) +(^i+l)(— 0 \^ ^ at,
after using § 16"6, we see that
Further, by differentiating the integral of § 16'6, it follows that
Exam'ple. Obtain these results from the ascending power series of 5^ 16 "5.
16"7. Properties of i)„ (z) when n is an integer.
When n is an integer, we may write the integral of § 16'6 in the form
at.
(- 0"+i
If now we write t = v — z, we get
(-r^'^.(e-n
a result due to Hermite*.
Also, if m and ?? be unequal integers, we see from the differential
equations that
fJn (z) Dm" {z) - D>„ {z) Dn" (z) + {m - n) A« (z) A. (z) = 0,
and so
{m - n) Dy, {z) Bn (z) dz =
J —00
D,{z)DJ{z)-D,„{z)D,:{z)
= 0,
by the expansion of § 16'5 in descending j)owers of z (which terminates
and is valid for all values of arg^^ when n is a positive integer).
Therefore if ni and n are unequal jjositive integers
r D,„{z)Dn{z)dz = 0.
J —00
* Comptes liendus, lviii. pp. 2G6-273.
16"61, 16'7] THE CONFLUENT HYPERGEOMETRIC FUNCTION 345
On the other hand, when m = n, we have
(n + l)r [l)n{z)Ydz
J —CD
= j Dn(z)\- Dn+,' (z) + I zDn+i (z)\ dz
= r {Dn^,{z)Ydz,
J —CO '
on using the recurrence formula, integrating by parts and then using the
recurrence formula again.
It follows by induction that
r [Bn{z)Ydz = n\r {D,{z)Ydz
J - X J —cc
= nl I e"^^ dz
— cc
= (27r)*w!,
by § 12-14 corollary 1 and § 12-2.
It follows at once that if, for a function /(^r), an expansion of the form
f{z) = cioDo (z) + a,D, {z) + ... + anDniz)+ ...
exists, and if it is legitimate to integrate term-by-term between the limits
— 00 and oc , then
««=-7^-,r Dn{t)f{t)dt
REFERENCES.
W. Jacobsthal, Math. Ann. Bd. lvi. \\\). 129-154.
E. W. Barnes, Trans. Camh. Phil. Soc. Vol. xx. pp. 253-279.
E. T. Whittaker, Bulletin American Math. Soc. Vol. x. pp. 125-134.
H. Weber, Math. Ann. Bd. i. pp. 1-36.
A. Adamoff, Ann.de rinstitut P oly technique cle St Petershourg, t. V. (1906), p^). 127-143.
E. T. Whittaker, Proc. London Math. Soc. Vol. xxxv. pp. 417-427.
G. N. Watson, Proc London Math. Soc. Ser. 2, Vol. viir. pp. 393-421.
H. E. J. CuRZON, Proc. London Math. Soc. Ser. 2, Vol. xii. pp. 236-2.")9.
A. Milne, Proc. Edinburgh Math. Soc. Vol. xxxii. pp. 2-14, Vol. xxxiii. pp. 48-64.
346 the transcendental functions [chap. xvi
Miscellaneous Examples.
1. Shew that
provided that the constants are chosen so that the integral converges.
2. Shew that
-^h, m (2) =s* "^ '" e ~ *^ lim F{^ + m - /(•, ^ + m - /(;+p ; 2m + 1 ; s/p).
3. Obtain the recurrence formulae
4. Prove that W,,^„,{2) is the integral of an elementary function when either of
the numbers k-^±77i is a negative integer.
5. Shew that by a suitable change of variables, the equation
, , . d-y , T . di/ , , ,
{a.2 + h^x) ^ + («! + 6, x) ^^ + (ffo + Kx) y=.0
can be brought to tiie form
%?+(«-« i-<.,=o,
and that this equation is derived from the eqviation for F {a,b; c; x) hy writing x—^jh and
making 6 -* x .
6. Shew that the cosine integral of Schlomilch and liesso {Giornale di Matematicke^
VI.), defined by the equation
Ci(.-)= /""~c?<
when I arg z \ < ^tt, may be written in the form
Ci(.)=iz--iei'^+i'^Mr_^^o(-i^)+|s-i«-i'~-^'^Mr_i_o (/.').
7. Express the functions
Si {z)=[' ^~ dt, Ei (z) =r '''-- dt
J 0 i J rj (
in terms of (Tj. „j functions.
8*. Shew that Sonine's polynomicil {Math. Ann. xvi. p. 41)
f » ^n - 1 «« - 2
T "■(z)= - I -
'"' ^ ^ H\im + n)lO] {a-l)l{m + n-l)\ll^{n-2)l{m + n-2)l2l '"'
where n is an integei', may be expressed in tlie form
* The results of examples 8, 9, 10 were communicated to us by Mr Bateman,
THE CONFLUENT HYPERGEOMETRIC FUNCTION 347
9. Shew that Abel's function <f>m (z) defined (Oeuvres, 1881, p. 284) as the coefficient of
A"» in the expansion of (1 -A)"ie~**Ai-ft) is expressed by the equation
10. Shew that the Pearson-Cunningham function {Proc. Royal Soc. Lxxxi. p. 310)
e-^(-^)"-^"^ L _ (n+im)(n-im)
{n + ^m) {n+hn — \) {n - ^m) {n-\m — 1 ) _ \
+ - "" 2 \7' ■ " j
may be expressed by the equation
11. Shew that, if | arg z \ < Jtt, and | arg (1 +0 | ^ tt,
(Whittaker.)
12. Shew that, if n be not a positive integer and if | arg z \ < Itt, then
and that this result holds for all values of arg z if the integral be / , the contours
enclosing the poles of T {-t) but not those of T {^t- ^n).
13. Shew that, if | arg a \ < ^tt,
['"+'.(i-'^)-'.-A.(^)^^-
J 00
?r a\n - m iri (m - h)
= ~-~ ;, ^^.F{-hi,hn + h;hn-hi + l; 1 -*«-').
r(-m)r(*m-i/i + l)a5('"+^)
14. Deduce from the preceding example that
f " e - 3^' z" A. + 1 (^) dz = (v/2) - 1 - '» r (m + 1) sin (^ - ^m) rr
if the integral converges. (Watson.)
15. Shew that, if ?i be a positive integer, and if
^n{^v) = j^^J-^'Uz-.i-r' D„{z)dz,
then En (x) = ± je^""' V(27r) r(n + l)c~ i'^'" /)_ „ _ i ( + iv),
the upper or lower signs being taken according as tlic imaginary part of .u is positive
or negative. (Watson.)
348 THE TRANSCENDENTAL FUNCTIONS [CHAP, XVI
16. Shew that, if w be a positive integer,
where fj, is ^n or ^{n- 1), whichever is an integer, and the cosine or sine is taken as n is
even or odd. (Adamoflf.)
17. Shew that, if n be a positive integer,
J)n (^) = ( - r (4^) ' * Wnf + 1 e^^' (J, + J,- J,),
where ^i = [ " e " " ('^ - 1)' '^^'^ (a^v^n) dv,
J -00 sin \ "* ' '
Jo= I (T (ij) . (xvJn) dv,
Jo sin ^ ^ ^ '
fO _„(j;_l)2C0S ,
^^^ I -^^ sin (•^■^>^**)^^''
and cr (i') = e*" (^ " '^'^ ." - e " " (^ ' l)'. (Adamoff.)
18. With the notation of the preceding examples, shew that, when .r is real,
1 _ 1 _ 4..T2 cos ,
sm
while ./Jj satisfies both the inequalities
Shew also that as v increases from 0 to 1, a{v) decreases from 0 to a minimum at
v=l—/ti and then increases to 0 at ?? = 1 ; and as v increases from 1 to co , o- (v) increases to
a maximum at 1 + /i2 and then decreases, its limit being zei'o ; where
2 \/(l) <^^< \/(Jn) ' I x/(l) < ^^^ < \/(2T.) '
and I o-(l-/ii) I < ^H~2^ o-(l+/<2) < An~^, where ^4 =-0742,... (Adamoff.)
19. By employing the second mean value theorem when necessary, shew that
D„ {.V) = ( - r ,/2 . (V»)" e - ^" P^'-' {.V ^In) + ^1 ,
^ ' |_sin V'^ J
where co„ (,i') satisfies both the inequalities
, , , 3'35... i-r- , /^s I 1-1
(Adamoff.)
20. Shew that, if n be positive but otherwise unrestricted, and if m be a positive
integer (or zero), then the equation in z
has m positive roots when 2m — 1 < n < 2m + 1. (Milne.)
CHAPTER XVII
BESSEL FUNCTIONS
17'1. The Bessel coefficients.
In this chapter we shall consider a class of functions known as Bessel
functions which have many analogies with the Legendre functions of
Chapter XV. Just as the Legendre functions proved to be particular
forms of the hypergeometric function with three regular singularities, so
the Bessel functions are particular forms of the confluent hypergeometric
function with one regular and one irregular singularity. As in the case of
the Legendre functions, we first introduce* a certain set of the Bessel functions
as coefficients in an expansion.
For all values of z and ^ (^ =0 excepted), the function
e
can be expanded by Laurent's theorem in a series of positive and negative
powers of t. If the coefficient of P\ where n is any integer positive or
negative, be denoted by /„ (z), it follows, from § 5*6, that
1 r(o+) 1 ^>z [ a —
Ztti J
To "express Jn {z) as a power series in z, write u = Itjz ; then
since the contour is any one which encircles the origin once counter-clockwise,
we may take it to be the circle j ^ j = 1 ; as the integrand can be expanded
in a series of powers of z uniformly convergent on this contour, it follows
from § 4-7 that
1 "^^ (_)»• /-^ \n+-ir /-(O+l
27ri^^o r! V^ / J
Now the residue of the integrand at ^ = 0 is {(n + r)]}~'^ by § 6'1, when
n + r is a positive integer or zero ; when n + /• is a negative integer the
residue is zero.
* This procedure is due to Scblomileh, Zeitschrlft filr Mdtlt. ii. (1857).
350 THE TRANSCENDENTAL FUNCTIONS [CHAP. XVII
Therefore, if w is a positive integer or zero,
'^^^'^- Z,r\{n + r)\
_ z^ { z" Z''
" Wn\ X 2M (w + 1) 2* A . 2 (n + \){n + 2)
whereas, when w is a negative integer equal to — m,
"^^ rZnr\{r-m)\ ,% (m + sjlsl '
and so J„ {z) = (-)'" /,„ (z).
The function Jn{z), which has now been defined for all integral values
of w, positive and negative, is called the Bessel coeffi^cient of order n; the
series defining it converges for all values of z.
We shall see later (§ 17 "2) that Bessel coefficients are a particular case of a class of
functions known as Bessel functions.
■Bessel coefficients were introduced in 1824 by Bessel in his Untersuchung des Theils der
planetarischen Stomngen welcheraus der Bewegung der Sonne entsteht {Berlin. Abk. 1824) ;
special cases of Bessel coefficients had, however, been previously considered by D. Bernoulli
in 1732, and by Euler.
In reading some of the earlier papers on the subject, it should be remembered that the
notation has changed, what was formerly written ,/„ (z) being now written J,, (22).
, Example 1. Prove that if
26(1 + ^2) _ ..... ...
then e"' sin hz = J 1 Jj (0) + J .^2 (2) + ^ a«^3 (2) + • • ■ •
(Math. Trip. 1896.)
[For, if the contour D in the '«-plane be a circle with centre ?t=0 and radius large
enough to include the zeros of the denominator, we have
the series on the right converging uniformly on the contour ; and so, using § 47 and
replacing the integrals by Bessel coefficients, we have
2
1 r ie(«--) ^^v^^'^W ^ 1 { i4«--)/^, A. A, \ ,
^r— . e- V "/ ^ o??< = „-- e ^ n/ I 1 + -i + -j + ... ) du
2m J D / 2a 1 \2 4b'^ 2,Tn J u \u- ?// u'^ J
1--- - + .
= A,Jy (z) + A.rh iz) + ^3-/-! (2) + . . . •
In the integral on the left write \i^u — u~'^) — a = t, so that as ti describes a circle of
radius e^, t describes an ellipse with semiaxcs cosh /3 and sinh^ with foci at — a±?' ; then
we have
the contour being the ellipse just specified, which contains the zeros of i" + b-. Evaluating
the integral by § 6'1, we have the required result.]
17 11] BESSEL FUNCTIONS 351
, Example 2. Shew that, when n is an integer,
Jn{y + Z)= 2 J,niy)Jn-m{z)-
TO = — 00
(K. Neumann and Schlafli.)
[Consider the expansion of each side of the equation
exp \^{^^z)[t- ^)| = exp l^y (t - J)| . exp ^z[t- J) j.]
' Example 3. Shew that
giecos* ^ j^ ^^^ _l_ ^. ^^g ^ j^ ^^^ _|_ 2^-2 ^^^ 20^2 (2) + ... •
• Example A. Shew that if r2=a;"^+2'^
Jo (r) = Jo (^0 Ji> (y) - 2^2 (^) J^ (y) + 2 J4('^^) ^4 (y) - . . . •
(K, Neumann and Lommel.)
17"11. BesseVs differential equation.
We have seen that, when n is an integer, the Bessel coefficient of order n
is given by the formula
From this formula we shall now shew that Jn{z) is a solution of the
linear differential equation
dhi Idy ^ r^\
which is called Bessel's equation for functions of order n.
For we find on performing the differentiations (§ 4'2) that
d'Jnjz) IdJnjz) ( n^
A.'dt
= 0,
since ^"'''"^ exp (^ — ^^/4^) is one- valued. Thus we have proved that
dz^ z dz \ z-J
The reader will observe that z= 0 is a regular point and ^ = oc an
irregular point, all other points being ordinary points of this equation.
Example 1. IJy differentiating the expansion
with regard to -a and with regard to t, shew tluit the Bessel coefficients satisfy Bessel's
equation. (St Jolin's, 1899.)
352 THE TRANSCENDENTAL FUNCTIONS [CHAP. XVII
Example 2. The function P„"' ( 1 — — g ) satisfies the equation defined by the scheme
{4n'^ 00 0 ^
^m n + 1 \in z^ Y ;
— \m -n — Jm J
shew that t/„j (z) satisfies the confluent forra»of this equation obtained by making n-^ao .
17*12. The connexion between Jn{z) and Wk,m functions.
The reader will verify without difficulty that, if in Bessel's equation we
write y = z~^ V and then write z = a;/2i, we get
d^v f 1 } — 7i^\ _ ^
dx' V 4 .'/- /'
which is the equation satisfied by Wo,»n(^')j i* follows that
Jn (z) = Az--M,,n{'^iz) + Bz " 4ifo,_,, (2t>).
Comparing the coefficients of z^"- on each side we see that
z-^
except in the critical cases when 2n is a negative integer; when n is half of
a negative odd integer, the result follows from Rummer's second formula
(§ 16-11).
17"2. The solution of Bessel's equation luhen n is not necessarily an
integer.
We now proceed, after the manner of § 15'2, to extend the definition of
Jniz) to the case when n is any number, real or complex. It appears by
methods similar to those of § 17"11 that, for all values of n, the equation
d-xj 1 dy /' n~\ _
dz- z dz \ Z'j
is satisfied by an integral of the form
y = z'' I r«-i exp U - ^)\ dt
provided that t~^''~^ exp (t — z"/4<t) resumes its initial value after describing G
and that differentiations under the sign of integration are justified.
Accordingly, we define Jn (z) by the equation
the expression being rendered precise by giving arg z its principal value and
taking | arg ^ | "$ tt on the contour.
1712-17-21] BESSEL FUNCTIONS • 353
To express this integral as a power series, we observe that it is an
analytic function of z ; and we may obtain the coefficients in the Taylor's
series in powers of z by differentiating under the sign of integration (§§ 5'32
and 4*44). Hence we deduce that
^ |; (-)r^n+2r
~ r^o 2»+^! r(7z-hr+l) '
by § 12'22. This is the expansion in question.
Accordingly , for general values of n, we define the Bessel function Jn{z)
by the equations
~ ,.ro 2'*+'-^''r!r(n + r + l) *
This function reduces to a Bessel coefficient when n is an integer; it is
sometimes called a Bessel function of the first kind.
The reader will observe that since Bessel's equation is unaltered by
writing — n for n, fundamental solutions are Jn (z), J_^ (z), except when
n is an integer, in which case the solutions are not independent. With this
exception the general solution of Bessel's equation is
where a and y8 a7'e arbitrary constants.
A second solution of Bessel's equation when n is an integer will be given
later (§ 17-6).
17'21. The recurrence formulae for the Bessel functions.
As the Bessel function satisfies a confluent form of the hypergeometric
equation, it is to be expected that recurrence formulae will exist, corresponding
to the relations between contiguous hypergeometric functions indicated in
§ 14-7.
To establish these relations for general values of n, real or complex, we
have recourse to the result of § 17*2. On writing the equation
r(o+) fj ( / ^2\)
0= _. sF^"? '-s)f*
i .IJ. 11 — O ..J « 1 1 I J. "
at length, we have
0=1 U-'' + ^ zH-'^-- - nt-""-' ]exY>it-^]dt
= 2r:i \(2z-^T-' Jn-l (2) + \z' {2z-'T+' Jn+, {z) " 'n(2^~')"-^ (^)
In
and so Jn-i{z) + Jn+-i{z)=— Jn{z) (A).
z
w. M. A. 23
354 . THE TRANSCENDENTAL FUNCTIONS [CHAP. XVII
Next we have, by § 4*44,
. , ■ i [-'Jn (.)) = 2„-i„. i\[*^ t-'-^ exp {t - Q dt .
= — Z Jn+1 \Z),
and consequently, if dashes denote differentiations with regard to z,
J^{z)='^^Jn{z)-Jn^,{z) (B).
From (A) and (B) it is easy to derive the other recurrence formulae
Jn{z)=\[Jn-r{z)-Jn^,{z)] (C),
and Jn {z) = Jn-i{z) - - Jn{z) (D).
Example 1. Obtain these results from the power series for Jn (z).
Example 2. Shew that ^ [z'^Jn 'z)} = 2" Jn _ i (2).
ciz
Example 3. Shew that i/q' (2) = —J\ {z)-
Example 4. Shew that
16J„i^ {z)=Jn-i (2) - 4J„_2 (2) + 6Jn {z)-Un + i{z) +Jn^i{z).
Example 5. Shew that
J,{z)-J,{z) = iJ^'{z).
Example 6. Shew that
J,{z)=J^'{z)-z-W^{z).
17*211. Relation between two Bessel functions whose orders differ by
an integer.
From the last article can be deduced an equation connecting any two
Bessel functions whose orders differ by an integer, namely
Z-^'-^Jn+r (Z) = i-r -^^y [z-^'Ju {Z)],
where n is unrestricted and r is any positive integer. This result follows at
once by induction from formula (B), when it is written in the form
Z-^-^J,,^,{z) = -^^{z-^^Jn{z)].
17'22. The zeros of Bessel functions tuhoae order n is real.
The relations of § 17"21 enable us to deduce the interesting theorem that
betiveen any two consecutive real zeros of z~^''Jn (z), tliere lies one and only one
zero* of z~"Jn+i{z).
* Proofs of this theorem have been given by Bocher, Dull, American Math. Soe. iv. (1897),
p. 206; Gegenbauer, Monatshefte fur Math. viii. (1897); and Porter, Ball. American Math.
Soc. IV. (1898), p. 274.
17-21 1-17-23] BESSEL FUNCTIONS S5S
; For, from relation (B) when written in the form
it follows from RoUe's theorem* that between each consecutive pair of zeros
of 2~^Jn (•2') there is at least one zero of z~^^Jn+i (z).
Similarly, from relation (D) when written in the form
it follows that between each consecutive pair of zeros of z^'''^^Jn+i(z) there is
at least one zero of 2"''"^J„(^).
Further z~^Jn(z) and -j- [z~^Jn(z)] have no common zeros; for the
former function satisfies the equation
and it is easily verified by induction on differentiating this equation that if
both y and -j- vanish for any value of z, all differential coefficients of y vanish,
and y is zero by § 5-4.
The theorem required is now obvious except for the numerically smallest
zeros ± ^ of z~''^Jn (z), since (except for z = 0), z~^^Jn (z) and ^^"'"^ Jn i^) have the
same zeros. But z=0 is a zero of z~^KTn+i (z), and if there were any other
positive zero of z~^Jn+i(z), say ^j, which was less than f, then z'^'^^Jn(z)
would have a zero between 0 and ^i, which contradicts the hypothesis that
there were no zeros of 2;"+^ J„ (z) between 0 and ^.
The theorem is therefore proved.
[See also § 17'3 examples 3 and 4, and example 19 at the end of the chapter.]
17-23. Bessel's integral for the Bessel coefficients.
We shall next obtain an integral first given by Bessel in the particular
case of the Bessel functions for which n is a positive integer ; in some respects
the result resembles Laplace's integrals given in § 15-23 and § 15-33 for the
Legendre functions.
In the integral of § 17'1, viz.
J^(^z) = J-A u—^e V ^^J du,
ZTTIJ
take the contour to be the circle | w | = 1 and write u = e*^, so that
ZirJ -„
* This is proved in Burnside and Panton's Theory of Equations (i. p. 157) for polynomials.
It may be deduced for any functions with continuous differential coefficients by using the First
Mean Value Theorem (§ 4-14).
23— -2
356 THE TRANSCENDENTAL FUNCTIONS [CHAP. XVII
Bisect the range of integration and in the former part write — ^ for ^ ;
we get
27rJo 27rJo
and so /„ (•2) = — cos (nO - z sin 6) dO,
which is the formula in question.
Example 1. Shew that, when z is real and n is an integer,
\Jn{z) 1^1.
Example 2. Shew that, for all values of n (real or complex), the integral
1 ["
y=- I cos («^ — s sin ^) c?^
satisfies
cPy 1 dy ^(^ 7^\^ ^_&vsxni! (\ n
dz^ z dz
which reduces to Bessel's equation when n is an integer.
[It is easy to shew, by differentiating under the integral sign, that the expression
on the left is equal to
1 {'" d {(n cos 6
75 H :
0 d6 \\z
sin (nd - z sin 6) \ dB.'\
17'231. The modification of Bessel's integral when n is not an integer.
We shall now shew that*, for general values of n,
J^ (^) = 1 r cos {nd - z sin 0) dO - ^^°^ f " e-ne-^smhe ^^ ^ _ /^x
when R{z)>0. This obviously reduces to the result of § 17*23 when n is
an integer.
Taking the integral of § 17"2, viz.
M /■(0+) / ^2\
and supposing that z is positive, we have, on writing t = ^uz,
1 r(<'+) [i I In)
But, if the contour be taken to be that of the figure consisting of the real
axis from — 1 to — go taken twice and the circle j m | = 1, this integral re-
presents an analytic function of z when R {zu) is negative as | w | -»- 00 on the
path, i.e. when | arg z\<^7r; and so, by the theory of analytic continuation,
the formula (which has been proved by a direct transformation for positive
values of z) is true whenever R (z) > 0,
* This result is due to Schlafli, Math, Ann. iii.
17*231] BESSEL FUNCTIONS 367
Hence
•^- (^) = h in + // /','} """"" ^"p K" - 5)} ''«•
where 0 denotes the circle | w | = 1, and arg u-= — it on the first path of
integration while arg u = + tt on the third path.
— 00 —1
Writing u = te^^ in the first and third integrals respectively (so that in
each case arg t = 0), and u = e^'* in the second, we have
Modifying the former of these integrals as in §17*23 and writing e^ for t
in the latter, we have at once
J^ (^) = 1 r cos {nd - z sin 6) dO + ^"L(^i±l)5 [" g-no-zsinhe ^q^
which is the required result, when ) arg z\<-'Tr.
When I arg 2 | lies between \nr and tt, since Jn {z) = e^'^'^^Jn{ — z), we have
±nni f r„ /•<" ]
J„(2) = !__J| cos(ri^+0sin^)rf^-sinw7r / e-***+^^'"^^(f^l (B),
the upper or lower sign being taken as arg 0 > ^tt or < - ^tt.
When n is an integer (A) reduces at once to Bessel's integral, and (B) does so when we
make use of the reduction J„ (z) = ( — )» J_n (2) which is true for integer values of 71.
Equation (A) is due to Schlafli, Math. Ann. m. (1871), and equation (B) was given by
Sonine, Math. Ann. xvi. (1880).
These trigonometric integrals for the Bessel functions may be regarded as corresponding
to Laplace's integrals for the Legendre functions. For (§ 17*11 example 2) ./,„(2) satisfies
the confluent form (obtained by making ?i-^oo ) of the equation for P^"* (1 - z^j^n^).
But Laplace's integral for this function is a multiple of
= \ jl+ — cos ^ + 0 (?4~2)|- COS m(f) d(f).
The limit of the integrand as to-»x is e*^°°^* cos ?«(/>, which exhibits the similarity of
Laplace's integral for P„"* (2) to the Bessel-Schlafli integral for t7'„, (2).
358 THE TRANSCENDENTAL FUNCTIONS [CHAP. XVII
Example 1. From the formula J'o (^) = ^r- / e~^'^'^°^^ d(f), by a change of order of
•^TT J — TT
integration, shew that, when % is a positive integer and cos ^ >0,—
i'„ (cos ^) = — .--^ - , I e-''''''^^J^{xiim6)x-^dx. (Callandreau.)
r(n + l}jo
Example 2, Shew that, with Ferrers' definition of P^™ (cos 6\
■ ' PrT (cos 6) = ^ _^^^_^^. /" " e-*°°«« J^ {x sin 6) x^dx
when n and m are positive integers and cos ^> 0. (Hobson.)
Yl'^^. Bessel functions whose order is half an odd integer.
We have seen (§ 17*2) that when the order n of a Bessel function Jn{z)
is half an odd integer, the difference of the roots of the indicial equation at
2^ = 0 is In, which is an integer. We now shew that, in such cases, Jn {z) is
expressible in terms of elementary functions.
^ r / N 2*2* L ^^ z" ] / 2 \* .
i^ or J i(z)= — J- \ 1 — w—F^ + r^— s — 'a — P — •••?•= — sm z,
^^ ' ttM 2.3 2.3.4.5 J \Trzj
and therefore (§ 17 '2 11) if A; is a positive integer
(-)^ (2^)^+4 d^ /sin^N
On differentiating out the expression on the right, we obtain the result that
Jn + i {z) = Pk sin z + Q^ cos z,
where P;;., Q;;. are polynomials in s~ 2.
Example 1. Shew that J_ j (2) = f — j cos z.
Example 2. Prove by induction that if k be an integer and n = k + ^, then
/2\ir , , ^ .{. (-)''(4n2-l2)(4?i2^.32)...{4n^-(4?'-l)2}l
•^"^^)=y r'^'~*""~^'^M''".!i — (2-^)12^;^^^^ — - — ^7
. , , , , (-)'-(4w2-l2)(4^2_32),_ I4^2_(4^_3X2}-|
+ sm {z-hnn-k^) 2^^^ (2^- 1) ! 20^-3^1- 1 ^J'
the summations being continued as far as the terms with the vanishing factors in
the numerators.
Example 3. Shew that z^+h -j-t^ic ( ) ^^ ^ solution of Bessel's equation for
^2»n + l y
Example 4. Shew that the solution of 2'""^* , 2m + 1+^"*^ ^^
2m
P
where Cqj Ci, ••• C2„i are arbitrary and oo, ai, ... a2,rt are the roots of
(Lommel.)
17-24, 17-3] BESSEL FUNCTIONS - 359'
17'3. Hankel's contour integral* for Jn{z)'
Consider the integral
2/ = ^« {f'-lf-^QOB{zt)dt,
J A
where -4 is a point on the right of the point t = l, and
arg(<-l) = arg(«+l) = 0
at A ; the contour may conveniently be regarded as being in the shape of
a figure of eight.
We shall shew that this integral is a constant multiple of Jn(z). It is
easily seen that the integrand returns to its initial value after t has described
the path of integration; for (^ — 1)" ~ ^ is multiplied by the factor e(2»-i)« after
the circuit (1+) has been described, and (^+1)""^ is multiplied by the
factor e"<2"~i^" after the circuit (— 1 -) has been described.
converges uniformly on the contour, we have (§ 4*7)
^ io (2r)! .L * ^* ^^ '^^-
To evaluate these integrals, we observe firstly that they are analytic
functions of n for all values of n, and secondly that, when ii ( ri + 2] >0, we
may deform the contour into the circles \t— 1\ = 8, \t + 1\ = 8 and the real
axis joining the points t= ±(1 — B) taken twice, and then we may make
S -* 0 ; the integrals round the circles tend to zero and, assigning to t—1
and ^ + 1 their appropriate arguments on the modified path of integration,
we get, if arg (1 — i^) = 0 and t- = u,
t'^(t'-lf-^dt
J A
= Q(n - 1) ni r "^ ^^r ^ _ ^,)n -lat + e- (" - ^) '^^ f t^'' (1 - t^f -^dt
= - 4t sin (n -l^-^i ^^'- ( 1 - t^T ~ ^ dt
= - 2i sin (n -^tt u'' " * (1 - wf '^du
= 2t-8in (n + ^) TT r (r + .^) T (n + -^Iv {n + r + 1).
* Math. Ami. i.
360 THE TRANSCENDENTAL FUNCTIONS [CHAP. XVII
Since the initial and final expressions are analytic functions of n for all
values of n, it follows from § 5*5 that this equation, proved when
ij(. + l)>0,
is true for all values of n.
Accordingly
y r=o (2r)!r(n + r + l)
= 2-+H- sin (n + |) irV [n + ^) V (^j /„ (z),
on reduction.
Accordingly, when "1 ^ [ 5 — w j i ^0,we have
'^n (-) = 2-^1- r (1) i^ («^ - 1 r - * COS {zt) dt.
Corollary. When i?(?i + |)>0, we may deform the path of integration, and obtain
the result
•'•<^)'2.r(,.+\)r(i)/',('-'')"'''°''(-'>'"
° 2— r (r+i) r (i) /o "'"'• '» '"^ <^ ""^ '^) '''^-
Example 1. Shew that, when /^ (% + J) > 0,
Example 2. Obtain the result
'^" ^'^ " 2-T{n+^)r{l) J I ''''^ ^' ''''^ *^^ ^''''" "^ '^'^'
when R{n)>0, by expanding in powers of 2 and integrating (§ 4-7) term-by-term.
Example 3. Shew that when - i < 7i < ^, J'„ (2) has an infinite number of real zeros.
[Let z = {m + i) IT where m is zero or a positive integer ; then by the corollary above
Jn {mtt -h ^tt) = 2»-ir(?!+.^)r(.^) ^2"" - ''^i + ^2 - . . . + ( - )™ u,n},
2r+l
where m^= rj"^^^ (1 _ f-y' - ^ cos {{m + ^) nt} dt I
ri/(w+A) f / 2r- lV]n-i
= jo {'-(^ + 2^+1)} «iM(- + i)-^}^^,
so, since n-^<0, u,n>u,n-i> itm-2> •-, and hence Jn{mn + \n) has the sign of (-)'».
This method of proof for n = 0 is due to Bessel]
Example 4. Shew that if 11 be real, J.^ (z) has an infinite number of real zeros ; and
find an upper limit to the numerically smallest of them.
[Use example 3 combined with § 17 "22.]
17'4, 17-5] BESSEL FUNCTIONS 361
17'4. Connexion between Bessel coefficients and Legendre functions.
We shall now establish a result due to Heine* which renders precise the statement of
§ 17*11 example 2, concerning the expression of Bessel coefficients as limiting forms of
hypergeomotric functions.
When I arg(l+2)| < tt, « is unrestricted and m is a positive integer, it follows by
differentiating the formula of § 15-22 that, with Ferrers' definition of Pn^{z),
^""(^^ = 2'».I^r"^r-"^m+l)(^-^^^"^^+^)^"-^(-^ + "^' « + l + m; m + \;\-^z\
and so, if | arg 2 | < ^tt, | arg (1 - Iz^jn^) \ <7r, we have
7Jm/i 2^\ r(» + m+l)2"'W-™/, 22\*"' , , , , o OS
Now make «-* +qo {n being positive, but not necessarily integral), so that, if d = «~S
5-»0 continuously through positive values.
Then —- f- ». 1, by S 13-6, and [l--r'»] ^1.
Further, the (r + l)th term of the hypergeometric series is
(-0^- mS) (1 + 8+mS+rS){l-(TO + 1)2 S2} {1 - (m+2)2 S2} ... {1 - (»i+7-)2 .
{¥r;
(m + l)(m + 2) ... (m + r). /• !
this is a continuous function of S and the series of which this is the (r + l)th term is
easily seen to converge uniformly in a range of values of S including the point S = 0; so,
by § 3*32, we have
limL-P-A--^^l= ^" g {-YikzY
w^ocL \ 2WJ 2»'.m!,.=o(»i + l)(m + 2) ...(m+r)r!
= ^41 (4
which is the relation required.
Example 1. Shew that t
lim r«-»'P„»"{cos-) =./,n(2).
Example 2. Shew that Bessel's equation is the confluent form of the equations
defined by the schemes
n ic \-\-ic z\, ei'p\ n \ 0 2I, pj ^/i \{c-n) 0 s^l
-n —ic ^ — ic J I - ^* f — 2ic 2ic-l J f. - ^Ji — |(c + n) ?i + 1 j
the confluence being obtained by making c -*• qo .
17"5. Asymptotic series for J^ (z) when \z\ is large.
We have seen (§ 17-12) that
1 • 3
where it is supposed that ; arg z < ir, — ^ tt < arg {'2iz) < -^tt.
* The apparently different result given in Heine's Kugelfunktionen is due to the difference
between Heine's associated Legendre function and Ferrers' function.
t The special case of this when 7/1 = 0 was given by Mehler, Crelle, lxviii.
362 THE TRANSCENDENTAL FUNCTIONS [CHAP. XVII
But for this range of values of z
by § 16'41 example 2, if — ^ tt < arg (— liz) < ^ir; and so, when | arg z\<ir,
{Iirzy-
But, for the values of z under consideration, the asymptotic expansion of
Tfo,«(±2i>) is
^ t - 8i> 2!(8i>f *••• -
■^ vv^izf "^^^^ ^r
and therefore, combining the series, the asymptotic expansion of /„ {z), when
I 2; I is large and | arg £; | < tt, is
Jn{z)'-[^^
cos I ■2 — o ^''"" ~ 4 *""
1 I \ %, i-y 14n2 - 12] {4'/i2 - 3-1 . . . {^n' - (4r - S)^
2 4
r=l
(2r-l)!2«'-='/'''-i
cos b - ^ ?i7r - ^ TT j . f7n (^) - siu (z - ^nTT-^Trj. Vn (z)
/2 ^i
\'7rz/
where Un (z), — Vn {z) have been written in place of the series.
The reader will observe that if n is half an odd integer these series
terminate and give the result of § 17"24 example 2.
Even when z is not very large, the value of J,^ {z) can be computed with great accuracy
from this formula. Thus, for all positive values of z greater than 8, the first three terms
of this asymptotic expansion give the value of i/q {z) and «/, (2) to six places of decimals.
This asymptotic expansion was given by Poisson* (for n = 0) and by Jacobit (for
general integral values of n) for real values of z. Complex values of z were considered by
Hankel+ and several subsequent writers. The method of obtaining the expansion here
given is due to Barnes §.
Asymptotic expansions for J^ {z) when the order n is large have been given by Debye
{Math. Ann. Lxvii. pp. bZb-bbS, Milnchen. Sitzungsberichte, xh. 1910, Abh. 5) and Nicholson
{Phil. Mag. 1907).
* Journal de VEc. Folytechnique (1), cah. 19 (1823).
t Astron. Nach. xxviii. p. 94.
+ J\I(Uh. Ann. i. p. 467.
§ Trans. Camb. Phil. Soc. xx. p. 274.
17*6] BESSEL FUNCTIONS 363
Example 1. By suitably modifying Hankel's contour integral (§ 17-3), shew that, when
r(n + i)(2,r2)*L yo \ 2?y
and deduce the asymptotic expansion of t/„ {z) when | 2 | is large and | arg z \ < ^ir.
[Take the contour to be the rectangle whose corners are ±1, ±l + tiV, the rectangle
being indented at ±1, and make JV-*-qo ; the integrand being (1 — «2)«-J e«/]
Example 2. Shew that, when | arg 2 | < ^tt and R{n+^)>0,
'^n (2) = ^r^\^" ■ I ^^ e-^'°'* cos'*-* (p cosec2« + 1 0 sin {2 - (?i - ^) c^} (^(^
[Write ?i = 22cot 0 in the preceding example.]
Example 3. Shew that, if | arg 2 | < ^n- and R{n + ^)>0, then
^e*'^2'' f " i;'»-i ( 1 + ivf-^ e--''' dv + ^e"^^ 2" /" " i''*"* (1 - ivf'^ e"'"" cfv
is a solution of Bessel's equation.
Further, determine A and B so that this may represent t/„ (2).
(Schafheitlin, Cre^^e, cxiv.)
17*6. The second solution of BesseVs equation when the order is an integer.
We have seen in § 17'2 that when the order n of Bessel's differential
equation is not an integer, the general solution of the equation is
aJn {z) + ^J-n {z),
where a and /3 are arbitrary constants.
When, however, n is an integer, we have seen that
Jn{z) = {-YJ^n{z),-
and consequently the two solutions /„ {z) and JL^ {z) are not really distinct.
We therefore require in this case to find another particular solution of the
differential equation, distinct from J^ {z), in order to have the general
solution.
We shall now consider the function
^ sin znir
which is a solution of Bessel's equation when 2n is not an integer. The
introduction of this function F„ {z) is due to Hankel*.
* Math. Ann. i. p. 472.
364 THE TRANSCENDENTAL FUNCTIONS [CHAP. XVII
When n is an integer, F^ is defined by the limiting form of this equation,
namely
F. (z) = lim 2,r6(«-^0 ^^ Jn^Az)ooB{n^-Ve^)-J-n-.{z)
e^o sm 2 (n + e) TT
= lim 4^ (/.+, {z) . i-Y - J_._. (.)}
e^-O Sin ZCTT
= lim e-^ { /„+, (^) - (-)« J-.n-e (^)}.
To express F„ (2^) in terms of Wk, m functions, we have recourse to the
result of § 17 '5, which gives
F„(^) = lim '"'
o(27r^^)2
remembering that W]c^m= Wjc^^m-
Hence, since* lim TFo,„+e(2i2') = Tfo,„(2i>), we have
This function (w being an integer) is obviously a solution of Bessel's
equation ; it is called a Bessel function of the second kind.
The asymptotic expansion for Yn{z), corresponding to that of § 17'5 for
Jn {z), is that, when | arg 2 j < tt and n is an integer,
o \ i r
Yn (2) '^ i^-jj sin [z -'^nTT-^Trj. Un (z) + COS U -I nir - j tt j . V,, {z)
where Un{z) and Vn{z) are the asymptotic expansions defined in §17*5, their
leading terms being 1 and (471^ - l)/8^ respectively.
Example 1. Prove that
where n is made an integer after diflferentiation. (Hankeh)
Example 2. Shew that if Yn{z) be defined by the equation of example 1, it is a
sokition of Bessel's equation when 11 is an integer.
17"61. The ascending series for F„ (z).
The series of § 17"6 is convenient for calculating Yn{z) when |^| is large.
To obtain a convenient series for small values of l^j, we observe that, since
the ascending series for J±(«+e) (^) are uniformly convergent series of analytic
functions -f- of e, each term may be expanded in powers of e and this double
series may then be rearranged in powers of e (§§ 5"3, 5"4).
* This is most easily seen from the uniformity of the convergence with regard to e of
Barnes' contour integral (§ 16-4) for W^^ ^,^(2iz).
f The proof of this is left to the reader.
17'61] BESSEL FUNCTIONS 365
Accordingly, to obtain F„ (z), we have to sum the coefficients of the first
power of 6 in the terms of the series
.=o^!r(n + e + r4-l) ^ ^ ^^„ r!r(-w-6 + r + 1)'
Now, if s be a positive integer or zero and t a negative integer, the
following expansions in powers of e are valid :
(a-^y '^"^ = (^^)"^"" |l + 6 log (^^) + ...| ,
|l-6(-7+ S m-M +
where 7 is Euler's constant (§ 12-1).
Accordingly, picking out the coefficient of e, we see that
+ (_)n s ^ ^ ^^7 — (-x-n+i r (/I - r),
and so
'*^i(^^)-«+«"(n-r-l)!
When n is an integer, fundamental solutions of Bessel's equations, regular
near z = ^, are /„ {z) and F„ (2').
Karl Neumann* takes as the second solution the function F'"''(^) defined
by the equation
F<«) {z) = \ F, (^) + Jn {z) . (log 2 - 7) ;
but Hankel's function is more useful for physical applications.
* Theorie der BesseVxchen Funktionen (Leipzig, 1867), p. 41.
366 THE TRANSCENDENTAL FUNCTIONS [CHAP. XVII
Example 1. Shew that the function Yn{z) satisfies the recurrence formulae
Y,:{z) = \{Yn.,{z)-Yn^M]-
Shew also that Neumann's function F(") {z) satisfies the same recurrence formulae.
[These are the same as the recurrence formulae satisfied by Jn (2).]
Example 2. Shew that, when | arg z \ < ^tt,
Yn (z) = f" sin (z sin e-ne)dd- T g-^^'-^^ {e'^e _ ( _ )« g-^^} dd.
(Schlafli.)
Example 3. Shew that
r(0)(2)=Jo(2)log2 + 2{J2(2)-K(^) + i«>^6(2)-...}.
17'7. Bessel functions with jmrely imaginary argumenf^ .
The function
is of ^frequent occurrence in various branches of applied mathematics ; in
these applications z is usually positive.
The reader should have no difficulty in obtaining the following formulae :
In
(i) In-i {z) - In+i (2) = -- In (z)-
(ii) ^^[zHn{z)]=Z-I,,_,(z).
(iii) ^-[z-In{z)]=z-I,,^,{z).
(^^) dz^-'^-z^:z -ii + ^j^'^(^)=«-
(v) When R (n + ^) > 0,
(vi) When — ^tt < arg 2^ < ^ tt, the asymptotic expansion of
In{z) is
"^ » ., {471^ - 1^1 {4n^ - 3^1 . . . 1471-'' - (2r - If]!
In (z) 'w
e-(n + l)^ig-z
{•Iirz)^
I {4/2^ - 1-} {4n' - 3^1 . . . {4n- - {2r - 1)-}'
the second series being negligible when jarg2^|<-7r. The result is easily
* This notation was introduced by Basset, Proc. Carnh. Phil. Soc. vi.
17 -7-1 7 -8] BESSEL FUNCTIONS 367
seen to be valid over the extended range — ^ tt < arg 2^ < „ ""■ i^ we write
g*(n+J)'" foj. g-(n+i)«^ ^jje upper or lower sign being taken according as
arg 2 is positive or negative.
17*71. Modified Bessel functions of the second kind.
When n is a positive integer or zero, /_„ (z) = In (z) ; to obtain a second
solution of the modified Bessel equation (iv) of § 17"7, we define* the function
Kn (z) for all values of n by the equation
-fir„(2) = f^j COSW7rIfo,n(2^), •
so that Kn (■2') = 2 ■"■ {^-n (^) — In (^)} COt WTT.
Whether n he an integer or not, this function is a solution of the modified
Bessel equation, and when j arg ^ j < ^ tt it possesses the asymptotic expansion
for large values of j 2^ | .
When n is an integer, Kn (z) is defined by the equation
Kn (z) = lim - TT {/_n_e (z) - In^, (z)} cot ire,
which gives (cf § 17"61)
as an ascending series.
Example. Shew that Kn (z) satisfies the same recurrence formulae as /„ (z).
17"8. Neumanns expansion^ of an analytic function in a series of Bessel
coeffijcients.
We shall now consider the expansion of an arbitrary function f{z),
analytic in a domain including the origin, in a series of Bessel coefficients, in
the form
f{z) = Ko^o {z) + ai/i {z) + a^J-, {z) + ...,
where a^, a^, a^, ... are independent of 2^.
* This notation is due to Gray and Mathews {Bessel Functions, p. 68) and is now generally
adopted (see example 40, p. 377). The function was first considered by Hankel, Math. Ann. i.
p. 498.
t K. Neumann, Jouriuil fiir Math, lxvii. p. 310 (1807). The exposition here given follows
Kapteyn, Ann. de VEcole Normale, (3) x. p. 106 (1893).
368 THE TRANSCENDENTAL FUNCTIONS [CHAP. XVII
Assuming the possibility of the expansion, let us determine the coefficients by com-
paring the Maclaurin expansion
f{z)=f{0) + zf'{0) + ~f"{0) + ...
with the assumed series ; we get
/(0) = ao, 2/'(0) = a,,...,
1 ! («-l) ! ' 2! (?i-2)! '
from which we iind without difficulty
and hence an is twice the residue of On{t) f{t) at t=0 where
2"-^?i! f z^ 2* "i
On{z)- ^„ + i [^'*'2(2?i-T)'''2.4(2?i-2)(2/i-4)"'""*J '
the series terminating with the term in 2" or 2»-i.
1 f(o+)
Thus a„=-. On{t)f{t)dt,
TTl J
the contour enclosing no singularities of f{t).
Example. Shew that
00(0=^-', Oi(0=-Oo'(0.
17'81. Proof of Neumanns expansion.
The method of § 17*8 merely determined the coefficients in Neumann's
expansion, on the hypothesis that the expansion existed and that its rearrange-
ment in powers of z was legitimate.
To obtain a proof of the validity of the expansion, we observe that *
0,, {t)=r \ t-^-' e-"^ [{w + {x' -f- f )*}" + {x- (x' + t')^''] dx.
io -^
Hence
Oo(0^o(^)+2 t On(t)Jn{2) = t-' t —. ^ Jn{z)e-^dx.
» = 1 >i=-ocJO ( f J
We shall now prove that the series on the left converges uniformly with regard
to t when j 2 | < 1 ^ [ = 1, and the interchange of summation and integration is justified.
When n ^ N, we have
Mz)\^ll-^,{^+K^),
■2»?i
where K depends only on N and 2 (not on n) and K^^-^0 as N-*- x ; tind if | < I = 1,
— =-^ — — '— < 2" (^4- !)"■.
* This result is obvious when 7i:=0 or 1 ; it is easily proved by induction for integer values
of n from the example of § 17'8.
17"81, 17-82] BESSEL FUNCTIONS 369
Hence
M -w r " ('^ 4-1 "1" 1 2 1 "
2 \On{t)Jn{z)\<^\t\-^ 2 -^^ V {-^+K^)e-'dx
n=N+\ n=N+l JO ^ '
M
<2e 2 i2|"(l + ^ ),
H=N+l
and, for any fixed value of z, this can be made arbitrarily small by taking If sufficiently
large. Hence the series in question converges uniformly on the circle | < | = 1 ; and since
both
2 0,,{t)Jn{z) and 2 N "^ ^ ' \ Jn{z)\e-'dx
W=iV+l n=N+l y 0 I I f I 1
tend to zero as iV-*- oo, it is not very difficult to see that the interchange of summation
and integration is justifiable (cf. § 4'7).
Consequently, if | ^^ | < j t j = 1, we have
Oo (0 J, (^) + 2 5 On it) Jn {Z) = t-^ r i {^^(^+^Y j^ ^^^ g_, ^^
n=l Jo M=-«) ( f J
= t~^ \ exp i- — x\ dx
= {t-z)-\
by § 17"1 ; and the series on the left converges uniformly with regard to t.
Hence, if f{t) be analytic when j i | :$ 1, we have, when | ^ | < 1,
= 2^ \f{t) jOo (0 ^0 {z) + 2 J^ 0„ it) Jn (^)| dt
= J,(z)f{0)+ 2 '^^^ \On{t)f{t)dt,
M = l TTl J
by § 4*7, the paths of integration being the circle | ^ j = 1 ; and this establishes
the validity of Neumann's expansion when | ^^ | < 1 and f(t) is analytic when
\t\^l.
Example 1. Shew that
cos 0 = Jo (2) — 2^2 {z) + 2Ji (2) - ...,
sin2 = 2Ji(2)-2J3(2) + 2J5(2)-.... (K. Neumann.)
Example 2. Shew that
(i2)»= j/^^ + '^)-^ + ^-^)'.4,..(2). (K. Neumann.)
17-82. Schlumilch's expansion of an arbitrary function in a series of Bessel coefficients
of order zero.
Schlomilch* has given an expansion of quite a different character from that of
Neumann, His result may be stated thus :
* Zeitschrift filr Math. u. Physik, 11. (1857).
W. .M, A. 24
370 THE TRANSCENDENTAL FUNCTIONS [CHAP. XVII
Any function f{a;), which has a contimious differential coefficient for all values of x in
the closed range (0, tt), may he expanded in the series
f{x) = aQ + aiJf^{x) + a2JQ{2x)-^a3jQ{Zx) + ...,
valid in this range; where
ao=/(0)+ - r u \ f {u sin 6) dddu,
'^ J 0 J 0
2 /""■ /"if
- I u COS nu f f {ti sin 6) dddu {n>0).
'"'Jo Jo
Schlomilch's proof is substantially as follows :
Let F{x) be the continuous solution of the integral equation
/(^)=- [^ F{xsm(f>)d({>.
^ J 0
Then (§ 11-81)
F{x)=f{0) + x f^'' f'{x Hind) d6.
J 0
In order to obtain Schlomilch's expansion, it is merely necessary to apply Fourier's
theorem to the function F{xsin(f)). We thus have
f(x) = — f d(ji <— I F{u)du+- 2 / cos nu cos (nx sin (b)F(u)duy
'"Jo {.'"Jo ^n=lJo )
= — / F{u)du+— 2 I cos mi F (u) Jo (nx) du,
""Jo ""n^ljo
the interchange of summation and integration being permissible by §§ 4*7 and 9'24.
In this equation, replace F{u) by its value in terms of/(w). Thus we have
/(:(;) = 1 f" j/(0) + u j^f {u sin 0) de\ du
+ -2 JQ{nx) I cos nu -If {0) + u f f ' (ii sin 6) d6\ du,
'" n=l Jo I ^0 J
which gives Schlomilch's expansion.
Example. Shew that, if 0 ^ .r ^ tt, the expression
is equal to x ; but that, if n ^.^^27r, its value is
■r + 27r arc cos {■irx~'^) — lix^ — n^),
where arc cos {irx''^) is taken between 0 and - .
Find the value of the expression when x lies between 2it and 3n.
(Math. Trip. 1895.)
17"9. Tabulation of Bessel functions.
Hansen used the asymptotic expansion (§ 17'5) to calculate tables of Jn{x) which are
given in Lommel's Studien uber die Bessel'schen Funktionen.
Meissel tabulated Jq{x) and J^ix) to 12 places of decimals from ;p=0 to a' = 15-5 {Ahh.
der Akad. zu Berlin, 1888), while the British Assoc. Report (1909), p. 33 gives tables by
which t/„ {x) and Y^,{x) may be calculated when x>\Q.
17 '9] BESSEL FUNCTIONS 371
Tables of J,{x), J Ax), J_Ax), J _Ax) are given by Dinnik, Archiv der Math, und
Phys. XVIII. (1911), p. 337.
Tables of the second solution of Bessel's equation have been given by the following
writers : B. A. Smith, Mess, of Math. xxvi. (1897), p. 98 ; Phil. Mag. XLV. (1898), p. 106 ;
Aldis, Proc. R. S. Lxvi. (1900), p. 32 ; Airey, Phil. Mag. xxii. (1911), p. 658,
The functions In{x) have been tabulated in the British Assoc. Reports, (1889) p. 28,
(1893) p. 223, (1896) p. 98, (1907) p. 94; also by Aldis, Proc. R. S. LXiv. (1899); by
Isherwood, Proc. Manchester Lit. and Phil. Soc. XLViii. (1904) ; and by E. Anding, Sechs-
ttellige Tafeln der BesseVschen Funktionen imagindren Argumentes (Leipzig, 1911).
Tables of J^ {xji), a function employed in the theory of alternating currents in wires,
have been given in the British Assoc. Reports, 1889, 1893, 1896 and 1912; by Kelvin, Math,
and Phys. Papers, iii. p. 493; by Aldis, Proc. R. S. Lxvi. (1900), p. 32; and by Savidge,
Phil. Mag. xix. (1910), p. 49.
Formulae for computing the roots of Jq (^) '''i^ere given by Stokes, Camh. Phil. Trans, ix.
and the 40 smallest roots were tabulated by Wilson and Peirce, Bull. American Math.
Soc. III. (1897), p. 153. The roots of an equation involving Bessel functions were computed
by Kalahne, Zeitschrift fitr Math, und Phys. liv. (1907), p. 55.
A number of tables connected with Bessel functions are given in British Assoc. Reports,
1910-1914, and also by Jahnke und Emde, Funktiontafeln (Leipzig, 1909),
KEFEKENCES,
R, LiPSCHiTZ, Crelle, LVl. (1859), pp, 189-196.
H. Hankel, Math. Ann. i. (1869), pp, 467-501.
K. Neumann, Theorie der BesseVschen Funktionen. (Leipzig, 1869.)
E. LoMMEL, Studien uber die BesseVschen Funktionen. (Leipzig, 1868.) Math. Ann.
III. IV,
H. E. Heine, Handbuch der Kugelfunktionen. (Berlin, 1878.)
R. Olbricht, Studien iiher die Kugel- und Cylinder-funktionen. (Halle, 1887.)
A. SoMMERFELD, Math. Ann. XLVii.
N. Nielsen, Hatidbuch der Cylinderfunktionen. (Leipzig, 1904,)
A. Gray and G, B. Mathews, A Treatise on Bessel Functions.
J, W. Nicholson, Quarterly Journal, xlil (1911), pp. 216-224,
Miscellaneous Examples.
1. Shew that
cos (2sin 6)=Jq («) + 2.^2 (2) cos 2^ + 2^4 {z) cos 45 + ...,
sin (zsin 5) = 2i7i (2) sin 5 + 2 J3 (2) sin .35 + 2/5(2) sin 55+ ....
(K. Neumann.)
2. By expanding each side of the equations of example 1 in powers of sin 5, express
z" as a series of Bessel coefficients.
24—2
372 THE TRANSCENDENTAL FUNCTIONS [CHAP. XVII
3. By multiplying the expansions for exp js'^f^"^)}" ^^^ ^xp \ - ^ z\t — )>■ and
considering the terms independent of t, shew that
{Jo {z)f + 2 Ki {z)f + 2 { ^2 {z)Y + 2 {J-3 (0)}2 + , . . = 1.
Deduce that, for the Bessel coefficients,
I Jo (2) 1^1, |J«(2)l<2-i, {n>\)
when z is real.
4. If
1 /"f
•^jn* (^'l = - I 2*^ cos* « cos {mu — zs,m.u)du
"^ J 0
(this function reduces to a Bessel coefficient when k is zero and m an integer), shew that
JJ{z)= i ;^,(i2)''i^^-».i,p,
where N --,„.% p is the 'Cauchy's number' defined by the equation
1 [■"
277 J -IT
Shew further that
J^ (z)=J^~\(z)+J^''2^{z\
m^ ' m-l \ ^ ' m+1 *> ''
and zC (z) = 2m j;+^ (^) - 2 (^+ 1) {j;_, (.) - J^^, (e)}.
(Bourget, Liouville (2) vi.)
5. If V and Jlf are connected by the equations
ir ET • 7P coaE-e , i i ,
M=E—esinL, cosij= :; =, where e<l,
shew that v = M+2(\-e'^)- 2 2 {^ef J J" (me)- sm mM,
m=l ft=0 ^i
where J,n'' (z) is defined as in example 4. (Bourget.)
6. Prove that, if m and n are integers,
P„-(cos^)=^j;„|(^H;y==)*
where 2=rcos^, x'^+y'^ — r'^sm^d, and c»'" is independent of z.
(Math. Trip. 1893.)
7. Shew that the solution of the differential equation
w-'^dz^XA w ~2rf^Uy 4 Iv^/ ^^dz\i/)-^v ~^) wr" '
where </> and ■v//' are arbitrary functions of z, is
y=(^y{^J,(f) + ^J_,(r/.)}.
8. Shew that
ji (.r)+ J3 (0,-)+ J5 (^)+ ... = 5 fjo (^) + r {-^0 it)+'^i (0} ^<- 1
2 L J 0
(Trinity, 1908.
9. Shew that
( - )" r (/x+ r + 2?i+ 1) (i2)'^+_';+2H
J^{z)J^{z)= 2
„=o ?i!r(;i+H+i)r(i/+7i+i)r(/Li+i'+n + i)
for all values of ^ and v.
(Schonholzer.)
BESSEL FUNCTIONS 373
10. Shew that, if n is a positive integer and m + 2n + l is positive,
(»n- 1) r x^Jn^i{x)J.^_y_ {x) dx^x^^-'{J^^i{x)Jn-i (x) - Jri^ (x)} +(m + 1) raf"JnHx)dx.
11. Shew that
(Math. Trip. 1899.)
12. Shew that
13. Shew that
^3(^) + 3^^)+4^>=0.
dz d^
Jn^xjz)^ U {\zf {\zf
Jn{z) K+1- « + 2- « + 3-...'
(Lommel.)
14. If "-!-y^^ be denoted by Qn (z), shew that
2«/re {Z)
}n(z)_l 2{n+l)
Qn{z) + z{Qn(z)}^
(K. Neumann.)
(K. Neumann.)
dz z
15. Shew that, if R'^ = r'^ + r^ — 2rri cos d and rj > r > 0,
Jo{R) = Jo{r)Jo{ri) + 2 2 J,, (r) J,, (r^) cos nd,
n = l
To (R) = Jq {r) To (r^) + 2 2 c4 (r) 7„ (r^) cos 7id.
n=i
16. Shew that, if R {n + ^) > 0,
/ J^n (22 cos e)de = \iT {Jn {z)Y.
J 0
17. Shew how to express z'^'^J^n (z) in the form
AJ2{2) + BJo{z),
where A, B are polynomials in z ; and prove that
J4(6^) + 3Jo(6*) = 0,
3Jo(30*) + 5^2(30*) = 0.
(Math. Trip. 1896.)
18. Shew that, if a t^,i3 and «->-!,
(oc ( d d ]
(a- - /32) I xJn (ax) J„ {^X) dx = X \ Jn (ox) j- Jn (^x) - Jn {^x) j- Jn (ax) V ,
2a2 r X {Jn {ax)Y dx= {a\v'- - n') {Jn (ax)}'- + ix ~ Jn {ax)y .
19. Prove that, if ?i> - 1, and Jn{a) = Jn (/S) = 0 while a ^ ^,
/ xJn{ax) Jn{^x)dx=0, and I x {Jn{ax)Y dx=h{Jn^i{a)}\
Jo Jo
Hence prove that, when ?i>-l, the roots of Jn{x) = 0, other than zero, are all real and
unequal.
[If a could be complex, take ^ to be the conjugate complex.]
(Lommel, Studien iiber die BesseV schen Funktiouen, p. 69.)
37-4 THE TRANSCENDENTAL FUNCTIONS [CHAP. XVII
20. Let x^ f{x) satisfy Dirichlet's conditions (§ 9'22) in the range 0 ^ a? ^ 1 ; let 5" be
a real constant and let n ^ 0,
Then, Mk^^k^,... denote the positive roots of the equation
k-'^{kJ^{k) + HJr,{k)}=0,
shew that/(;r) can be expanded in the form
where A,.— I x {Jn (kr^)}^ dx \ I xf{x)Jn(krX)dx.
In the special case when H= — n, ki is to be taken to be zero, the equation deter-
mining ^1, k2, ... being Jn + i {k)==0, and the first term of the expansion is AqX"' where
Ao={2n + 2) I af^-*-^f{x)dx.
Jo
Discuss, in particular, the case when H is infinite, so that Jn {k) = 0, shewing that
J 0
[This result is due to Hobson, Proc. London Math. Soc. (2) vii. p. 349 ; the formal
expansion was given (when n=0) by Fourier and (for general values of n) by Lommel.
The formula when H——n was given incorrectly by Dini, Serie di Fourier^ the term ^o-^"
being printed as Jq, and this error was not corrected by Nielsen. See Bridgeman, Phil.
Mag. (6) xvi. p. 947 and Chree, Phil. Mag. (6) xvii. p. 330.]
21. Prove that, if the expansion
exists as a uniformly convergent series when —a^x^a, where Xi, X2, ... are the positive
roots of Jo(^o^) = 0, then
An= 8 {aX„3 Jj (X„a)}-i. (Clare, 1900.)
22. If ^'1, X*2) ••• f^i's the positive roots of J„(^a) = 0, and if
this series converging uniformly when 0 < .r ^ a, then
Ar = ^~^ (4^ + 4 -aV) ^^^^4^.
k/ 'da
23. Shew that
(Math. Trip. 1906.)
Jn {X) = ^n-.n-X^^,^_,^^ j ^ J^ioC siu 6) COs2»-2»»-l 6 sin»» ^^ 6d6
when n>TO> — 1.
24. Shew that, if a- > 0,
/;o..3-.).=3'^:{...(^)...,(|!)}.
(Nicholson.)
BESSEL FUNCTIONS 375
26, If m be a positive integer and u>0, deduce from Bessel's integral formula that
e-a.lnhu J-^ (_y) c;_y_g-mu gech M.
(Math. Trip. 1904.)
26. Prove that, when ^ > 0,
2 r*
Jq(x)=- I ain{xcoa'ht)dt.
T j 0
[Take the contour of § 17'1 to be the imaginary axis indented at the origin and a
semicircle on the left of this line.]
Prove also that
l^o(^)««i— 2 I cos (^ cosh f) 0?^ (Sonine.)
Jo
27. Shew that
/ x~ '^jQ{xt) sin xdx = ^7r 0<^<1
I
=arccosec< ^>1 j
/ x-^ Ji{xt) sin xdx = t-'^ {I- {I- i^)^} 0<t<l
= «-! ^>1
28. Shew that
(Weber.)
M- /" " e"'-^'"^ ^{A+B log (r sin2 $)} d6
is the solution of
^^ + Y£-nH^O. (Trinity, 1886.)
29. Prove that no relation of the form
8=0
can exist for rational values of N^, n and x. (Math. Trip. 1901.)
[Express the left-hand side in terms of Jn{x) and Jn + x{x)^ and shew by example 12
that Jn + \{x)jJn{x) is irrational when n and x are rational.]
30. Prove that, when R{n)> — \,
-"" ^'^- 2"-ir(,^+i)r(i) V^ + cp; \-Y~) '
[(..-)-. ea„. ,.^i-.(.i.Mizl> ...... W..^^./>..]
(Hargreave, Phil. Trains. 1848 ; Macdonald, Proc. London Math. Sac. xxix.)
31. Shew that, when R{m + \)>0,
/2\i th^
[ — ) \ ^w(2sin(9)sin™ + i^c?^=2-i J^ + j(4
(Hobson.)
376 THE TRANSCENDENTAL FUNCTIONS [CHAP. XVII
32. Shew that, if 2n + l>m>-l, ' ■
jo r(«-|m + ^)
(Math. Trip. 1898.)
33. Shew that
(Lommel.)
34. In the equation
n is real; shew that a solution is given by
, , , °° ( - )"* 2^'" COS (tt™ — « log Z)
COS w log z)- 2 ^-^ . ^'". ^-^ .,
m=i a^w'm! (l+Ti'y (4 + «2)* (m2 + %2)*
?»
where Um denotes 2 arc tan (n/r).
(Math. Trip. 1894.)
35. Shew that, when n is large and positive,
J^{n) = 2-i3-in-^r{l)n--s + o{n-'^).
(Nicholson, Phil. Mag. 1908.)
36. Shew that
(Mehler.)
37. Shew that
eAcose^2»-ir(«) 2 (7i + ^)C;(cos^)X-"/„ + ;t(X).
(Math. Trip. 1900.)
38. Shew that, if
Tf= I Jm{ax)Jm{bx)J„,{cx)x^-'"dx;
a, b, c being positive, and m is a j)ositive integer or zero, then
W=0 (a-bf>c\
W= - , — {2262c2-2an™-i (a + 6)2>c2>(a-6)2,
23m-l^ir(m + ^)'
W=0 {a + by>c^. (Sonine, Math. Ann. xvi.)
39. Shew that if ?i>— 1, m>-| and
W= I Jn (ax) Jn (bx) Jm (ex) x^ ~ '" dx,
Jo
a, b, c being positive, then
W=0 {a-bf>c\
Pf=(2,r)-ia'»-i6»-ic-»^(l-/i2)i(2'»-i)pi-^(^) {a + bf>c^>{a-b)\
TT W— 5
c^>{a+bf,
where ii = {aP' + b^-G'^)l2ab, ixi= - fx.
(Macdonald, Froc. London Math. Soc. (2) vii.)
BESSEL FUNCTIONS 377
40. Shew that, if R{m+^)>0,
^-^^>= 2>»r(m+^)r(i) jo coah(.co80)8in2-</.#,
and, if |arg2|<^»r,
„ . , z^T (A) cos wtt /■" , ^
^^ ^') = 2^r{m + ^) j „ «~'°''^* 8inh2-(/.c^.
Prove also that
jr,„(3) = 7r~*2'»2-'"r(m + |)cosm7r I (m2 + 22)-'^-*cosmc^m.
(Math. Trip. 1898. Cf. Basset, Proc. Camb. Phil. Soc. vi.)
[The first integral may be obtained by expanding in powers of z and integrating term-
by-term. To obtain the second, consider
/•(1+, -1+) _ ,
where initially arg(<- l)=arg(<-f-l) = 0. Take \t\>\ on the contour, expand (<2-l)"*~* in
descending powers of t, and integrate term by term. The result is
aze^'""" sin (2m7r) V (2m) 2-^ r (1 - m) /_,„ (2).
Also, deforming the contour by flattening it, the integral becomes
2ie2'»«0msin2«i7r f e-^« (i;2-l)'""*(^; + 2ie2'"''*> cosmTr P e-'^ (l-f)"'-^ df,
and consequently
41. Shew that (?„ (2) satisfies the differential equation
where ^^^ = 2-1 l^n even), gn=nz-'^ (n odd).
(K. Neumann.)
42. If f(z) be analytic throughout the ring-shaped region bounded by the circles c, C
whose centres are at the origin, establish the expansion
/(s) =Uo Jo {^) + aiJi {z) + aoJ2 (2) + . ..
+ i^oOo{z)+^iOi{z)+^,0^{z) + ...,
where «« = ^- f / (0 ^» (0 ^^, /3» = -• f / (^) '^^ (^) ^^-
m J c 'rri J c
(K. Neumann.)
43. Shew that, if x and y are positive,
j^ ~-^Joi.ky)kdk = ~~-,
where r= +sI{^'^-\-y^) and ^= +J(Jc^ - 1) or i J{\ - B) according as ^> 1 or ^< 1.
(Math. Trip. 1905.)
44. Shew that, with suitable restrictions on the form of the function /(^),
f{x) = [ J^ {tx) 1 1 f f{x') ./o {tx') x' dx\ dt.
[A proof with an historical account of this important theorem is given by Nielsen,
Cylinderfunktionen, pp. 360-363.]
378 THE TRANSCENDENTAL FUNCTIONS [CHAP. XVII
45. If C be any closed contour, and m and n are integers, shew that
j^J^ {z) J^ {z) dz={ 0^ {z) 0„ {z) dz=ij^ {z) 0„ {z) dz=0,
unless C contains the origin and m = n ; in which case the first two integrals are still zero,
but the third is equal to iri (or 2ni if m = 0) if C encircles the origin once counter-
clockwise. (K. Neumann.)
46. Shew that, if
{-Y _
and if ?i be a positive integer, then
n
m=l
n-1
while si-2"=a„_i,„_iOo(^) + 2-2 a„_^_i,„+„i_i 02^(4
m=\
(K. Neumann.)
. „ ^ „ / N " 22"* (m !)2 7^2 1^2 _ 12J 1^2 _ 22} . . . {?l2 - (?» - 1)2}
47. it 0«(y)=^2^-2^^^j- ^^^ ,
shew that
(^2 _ ^2)_i = Q^ (3^) {j^ (^)}2 + 2 i Q„ (3^) {/„ (a;)}2
when the series on the right converges. (K. Neumann, Math. Ann. ill.)
48. Shew that, if c> 0, ^ (%) > - 1 and R{a± hf > 0, then
J, (a) J„ {b)=^. f '"^" ' «-i exp {(<2 - a2 _ 52)/(2^)} . /„ (abjt) dt.
(Macdonald, Proc. London Math. Soc. xxxil.)
49. Deduce from example 48, or otherwise prove, that
(a2 + 62_2a6cos^)-*'V„{(a2 + 62-2a6cos^)*}
= 2«r(«) 2 (m + ?i) a-'»6-»J^ + „ («)/„+„ (6) C^» (cos ^).
m=0
(Gegenbauer, Wien. Sitzungsberichte, LXix, lxxiv.)
50. Shew that
y=(jm{t)Jn{tzi)t^-^dt
satisfies the equation
if kt^ Jm (0 Jn {tZ^) -i"*' Jm' (0 ^n {tZ^ + zh"^^ J^ {t) J„' (tZ^)
resumes its initial value after describing the contour.
Deduce that
(Math. Trip. 1903.)
CHAPTER XVIII
THE EQUATIONS OF MATHEMATICAL PHYSICS
18'1. The differential equations of mathematical physics.
The functions which have been introduced in the preceding chapters are
of importance in the applications of mathematics to physical investigations.
Such applications are outside the province of this book ; but most of them
depend essentially on the fact that, by means of these functions, it is possible
to construct solutions of certain partial differential equations, of which the
following are among the most important :
(I) Laplace's equation
which was originally introduced in a memoir* on Saturn's rings.
If {x, y, z) be the rectangular coordinates of any point in space, this equation is
satisfied by the following functions which occur in various branches of mathematical
physics :
(i) The gravitational potential in regions not occupied by attracting matter.
(ii) The electrostatic potential in a uniform dielectric, in the theory of electro-
statics.
(iii) The magnetic potential in free aether, in the theory of magnetostatics.
(iv) The electric potential, in the theory of the steady flow of electric currents in
solid conductors.
(v) The temperature, in the theory of thermal equilibrium in solids.
(vi) The velocity potential at points of a homogeneous liquid moving irrotationally,
in hydrodynamical problems.
Notwithstanding the physical differences of these theories, the mathematical investi-
gations are much the same for all of them : thus, the problem of thermal equilibrium in a
solid when the points of its surface are maintained at given temperatures is mathe-
matically identical with the problem of determining the electric intensity in a region
when the points of its boundary are maintained at given potentials.
* Mem. de VAcad. des Sciences, 1787 (published 1789), p. 252.
380 THE TRANSCENDENTAL FUNCTIONS [CHAP. XVIII
(II) The equation of wave motions
dx" ^ df "^ dz' ~ c' dt^ '
This equation is of general occurrence in investigations of undulatory disturbances
propagated with velocity c independent of the wave length ; for example, in the theory of
electric waves and the electro-magnetic theory of light, it is the equation satisfied by each
component of the electric or magnetic vector ; in the theory of elastic vibrations, it
is the equation satisfied by each component of the displacement ; and in the theory
of sound, it is the equation satisfied by the velocity potential in a perfect gas.
(III) The equation of conduction of heat
dx" "^ By"" "*" dz^ ~ k dt '
This is the equation satisfied by the temperature at a point of a homogeneous isotropic
body ; the constant k is proportional to the heat conductivity of the body and inversely
proportional to its specific heat and density.
(IV) A particular case of the preceding equation (II), when the
variable z is absent, is
a^ "*" dy- ~ (? dt' ■
This is the equation satisfied by the displacement in the theory of transverse vibrations
of a membrane ; the equation also occurs in the theory of wave motion in two dimensions.
(V) The equation of telegraphy
This is the equation satisfied by the potential in a telegraph cable when the inductance
L, the capacity K, and the resistance R per unit length are taken into account.
It would not be possible, within the limits of this chapter, to attempt
an exhaustive account of the theories of these and the other differential
equations of mathematical physics ; but, by considering selected typical
cases, we shall expound some of the principal methods employed, with
special reference to the uses of the transcendental functions.
18"2. Boundary conditions.
A problem which arises very frequently is the determination, for one of the
equations of § 18"1, of a solution which is subject to certain boundary con-
ditions ; thus we may desire to find the temperature at any point inside a
homogeneous isotropic conducting solid in thermal equilibrium when the
points of its outer surface are maintained at given temperatures. This
amounts to finding a solution of Laplace's equation at points inside a given
surface, when the value of the solution at points on the surface is given.
A more complicated problem of a similar nature occurs in discussing
small oscillations of a liquid in a basin, the liquid being exposed to the
atmosphere ; in this problem we are given, effectively, the velocity potential
18*2, 18*3] THE EQUATIONS OF MATHEMATICAL PHYSICS 381
at points of the free surface and the normal derivate of the velocity potential
where the liquid is in contact with the basin.
The nature of the boundary conditions, necessary to determine a solution
uniquely, varies very much with the form of differential equation considered,
even in the case of equations which, at first sight, seem very much alike.
Thus a solution of the equation
(which occurs in the problem of thermal equilibrium in a conducting
cylinder) is uniquely determined at points inside a closed curve in the
ary-plane by a knowledge of the value of V at points on the curve ; but
in the case of the equation
(which effectively only differs from the former in a change of sign), occurring
in connexion with transverse vibrations of a stretched string, where V
denotes the displacement at time t at distance x from the end of the
string, it is physically evident that a solution is determined uniquely only if
dV
both V and -^ are given for all values of x such that O^x^l, when ^ = 0
(where I denotes the length of the string).
Physical intuitions will usually indicate the nature of the boundary
conditions which are necessary to determine a solution of a differential
equation uniquely; but the existence theorems which are necessary from
the point of view of the pure mathematician are usually very tedious and
difficult*.
18"3. A general solution of Laplace's equation f.
It is possible to construct a general solution of Laplace's equation in
the form of a definite integral. This solution can be employed in solving
problems involving boundary conditions of a certain type.
Let V (x, y, z) be a solution of Laplace's equation which can be expanded
into a power series in three variables valid for points of {x, y, z) sufficiently
near a given point {x^, y^, Zq). Accordingly we write
X=Xo + X, y = y^+Y, z = Zo + Z;
and we assume the expansion
F= ao + a,X + b,Y+c,Z+ a^X'' + kY^ + c^Z"-
+ U^YZ-v'ie^ZX + 2/;ZF+ ...,
it being supposed that this series is absolutely convergent whenever
IZp + IFj^+lZl^^a,
* See e.g. Forsyth, Theory of Functions, pp. 442-459, where an apparently simple problem
is discussed.
t Whittaker, Math. Ann. lvii. (1902), p. 333.
382 THE TRANSCENDENTAL FUNCTIONS [CHAP. XVIII
where a is some positive constant*. If this expansion exists, V is said to
be analytic at {x^, y^, z^). It can be proved by the methods of §§ 3'7, 47
that the series converges uniformly throughout the domain indicated and
may be differentiated term-by-term with regard to X, Y or Z any number of
times at points inside the domain.
If we substitute the expansion in Laplace's equation, which may be
written
and equate to zero (§ 3*73) the coefficients of the various powers of X, Y
and Z, we get an infinite set of linear relations between the coefficients,
of which
(12 + ^2 + C2 = 0
may be taken as typical.
There are ^n{7i — l) of these relations f between the ^(n + 2){n + l)
coefficients of the terms of degree n in the expansion of V, so that there
are only ^ (n + 2) (n + 1) — ^'^^ (n — 1) = 2n + 1 independent coefficients in
the terms of degree n in V. Hence the terms of degree n in V must be
a linear combination of 2n + 1 linearly independent particular solutions of
Laplace's equation, these solutions being each of degree n in X, Y and Z.
To find a set of such solutions, consider (Z + iX cos u + iY sin u)'^; it is
a solution of Laplace's equation which may be expanded in a series of sines
and cosines of multiples of u, thus :
n n
S gm {X, Y, Z) cos mu + % hm {X, Y, Z) sin mu,
vi=0 m=l
the functions g^ (X, Y, Z) and h^ {X, Y, Z) being independent of u. The
highest power of Z in g^n {X, Y, Z) and h^ (X, Y, Z) is Z^~^ and the former
function is an even function of Y, the latter an odd function ; hence
the functions are linearly independent. They therefore form a set of
2n + 1 functions of the type sought.
Now by Fourier's rulej (§ 9"12)
irgm (X,Y,Z)=l {Z + iX cos u + iF sin uY cos mudu,
J -TT
Trillin (X, Y, Z)= \ (Z + iX cos u + iY sin uY sin mudu,
J —IT
* The functions of applied mathematics satisfy this condition.
t If a^,t,t (where r + s-\-t = n) be the coefBcieiat of Xi'Y^Z^ in V, and if the terms of degree
71 - 2 in jT— ^ + ^-^2 + s yl ^^ arranged primarily in powers of X and secondarily in powers of Y,
the coetticient a,.,8,< does not occur in any term after X^-'^Y'Z^ (or X^Y'~^Z* if r = 0 or 1), and
hence the relations are all linearly independent.
X 27r must be written for tt in the coefficient of go (X, Y, Z).
18-3] THE EQUATIONS OF MATHEMATICAL PHYSICS 883
and so any linear combination of the 2n+l solutions can be written in the
form
/:
(Z 4- iX cos u + iY sin w)"/n (w) du,
where /„ (u) is a rational function of e*".
Now it is readily verified that, if the terms of degree n in the expression
assumed for V be written in this form, the series of terms under the integral
sign converges uniformly if |Zp+|Fp+jZp be sufficiently small, and so
(§ 4*7) we may write
V= j t {Z + iX cos u + iY sin w)"/„ {u) du.
J -w n-0
But any expression of this form may be written
V = I F(Z + iX cos u + iF sin u, u) du,
J —w
where F is a function such that differentiations with regard to X, Y or Z
under the sign of integration are permissible. And, conversely, if F be any
function of this type, F is a solution of Laplace's equation.
This result may be written
V = I f(z + ix cos u + iy sin u, u) du,
J —77
on absorbing the terms —2o — '^^a cos u — iy^ sin u into the second variable ;
and, if differentiations under the sign of integration are permissible, this
gives a general solution of Laplace's equation ; that is to say, every solution
of Laplace's equation which is analytic throughout the interior of some
sphere is expressible by an integral of the form given.
This result is the three-dimensional analogue of the theoreni that
V=f{x-\-iy)+g{x-iy)
is the general solution of
dx^ dy'^
[Note. A distinction has to be drawn between the primitive of an ordinary differential
equation and general integrals of a partial dififerential equation of order higher than the
first*.
Two apparently distinct primitives are always directly transformable into one another
by means of suitable relations between the constants ; thus in the case of i o+y — ^, ^ve
can obtain the primitive Csin(A' + e) from A aoa x + B ^m x by defining C and e by the
equations Csin e = A, Gcose = B. On the other hand, every solution of Laplace's equation
is expressible in each of the forms
/ f{x cos t+y sin t + iz, t) dt, I g {i/ cos u + z sin u + ix, u) du ;
* For a discussion of general integrals of such equations, see Forsyth, Theory of Differential
Equations, Part iv. (Vol. vi.) Chap. xii.
384 THE TRANSCENDENTAL FUNCTIONS [CHAP. XVIII
but if these are known to be the same solution, there appears to be no general analytical
relation, connecting the functions / and g, which will directly transform one form of
the solution into the other.]
Example 1. Shew that the potential of a particle of unit mass at (a, 6, c) is
1 /""• du
2t ./ —:r{z — c)-\-i{x — a)cosu + i{i/-b)s.\nu
at all points for which z>c.
Example 2. Shew that a general solution of Laplace's equation of zero degree in
X, y, z is
/ log {x COS. t->rys.m.t + iz)g{t)dt, if /" g{t)dt = 0.
Express the solutions — — and log — ^ in this form, where r^^x^ + v^ + z^.
z+r r — z ' ' ^ '
Example 3. Shew that, in the case of the equation
^2 + g'' = X 4- ?/
(tiz 'bz\
where p = - , j = ^ j , integrals of Charpit's subsidiary equations (see Forsyth, Differential
Equations, Chap, ix.) are
(i) p^-x=y-q^=a,
(ii) p = q + a'^.
Deduce that the corresponding general integrals are derived from
(i) z=\{x + af + ^ij,-af + F{a)\
(ii) Az=l (.r+y)3 + 2a.2 (x -y) - a* {x+yy^ + G {a)\
0 = 4a (.r - y) - 4a3 (^•+3/)-i + 6-" (a)
and thence obtain a differential equation determining the function O (a) in terms of the
function F (a) when the two general integrals are the same.
18'31. Solutions of Laplace's equation involving Legendre functions.
If an expansion for V, of the form assumed in § 18-3, exists when
^0 — yo ~ ■2'o == ^j
we have seen that we can express V as a series of expressions of the type
I {z + ix cos u + iy sin u^ cos mu du, {z + ix cos u + iy sin uY sin mudu,
where n and m are integers such that O^vn^n.
We shall now examine these expressions more closely.
If we take polar coordinates, defined by the equations
X = r sin 6 cos 0, y = r sin 6 sin 0, z = r cos 6,
18-31] THE EQUATIONS OF MATHEMATICAL PHYSICS 385
we have
1 (z + ix cos u + iy sin w)" cos mudu
J -w
= r" I {cos d + i sin 6 cos (u — ^)}" cos mudu
J -n
= r'' {cos ^ + 1 sin ^ cos i/rj" cos m{<j> + yfr) d-yjr
J -TT — (J)
= r" I {cos 0 +i sin ^ cos yjr}^ cos ??i (0 + i/r) c?\/r
J -TT
= r" cos m<^ I {cos 6 + i sin ^ cos i/r}" cos mi/rc^-v^,
since the integrand is a periodic function of yfr and
(cos ^ + i sin d cos -v/r)" sin mi/r
is an odd function of yjr. Therefore (§ 15-61), with Ferrers' definition of the
associated Legendre function,
f "" . . . 27ri"'' . n '
I (z + ix cos 'M + iy sin m)"' cos mudu — ~, '— r''^Pn^ (cos 6) cos to</>.
Similarly
(z + ix cos w + iy sin m)** sin mudu = 7 ^i r'^PJ^ (cos ^) sin 7716.
;-,r -^ ' {n + m)\ ' ^ ^ ^
Therefore 7'"P,j™ (cos ^) cosm0 and r^P^^ (cos 6) sin mcj) are polynomials
m X, y, z and are particidar solutions of Laplace's equation. Further, by
§ 18'3, every solution of Laplace's equation, which is analytic near the origin,
can be expressed in the form
F = S r'' {AnPn (cos e)+ t (^,,'"'' cos m(f> + 5„<"*> sin m4>) P„"* (cos ^)| .
n = 0 t m=l J
Any expression of the form
AnPn (cos 6)+ 1 (^„<"'* cos mcf) + Bn^'^^ sin m(f)) Pn"^ (cos 6),
ni — \
where n is a positive integer, is called a surface harmonic of degree n ;
a surface harmonic of degree n multiplied by ?•"■ is called a solid harmonic
of degree n.
A solid harmonic of degree n is evidently a homogeneous polynomial of degree n in
X, y, z and it satisfies Laplace's equation.
It is evident that, if a change of rectangular coordinates* is made by rotating the axes
about the origin, a solid harmonic (or a surface harmonic) of degree n transforms into
a solid harmonic (or a surface harmonic) of degree n in the new coordinates.
92 ga 92
* The o2)erator ,— .^ + ^— ^ + 5-^ is invariant for changes of rectangular axes.
W. M. A. 25
386 THE TRANSCENDENTAL FUNCTIONS [CHAP, XVII I
Example. If coordinates r, 6, 0 are defined by the equations
x = rcoii6, y = (?-2 - 1 )* sin ^ cos ^, 2=(r2— l)*sin^sin^,
shew that P^'" (»') Pn"' (cos 6) cos mtf) satisfies Laplace's equation,
18"4. The solution of Laplace's equation which satisfies assigned boundary
conditions at the surface of a sphere.
We have seen (§ 18"31) that any solution of Laplace's equation which
is analytic near the origin can be expanded in the form
V(r,e,(}>)= i r'^j^^P, (cos ^)
+ 2 (AJ^^^ cos m(f) + Bn^""^ sin m(j>) Pn"" (cos 6)\ ;
and, from § 3*7, it is evident that if it converges for a given value of r,
say a, for all values of 6 and (f> such that O^^^tt, 0-^<f)^ 27r, it converges
absolutely and uniformly when r< a.
To determine the constants, we must know the boundary conditions
which V must satisfy. A boundary condition of frequent occurrence is
that F is a given bounded integrable function of 0 and cf), say f(0, </>), on
the surface of a given sphere, which we take to have radius a, and V is
analytic at points inside this sphere.
We then have to determine the coefficients An, -4„<'"', 5„<"'> from the
equation
f{6, </>) = 2 a'* \ AnPn (cos 6)^ t (^,,<'»> COS m<i> + 5„<'^» sin m<^) Pn'" (cos 6)
n=0 ( m=l
Assuming that this series converges uniformly* throughout the domain
O^d^TT, 0 ^ (/> ^ 27r,
multiplying by
Pn"^ (cos ef^^mcb,
sm ^
integrating term-by-term (§ 4'7) and using the results of §§ 15'14, 15'51 on
the integral properties of Legendre functions, we find that
r {^ f{d', <h') P„'« (cos 6') cos 7n<b' sin d'de'd<i>' = ira'' ^r^ . ("Ll^ A,:^\
J-nJi) zn+1 {n — m)l
r r f{e', f) Pn'^ (cos e') sin m<b' sin eWdS' - Tra'^ ,^-^, . ^''±'^^; 5,/"",
J —ttJ 0 zn + l (w — m)!
f " rf(d', cf>') Pn (cos 6') sin e'de'd<^' = 27ra" ^^-^ An.
J -TT J 0 zn + i
* This is usually the case in physical problems.
18-4] THE EQUATIONS OF MATHEMATICAL PHYSICS 387
Therefore, when r < a,
V{r, 6, «/,) = IJ^ (^)" f[Jlf(^' ^') {^n(«o« ^) ^«(co8 &)
+ 22 (l!li:!^ P^r^ (cos 6) Pn*" (cos 6') COS m (<^ - (^o[ sin ffdO'di^'.
The series which is here integrated term-by-term converges uniformly
when r < a, since the expression under the integral sign is a bounded
function of 6, 6', <^, <^', and so (§ 4*7 )
^'irV{r,d,<^)=r \^ f{e\4>') i (2n + l)(-Y'|p„(cos^)P,(cos^)
+ 22 ^^~^^'. PrT (cos (9) P„™ (cos d') COS m (<^ - <^')[ sin d'd&d4>'.
Now suppose that we take the line {6, (f>) as a new polar axis and let
i^iy 4>i) be the new coordinates of the line whose old coordinates were (6\ (j>') ;
we consequently have to replace P„ (cos 6) by 1 and P^"' (cos 6) by zero ; and
so we get
4>7rV(r,e,cf>)= f{0',4>') 2 (2w + l) -1 Pn{cos 6,') sin 6,' de,'d(f>,'
J-jrJo n=0 V'^/
7(^'> f) S (2n + l) - Pn{cose,')sm0'dd'd<}>'.
J— IT Jo »=0 \^/
If, in this formula, we make use of the result of example 23 of Chapter xv
(p. 326), we get
J -TT. 0 (r^ - 2ar cos ^i + a^f
and so
47rF(r, ^, 0)
/(^', (f>') sin 6''c^^'c^(^'
= a (a2 - ?'2)
77 fTT
-TT./ 0 [r^ - 2ar {cos ^ cos 6' + sin ^ sin 6' cos (<^ - <^')} + <^^]*'
In this compact formula the Legendre functions have ceased to appear
explicitly.
The last formula can be obtained by the theory of Greenes fimctions. For properties
of such functions the reader is referred to Thomson and Tait, Natural Philosophy,
§§ 499-519.
[Note. From the integrals for V {r, 6, (p) involving Legendre functions of cos ^i' and
of cos d, cos 6' respectively, we can obtain a new proof of the addition theorem for the
Legendre polynomial.
For let
Xn {ff, 0') - P„ (cos ^i') - IPn (cos 0) Pn (cOS d')
+ 22 ^/* ~ '"? ; Pn'" (cos 6) iV" (cos ff) COS m (<i - (b')\ ,
25—2
388 THE TRANSCENDENTAL FUNCTIONS [CHAP, XVIII
and we get, on comparing the two formulae for V{r, 0, (p),
J -TT J 0 n = 0 \"'/
If we take f{ff, 0') to be a surface harmonic of degree n, the term involving ?•" is the only
one which occurs in the integrated series ; and in particular, if we take/(^', 0') = _;(„ {6', 0'),
we get
fl„ jl ^^» ^^'' '^')^'^'" o'de'dcj>'=o.
Since the integrand is continuous and is not negative it must be zero; and so
Xn {&', 4>') = ^> that is to say we have proved the formula
Pn (cos 0^') = Pn (cos 6) Pn (cOS 6') + =l 2 ,' P,,"* (COS 6) P^T (cOS 6') COS m ((/) - </)'),
wherein it is obvious that
cos $1 — cos 6 cos ^' + sin ^ sin 6' cos {(f> — (f)'),
from geometrical considerations.
We have thus obtained a physical proof of a theorem proved elsewhere* (§ 15-7) by
purely analytical reasoning.]
Example 1. Find the solution of Laplace's equation analytic inside the sphere r=l
which has the value sin 3^ cos (p at the surface of the sphere.
[~^r^Ps^ (cos 6) cos (f) - ^rP^^ (cos 6) cos 0.]
Example 2. Let /„ (r, d, (f)) be equal to a homogeneous polynomial of degree n
in X, i/, z. Shew that
I fn (a, 6, (f)) Pn {cos 0 cos ^'+sin 6 sin 6' cos (0 - 0')} a^ sin 6ddd(p
-T J 0
4na^
= 2^1-^«^'''^''^^-
[Take the direction (d', 0') as a new polar axis.]
18'5. Solutions of Laplace's equation which involve Bessel coefficients.
A particular case of the result of § 18"3 is that
is a solution of Laplace's equation, k being any constant and m being any
integer.
Taking cylindrical-polar coordinates (p, </>, z) defined by the equations
X = p cos (/), y = p sin 0,
the above solution becomes
kz I ik p con (u- (i>) 7 kz I ikpcosv / , i\ 7
' e '^ ^ ' cos maau= e e cos m {v + (p) . dv
ct kz i ikp cos J' I 7
= ze / 6 cos mv COS) ncpdv
J 0
o /t'^ / I \ / ?/rpcos I' 7
= ie cos (m<p) e cos 7Hvay,
Jo
* The absence of the factor ( - )'» which occurs in § 15-7 is due to the fact that the functions
now employed are Ferrers' associated functions.
18-5, 18-51] THE EQUATIONS OF MATHEMATICAL PHYSICS 389
and so, using § 171 example 3, we see that 2'iri^"' ^'^ cos (m<l>) . Jm (kp) is a
solution of Laplace's equation analytic near the origin.
Similarly, from the expression
r gk iz+ixcosu+iysinu) gj^ mudu,
J -ir
where m is an integer, we deduce that 27ri"* e*^ sin (m^) . Jm {kp) is a solution
of Laplace s equation.
18"51. The periods of vibration of a uniform memh'ane*.
The equation satisfied by the displacement V at time ^ of a point {x, y) of a uniform
plane membrane vibrating harmonically is
827 dW_ 1 dW
dx^ ■*" 8^2 - c'i 8^2 '
where c is a constant depending on the tension and density of the membrane. The
equation can be reduced to Laplace's equation by the change of variable given by z = cti.
It follows, from § 18"5, that expressions of the form
T n \ '^OS , cos ,
satisfy the equation of motion of the membrane.
Take as a particular case a drum, that is to say a inembrane with a fixed circular
boundary of radius R.
Then one possible type of vibration is given by the equation
V=Jm (kp) cos m(f) cos ckt,
provided that F=0 when p = R; so that we have to choose k to satisfy the equation
This equation to determine k has an infinite number of real roots (§ 17 '3 example 3),
^1, ^2) '^S) ••• say. A possible type of vibration is then given by
F— /to (^rp)cos??i(^ cos c^^<. (»• = !, 2, 3, ...)
This is a periodic motion with period 2Tr/{ckr) ; and so the calculation of the periods
depends essentially on calculating the zeros of Bessel coeflficients (see § 17 "9).
Example. The equation of motion of air in a circular cylinder vibrating per-
pendicularly to the axis OZ of the cylinder is
82 F 82 F_ 1 82F
dx^'^d^^~~c^W'
V denoting the velocity potential. If the cylinder have radius R, the boundary condition
is that ^;— = 0 when p = R. Shew that the determination of the free periods depends on
finding the zeros of Jm {0 = ^-
* Poisson, Mem. de V Academic, viii. ; Bourget, Ann. de I'Ecole Normale, in. For a detailed
discussion of vibrations of membranes, see also llayleigh, Theory of Sound, Chapter ix.
390 THE TRANSCENDENTAL FUNCTIONS [CHAP. XVIII
18'6. A general solution of the equation of wave motions.
It may be shewn* by the methods of § 18"3 that a general solution of
the equation of wave motions
'ds^'^ dy^'^ dz^ ~ 6' dt^
IS
V=j I f(xsmucosv + ysiD.usinv + zcosu + ct,u,v)dudv,
J — n J — 7r
where /is a function (of three variables) of the type considered in § 18'3.
Regarding an integral as a limit of a sum, we see that a physical
interpretation of this equation is that the velocity potential V is produced
by a number of plane waves, the disturbance represented by the element
f(x sin u cos V + y sin u sin v +z cos u + ct, u, v) huhv
being propagated in the direction (sin u cos v, sin u sin v, cos u) with velocity c.
The solution therefore represents an aggregate of plane waves travelling in
all directions with velocity c.
18"61. Solutions of the equation of wave motions which involve Bessel
functions.
We shall now obtain a class of particular solutions of the equation of
wave motions, useful for the solution of certain special problems.
In physical investigations, it is desirable to have the time occurring by
means of a factor sin ckt or cos ckt, where k is constant. This suggests that
we should consider solutions of the type
V= i I gik ('^ ^^^u cosv-i- y sin u sin v+z cos u+ct) f/^ v) dudv
J -ttJ 0
Physically this means that we consider motions in which all the elementary waves
have the same period.
Now let the polar coordinates of (x, y, z) be (r, d, </>) and let (cu, ^\r) be the
polar coordinates of the direction {u, v) referred to new axes such that
the polar axis is the direction {6, <^), and the plane -y^^Q passes through
OZ; so that
cos ft) = COS 6 cos w + sin ^ sin u cos (^ — v),
sin w sin (^ — v) = sin tt> sin t/t.
Also, take the arbitrary function /(w, v) to be 8n{u,v)s\nu, where 8^
denotes a surface harmonic in u, v of degree n; so that we may write
S,Xu, v) = Sn(0,<f>; Q), yjr),
where (§ 18'31) S^ is a surface harmonic in co, i/r of degree n.
* See the paper previously cited, 3Iath. Ann. lvii. pp. 342-345, or Messenger of Mathe-
matics, XXXVI. pp. 98-106.
18*6-18-611] THE EQUATIONS OF MATHEMATICAL PHYSICS 391
We thus get
F= B^^ r r e**^«'^" S^ (0, <f>; 0), i/r) sin (cdcody^.
. J -irJ 0
Now we may write (§ 18*31)
S„ {6, <j>; co,ylr) = A, (6, cf>) . P„ (cos a>)
+ S {^n""* (6, <f>) cos my}r + £„<"" (^, <^) sin myjr} P^"* (cos w),
OT=l
where A„ (0, (j>), ^„<»»> {0, (f>) and 5„'"*> (0, <f>) are independent of i|r and «.
Performing the integration with respect to i/r, we get
V = 27re'*^^ A,, (0, <f>) r e'«^»-cos<- p^ (cos «) sin wc^w
J 0
= 27re'*'^^ J„ (^, (/,) j' ^ e''^'^ P. (/.) df.
= 2W*^^^„(^, <^)/'/^''^'^ 2^1^ (^^-ir^/*>
by Rodrigues' formula (§ 15'11); on integrating by parts n times and using
Hankel's integral (§ IT'S corollary), we obtain the equation
27r ,...„, . _ „. fi
F =
;p^ e''^' A, {0, 0) (iA;rr j ^ e^*'"'^ (1 - ^Lt^f d^l
= (27r)* *• V*''' (A;r) " ^ /^^^ (^r) J, (^, (^),
and so F is a constant multiple of e''"'*^ r'^J (kr) A^ {0, ^).
Now the equation of wave motions is unaffected if we multiply x, y, z
and t by the same constant factor, i.e. if we multiply r and t by the same
constant factor leaving 0 and 0 unaltered; so that il„(^, 4>) may be taken
to be independent of the arbitrary constant k which multiplies r and t
Hence lim e''''^* r k~^~ J^ , (kr) A „ (0, cf)) is a solution of the equation
of wave motions; and therefore r'^An(0, (f)) is a solution (independent of t)
of the equation of wave motions, and is consequently a solution of Laplace's
equation; it is, accordingly, permissible to take A,^{0, <^) to be any surface
harmonic of degree n ; and so we obtain the result that
r~^J , (kr) P,r (cos 0) ^°^ m6 ^°^ ckt
n + l^ > '' ^ ^ sm ^ sin
is a particular solution of the equation of wave motions.
18'611. Application of § 18"61 to a physical problem.
The solution just obtained for the equation of wave motions may be used in the
following manner to determine the periods of free vibration of air contained in a rigid
sphere.
392 THE TRANSCENDENTAL FUNCTIONS [CHAP. XVIII
The velocity potential V satisfies the equation of wave motions and the boundary
dV
condition is that -=r- = 0 when r = a, where a is the radius of the sphere. Hence
a?'
V= r'^J ,. ikr) P,™ (cos 6) ^^^ md) ^^^ ckt
'H-i \ ' 1^ \ ^ sin ^ sin
gives a possible motion if k is so chosen that
This equation determines k ; on using § 17*24, we see that it may be written in
the form
tan^a=/„(^a),
where /„ {ka) is a rational function of ka.
In particular the radial vibrations, in which V is independent of 6 and 0, are given by
taking n = 0; then the equation to determine k becomes simply
tan ka = ka ;
and the pitches of the fundamental radial vibrations correspond to the roots of this
equation.
REFERENCES.
J. Fourier, Theorie Analytique de la Chaleur. (Translated by A. Freeman.)
W. Thomson and P. G. Tait, Natural Philosophy.
Lord Rayleigh, Theory of Sound.
F. PocKELS, ilber die partielle Differentialgleichtmg Am + Fm = 0. (Leipzig, 1891.)
H. BuRKHARDT, EntivickeluTigen nach oscillirenden Funktionen. (Leipzig, 1908.)
H. Bateman, Electrical and Optical Wave-motion.
E. T. Whittaker, History of the Theories of Aether and Electricity.
A. E. IT. Love, Proc. London Math. Sac. xxx. pp. 308-.321.
H. Bateman, Proc. London Math. Soc. (2) i. pp. 451-458.
L. N. G. Filon, Philosophical Magazine (6) vi.' (1903), pp. 193-213.
Miscellaneous Examples.
1. If V be a solution of Laplace's equation which is symmetrical with respect to OZ,
and if V=f{z] on OZ, shew that if /{t} be a function which is analytic for a suitable
range of values of the complex variable ^, then
1 /"""
V=- f{z + i (.v- + y-f cos (j)] d(f)
'"'Jo'
at any point of a certain three-dimensional region.
THE EQUATIONS OF MATHEMATICAL PHYSICS 393
2. Deduce from example 1 that the potential of a uniform circular ring of radius c
and of mass M lying in the plane XO Y with its centre at the origin is
^n- - * r [cii + {a + 1 {x'- +/)i cos c/)}*]"* di\>.
J 0
3. If u be determined as a function of x, y and z by means of the equation
Ax-\-By-\-Cz = \,
where A, B, C are functions of u such that
A^-VB^ + C^ = 0,
shew that (subject to certain general conditions) any function of u is a solution of
Laplace's equation.
(Forsyth, Messenger of Mathematics, xxvii.)
4. A, B are two points outside a sphere whose centi'e is C. A layer of attracting
matter on the surface of the sphere is such that its surface density o-p at P is given by
the formula
o-pQC {AP.BP)-K
Shew that the total quantity of matter is unaffected by varying A and B so long as
CA . CB and ACB are unaltered ; and prove that this result is equivalent to the theorem
that the surface integral of two harmonics of different degrees taken over the sphere
is zero.
(Sylvester.)
5. Let V{x, y, z) be the potential function defined analytically as due to particles
of masses X + i'/x, X — ifi at the points {a + ia', b + ib', c + ic') and {a — ia, b — ib', c-ic')
respectively. Shew that V {x, y, z) is infinite at all points of a certain real circle, and
if the point {x, y, z) describes a circuit intertwined once with this circle the initial and
final values of V (x, y, z) are numerically equal, but opposite in sign.
(Appell.)
6. Find the solution of Laplace's equation analytic in the region for which a<r<A,
it being given that on the spheres r = a and r=A the solution reduces to
2 c„A(cos<9), 2 C„P„ (cos <9),
n=o M=0
respectively.
7. Let 0' have coordinates (0, 0, c), and let
Pdz=d, P(yZ=6', PO = r, PO'^r'.
Shew that
Pn (cos 6') ^ P,^(co^) , 1 N c^«+jt (cos_^) (n + l)(7t + 2) c^Pn^^ (cos^) ,
_ (_}_ rP, icosj) (n + l) (,^+2) r^-P, {cos 0)
according as r>c or r<.c.
Obtain a similar expansion for r'"/*,/ (cos ^). (Trinitj', 1893.)
8. At a point (r, 0, (f)) outside a uniform oblate spheroid whose semi-axes are a, b and
whose density is p, shew that the potential is
4n pa'^b
' 1 m' p., (cos 6) ?>i^ P^ (cos i
3r 3.5 /••' 5.7 ?"^
where m-=a'^ — b^ and r>m. Obtain the potential at points for which r<m.
(St John's, 1899.)
394 THE TRANSCENDENTAL FUNCTIONS [CHAP. XVIII
9. Shew that
gir cose_ 2 i™ (277)* {2n + 1) r"* P„ (cos 6) J„,. (r).
n=0 *
(Bauer, Crelle, LVi.)
10*. Shew that ]i x±iy=h cosh (| ± ?■»;), the equation of two-dimensional wave motions
in the coordinates | and 77 is
927 927 ^2 927
-9|2+s,-F=^(«o«h2^-cos^)-g^.
(Lame.)
11. Let x=-{c + r COS. 6) cos (l>, 1/ = {c + r cos 6) sin (f), z=rsm6;
shew that the surfaces for which r, d, <b respectively are constant form an orthogonal
system ; and shew that Laplace's equation in the coordinates r, 6, (p is
g-|r(c+rcos^)g^| + -^g^|(c + rcos^)g^^+^-p^:^^g^ = 0.
(W. D. Niven.)
12. Let P have Cartesian coordinates {x, y, z) and polar coordinates (r, 6, 0). Let
the plane F 02 meet the circle ;2;2-|-?/2=X-2, 2 = 0 in the points a, y ; and let
A
aPy = <u, log {PajPy) = o".
Shew that Laplace's equation in the coordinates a; <b, (f) is
B f sinh a- d V\ d ( sinho- dV] 1 ^-n-
30- \cosh o- — cos (i) da- j Bat (cosh a — cos to Ceo J sinh o- (cosh cr — cos «) 8<^^ '
F= (cosh a — cos co) cos 7i(o cos m0 P"''_ (cosh a).
and shew that a solution is
ish a- — cos ft)) C(
(Hicks, PAi7. Trans. CLXXii. p. 617 et seq.)
13. Shew that
(ii;2^.p2_27?pcos0 + c2)-i= 2 dk / e-"" J^ (kp) 6'^^'°''' cos mudu,
and deduce an expression for the potential of a particle in terms of Bessel functions.
14. Shew that if a, b, c are constants and X, n, v are confocal coordinates, defined as
the roots of the equation in e
.*'2 «2 ^2
a- + e b^+f c^ + f
then Laplace's equation may be written
where A^ = {a^ + X)^ (b^ + X)^ {c''- + \)K
(Lame.)
Examples 10, 11, 12 and 14 are most easily proved by using Lamp's result (Journal de
VEcole Polyt. xiv.) that if (X, /x, v) be orthogonal coordinates for which the line-element is given
by the formula {Sx)^ + (5ij)^+{5z)^ = {Hid\y^ + (H25ny^ + lH3Sv)'^, Laplace's equation in these
coordinates is
A /^Ha dV\,'d_ (HsH^ dV\ _a_ fH.H^ SV\_(.
d\\ Hi d\J^bfx\ Ho OfMj'^d.^y Hs dvj •
A simple method (due to W. Thomson, Camh. Math. Journal, iv.) of proving this result is
reproduced by Lamb, H^jdrodynamics (1906), p. 141.
THE EQUATIONS OF MATHEMATICAL PHYSICS 895
15. Shew that a general solution of the equation of wave motions is
V= j F(xGoa6+yBin6-\-iz, ^+izH'm d+ct COB 0, 6) dd.
(Bateman.)
16. If U=f{x, 1/, z, t)he & solution of
a* dt 8^-2 "'■ df "^ dz'^ '
prove that another solution of the equation is
^='-v(?'f'-:'-7)-K-^-'^>
17. Shew that a general solution of the equation of wave motions, when the motion is
independent of ^, is
I f{z-\-ip COS 6, ct+p sin 6) dd
+ / / arc smh ( ^V—; F(a, 6) dOda,
where p, cf), z are cylindrical coordinates and a, b are arbitrary constants.
(Bateman.)
18. If V=f{x, y, z) is a solution of Laplace's equation, shew that
1 ./r^-aP- r^-\-d?- az \
{x — iy)^ \^{x-iy) 2i{x — iy) x — iyj
is another solution. (Bateman.)
19. If U'=f(x, y, z, t) is a solution of the equation of wave motions, shew that
another solution is
1 / X y r^- 1 r^ + l \
~ z-cr\z-ct'' z-ct' 2{z-ct)' 2c{z-ct))'
(Bateman.)
20. If l=x — iy, 7n = z + iw, n=^x^+y^ + z^ + ^v^,
\=x + ty, p. = z-iw, v=-l,
so that l\+7np. + nv=0,
shew that any homogeneous solution, of degree zero, of
dH7 dHJ d'^U (>'^U_Q
dx"^ dy'^ dz'^ dvP'
d'^U d'^U d'-U ^
satisnes ;rj^^ + ~ — -- + ^ >" = ^ j
oloX dm Op. oncv
and obtain a solution of this equation in the form
ia, h, c \
«, i3, y, c[,
where ZX = (6-c) (f-a), 7np = {c-a){(-b), nv = {a-b){C-c).
(Bateman, Proc. London Math. Soc. (2) vii.)
<J96 THE TRANSCENDENTAL FUNCTIONS [CHAP. XVIII
21*. If (r, 0, cf)) are spheroidal coordinates, defined by the equations
x=c(r^ + lfsmd cos(f), y=c {r^ + lf sin 0 sin cf), z=crcos6,
where x, y, z are rectangular coordinates and c is a constant, shew that, when n and m are
integers,
/■f „ f xcost + y sin t+iz\ cos ^ , _, (n — m)! „,„,.> „ ™ / /,\ cos ,
/ Pn[ ^^ ^'— ) . mtdt = ^iT\ (- P^'" (zr) Pn™ (cos (9) . md).
(Blades, Proc. Edinburgh Math. Soc. xxxiii.)
22. With the notation of example 21, shew that, if 2=t=0,
l" Q,, A-co^^+y^in^+M cos (^^-m^ ^^^ cos ^
j _,r V c / sm (?i. + m) ! ^" v ; n v '' sin ^
(Jeffery.)
* The functions introduced in examples 21 and 22 are known as internal and external
spheroidal harmonics respectively.
CHAPTER XIX
MATHIEU FUNCTIONS
19*1. The differential equation of Mathieu.
The preceding five chapters have been occupied with the, discussion of
functions which belong to what may be generally described as the hyper-
geometric type, and many simple properties of these functions are now well
known.
In the present chapter we enter upon a region of Analysis which lies
beyond this, and which is, as yet, only very imperfectly explored.
The functions which occur in Mathematical Physics and which come
next in order of complication to functions of hypergeometric type are
called Mathieu functions ; these functions are also known as the functions
associated with the elliptic cylinder. They arise from the equation of two-
dimensional wave motion, namely
dx^ dy"^ c- df^
This partial differential equation occurs in the theory of the propagation of electro-
magnetic waves ; if the electric vector in the wave-front is parallel to OZ and if E denotes
the electric force, while {H^, Hy, 0) are the components of magnetic force. Maxwell's
fundamental equations are
}_dE^_dHy_dff^ dH^__dE dHy_dE
c^ dt dx cy ' ct 8y ' 3« ~ 3a; '
c denoting the velocity of light ; and these equations give at once
C^ Ct'^ ex- C'tf' '
In the case of the scattering of waves, propagated parallel to OX, incident on an
elliptic cylinder for which OX and OY are axes of a principal section, the boundary
condition is that E should vanisli at the surface of the cyliuder.
The same partial differential equation occurs in connexion with the vibrations of
a uniform i)lane membrane, the dependent variable being the displacement perpendicular
to the membrane ; if the membrane be in the shape of an ellipse with a rigid boundary,
the boundary condition is the same as in the electromagnetic problem just discussed.
398 THE TRANSCENDENTAL FUNCTIONS [CHAP. XIX
The differential equation was discussed by Mathieu* in 1868 in connexion
with the problem of vibrations of an elliptic membrane in the following
manner :
Suppose that the membrane, in the plane XOY, is vibrating with
frequency^. Then writing V=u(x, y)cos{pt + €), the equation becomes
d-u d^u p" _
Let the foci of the elliptic membrane be (+ A, 0, 0), and introduce new
real variables f f, t] defined by the complex equation
x-\-iy = h cosh {^ + i-q),
so that x = h cosh ^ cos r), y = h sinh ^ sin 97.
The curves, on which ^ or 77 is constant, are evidently ellipses or hyper-
bolas confocal with the boundary ; if we take ^ ^ 0 and — tt < 77 < tt, to each
point {x, y, 0) of the plane corresponds one and only one:}: value of (^, rf).
The differential equation for u transforms into§
P + 0 + -*y(cosh'|-cos',)» = O.
If we assume a solution of this equation of the form
where the factors are functions of ^ only and of 77 only respectively, we see
that
Since the left-hand side contains f but not 77, while the right-hand side
contains 77 but not ^, F(^) and 6^(77) must be such that each side is a constant,
A, say, since f and 77 are independent variables.
We thus arrive at the equations
drj''
-(-^cos^V-A)g(v) = 0.
* Liouville^s Journal, ser. 2, t. xiii. p. 137.
t The introduction of these variables is due to Lam(^, who called f the thermometric ])ara-
meter. They are more usually known as confocal coordinates. See Lam^, Sur les fonctions
inverses des transcendantes, lere Lecjon.
J This may be seen most easily by considering the ellipses obtained by giving f various
positive values. If the elUpse be drawn through a definite point (f, rj) of the plane, 7? is the
eccentric angle of that point on the ellipse.
§ A proof of this result, due to Lame, is given in numerous text-books ; see p. 394, footnote.
1911, 1912] '■^ MATHIEU FUNCTIONS 399
By a slight change of independent variable in the former equation, v/e see
that both of these equatimis are linear differential equations, of the second
order, of the form
d^u
-T-j + (a + IQq cos 2z) w = 0,
where a and q are constants*. It is obvious that every point (infinity ex-
cepted) is a regular point of this equation.
This is the equation which is known as Mathieu's equation and, in certain
circumstances (§ 19'2), particular solutions of it are called Mathieu fvmctions.
19 "11. The form of the solution of Mathieu's equation.
In the physical problems which suggested Mathieu's equation, the constant
a is not given a priori, and we have to consider how it is to be determined.
It is obvious from physical considerations in the problem of the membrane
that u{x, y) is a one-valued function of position, and is consequently unaltered
by increasing 77 by 27r ; and the conditionf G(7} + 27r) = G (97) is sufficient to
determine a set of values of a in terms of q. And it will appear later (§§ 19'4,
19*41) that, when a has not one of these values, the equation
G{r} + 2'7r) = G{v)
is no longer true.
When a is thus determined, q (and thence p) is determined by the fact
that F{^) = 0 on the boundary; and so the periods of the free vibrations of
the membrane are obtained.
Other problems of Mathematical Physics which involve Mathieu functions in their
solution are (i) Tidal waves in a cylindrical vessel with an elliptic boundary, (ii) Certain
forms of steady vortex motion in an elliptic cylinder, (,iii) The decay of magnetic force in
a metal cylinder J. The equation also occurs in a problem of Rigid Dynamics which
is of general interest §.
1912. HiWs equation.
A differential equation, similar to Mathieu's but of a more general nature,
arises in G. W. Hill's || method of determining the motion of the Lunar
Perigee, and in Adams' IT determination of the motion of the Lunar Node.
Hill's equation is
d'hi
dz'
+ ($0 + 2 ^ On cos 27iz\ u = 0.
* Their actual values are a — A- h^p^l(2c^), q = h'^2^"l{32c^) ; the factor 16 is inserted to avoid
powers of 2 in the solution.
t An elementary analogue of this result is that a solution of -^ +au — 0 has period 27r if, and
only if, a is the square of an integer.
X li. C. Maclaurin, Trans. Camb. Phil. Soc. xvii. p. 41.
§ A. W. Young, Proc. Edinburgh Math. Soc. xxxii. p. 81.
Ii Acta Mathematica, viii. (1886). Hill's memoir was originally published in 1877 at
Cambridge, U.S.A.
11 Monthly Notices R. A. S. xxxviii. p. 43.
400 THE TRANSCENDENTAL FUNCTIONS [CHAP. XIX
The theory of Hill's equation is very similar to that of Mathieu's (in spite
of the increase in generality due to the presence of the infinite series), so the
two equations will, to some extent, be considered together.
In the astronomical applications 6q, 6^, ... are known constants, so the
problem of choosing them in such a way that the solution may be periodic
does not arise. The solution of Hill's equation in the Lunar Theory is, in
fact, not periodic.
19'2. Periodic solutions of Mathieu's equation.
We have seen that in physical (as distinguished from astronomical)
problems the constant a in Mathieu's equation has to be chosen to be such
a function of q that the equation possesses a periodic solution.
Let this solution be G {z) ; then G {z), in addition to being periodic, is an
integral function of z. Three possibilities arise as to the nature of G (z) :
(i) G (z) may be an even function of z, (ii) G (z) may be an odd function of z,
(iii) G (z) may be neither even nor odd.
In case (iii), ^ [Gr {^) + Gr (— z)}
is an even periodic solution and
^\G{z)-G(-z)}
is an odd periodic solution of Mathieu's equation, these two solutions forming
a fundamental system. It is therefore sufficient to confine our attention to
periodic solutions of Mathieu's equation which are either even or odd. These
solutions, and. these only, will be called Mathieu functions.
It will be observed that, since the roots of the indicia! equation at 2 = 0 are 0 and 1,
two even (or two odd) periodic solutions of Mathieu's equation cannot form a fundamental
system. But, so far, there seems to be no reason why Mathieu's equation, for special
values of a and q, should not have one even and one odd periodic solution ; for com-
paratively small values of \ q \ it can be seen [§ 19-3 example 2, (ii) and (iii)] that
Mathieu's equation has two periodic solutions only in the trivial case in which q — 0; but
for larger values oi \q\ there may be pairs of periodic solutions, though no such pairs
have, as yet, been discovered.
19'21. An integyxil equation satisfied by even Mathieu functions* .
It will now be shewn that, if G (77) is any even Mathieu function, then
G (r]) satisfies the homogeneous integral equation
J —IT
where k = \J{^2q). This result is suggested by the solution of Laplace's
equation given in § 18"3.
* This integral equation and the expansions of § 19-3 were published by Whittaker, Proc.
Int. Congress of Math. 1912. The integral equation was known to him as early as 1904, see
'Trans. Cavih. Phil. Sac. xxr. p. 193.
19-2-19-22] MATHIEU FUNCTIONS 401
For, if x+ iy = h cosh (| + ir]) and if F(^) and 0 (rj) are solutions of the
differential equations
^^ -(A + m^h^ cosh« ^)F(^) = 0,
_^^ + (^ + ^.;,. cos^ ^) (r (v) = 0,
then, by § 19-1, F(^) G (v) e"*'^ is a particular solution of Laplace's equation.
If this solution is a special case of the general solution
I f(h cosh ^ cos r)Cosd + h sinh ^ sin ?; sin ^ + 1^;, 0) dO,
•I —IT
given in § 18-3, it is natural to expect that*
where j> (0) is a function of 0 to be determined. Thus
F (f ) G (7}) e^i' = r F{0) (f> (6) exp [mh cosh ^ cos v cos ^
J — TT
+ 7yi^ sinh | sin ?; sin ^ + miz] dd.
Since | and 77 are independent, we may put ^ = 0 ; and we are thus led to
consider the possibility of Mathieu's equation possessing a solution of the
form
G(7))= I e™^°°«''^°«« (P (d) dd.
J —TT
19-22. Proof that the even Mathieu functions satisfy the integral equation.
It is readily verified (§ 5-31) that, if 0 {d) be analytic in the range (- tt, tt)
and if G^ (77) be defined by the equation
G (77) = I e^w'^cosrjcose ^ (^Q^ fi0^
then G (77) is an even periodic integral function of 77 and
d'G (ri) , , , _
-^H^ + {A+ nfh- cos^ 77) G (v)
= I [m'/i- (sin2 77 cos"-^ 0 + cos'-' 77) - mh cos 77 cos ^ + ^} g'"'* cos,, cose ^ (^^ ^^c^
= - |m/i sin 6 cos 'r](f)(6)+ ^' {6)] g^A cos,, cose |
L J-TT
+ I {</)" (^) + ( J + m^/i^ cos- 0) 0 (6')} e'«/'^°«')cose ^iq^
J —TT
on integrating by parts.
* The constant i^(0) is inserted to simplify the algebra.
W. M. A. 26
402 THE TRANSCENDENTAL FUNCTIONS [CHAP. XIX
But if (f> (6) be a -periodic function (with period 27r) such that
0" ($) + (A+ m'h' cos^ 6) </> (6) = 0,
both the integral and the integrated part vanish ; that is to say, G {-q), defined
by the integral, is a periodic solution of Mathieu's equation.
Consequently G (r)) is an even periodic solution of Mathieu's equation if
<j) {0) is a periodic solution of Mathieu's equation formed with the same con-
stants ; and therefore <p (^) is a constant multiple of G (6) ; let it be \G (6).
[In the case when the Mathieu equation has two periodic solutions, if this case exist,
we have (p {0) = }<& (6) + Oi (6) where (rj (6) is an odd periodic function ; but
vanishes, so the subsequent work is unaffected.]
If we take a and q as the parameters of the Mathieu equation instead of
A and mh, it is obvious that mh — \J{'.i2q) = k.
We have thus proved that, if G{r^) be an even periodic solution of
Mathieu's equation, then
G{'n)^\ r e^^^^^^''^^ G{d)de,
J -TT
which is the result stated in § 19'21.
From I 11 •23, it is known that this integral equation has a solution only
when \ has one of the ' characteristic values.' It will be shewn in § 19'3 that
for such values of A,, the integral equation affords a simple means of con-
structing the even Mathieu functions.
Example 1. Shew that the odd Mathieu functions satisfy the integral equation
G{r]) = X j sin (/• sin rjHm6)G(d) d6.
Example 2. Shew that both the even and the odd Mathieu functions satisfy the
integral equation
G{r^) = \i" e'''^''''>^'''^G{d)dd.
Example 3. Shew that when the eccentricity of the fundamental ellipse tends to zero,
the confluent form of the integral equation for the even Mathieu functions is
J,, {j;) =~ I" e'^ ^o** ^ cos n6 dd.
19"3. The construction of Mathieu functions.
We shall now make use of the integral equation of § 19"21 to construct
Mathieu functions; the canonical form of Mathieu's equation will be taken as
d-u ^ ^, _, ^
-r-:, + [a + Ibq COS 2z) u = 0.
19-3] MATHIEU FUNCTIONS 403
In the special case when q is zero, the periodic solutions are obtained by
taking a - ti^, where n is any integer; the solutions are then
1, cos Z', cos 23', ...,
sin z, sin 2z, ....
The Mathieu functions, which reduce to these when ^ -^ 0, will be called
ceo {z, q), cei {z, q), ce^ (z, q), ...,
se^{z, q), se^iz, q), ....
To make the functions precise, we take the coefficients of cos nz and sin nz
in the respective Fourier series for ce„ (z, q) and sen {z, q) to be unity. The
functions cen{z, q), sen{z, q) will be called Mathieu fanctions of order n.
Let us now construct ce^ (z, q).
Since ceo(^, 0) = 1, we see that X-^(27r)~^ as ^-^0. Accordingly we
suppose that, for general values of q, the characteristic value of X which gives
rise to ce^ {z, q) can be expanded in the form
(27rX)-i =l+G,q+a,q'' + ...,
and that ceo {z, q) = l+ q^^ {z) + q^^o_{z) + ...,
where a^, a^, ... are numerical constants and iS-^{z), ^^{z), ... are periodic
functions of z which are independent of q and which contain no constant
term.
On substituting in the integral equation, we find that
{l + a,q + a,q'+...)[l-^q^,{z) + f^,{z)-\-...]
1 r
= ^\ {1 + \/(325) . cos ^ cos ^ + lQqGo^^zcos''6 + ...]
X [i + q^,{d)-^f^,{d) + ...]de.
Equating coefficients of successive powers of q in this result and making
use of the fact that /3i {z), ^^ {z), . . . contain no constant term, we find in
succession
a^ = 4, /?! {z) = 4 cos 2z,
02 = 14, /32 {z) = 2 cos 4^,
and we thus obtain the following expansion :
/ N -, (a oo , 2^29 . \ ^ /^ , 160 , \
ce^{z, q) = l + [^q - 2,%q^ -^ ——- q' - ...jcos2z+ i2q--~^q'+ ...j cos 4^
+ (-^q^ - -w- q"^ + . . .] cos Gz + ( |o?^- •••) cos 8^
/" 1 '^
,225
the terms not written down being 0 (r/) as q ^0.
910 29
The value of a is -32g-+2245*- ' q*' + 0{q^); it will be observed
that the coefficient of cos 2z in the series for cBq (z, q) is — a/(Sq).
26—2
404 THE TRANSCENDENTAL FUNCTIONS [CHAP. XIX
The Mathieu functions of higher order may be obtained in a similar
manner from the same integral equation and from the integral equation of
§ 19'22 example 1. The consideration of the convergence of the series thus
obtained is postponed to § 19'61.
Example 1. Obtain the following expansions* :
« -»(^. «) = ' + ..!, {ttJt - ^rltXtty'^Oif*')] cos 2.,
(ii) ce, {z, q) = cosz+ ^l^\^^r^Y)lVl " (r+ 1) ! (.+ 1) !
(iii) ..,(.,^) = sm. + ^2^|^-^;^-^^, + ^_^^^,^^._^^^,
(iv) ce.2 {z, ?) = I - 2(? + ^ j3 + 0 (25)| + cos 2^
'^4iV!(r + 2)!+ 32.(r + 2)!(r + 3)! +^^^ )j cos (2r+2).,
where, in each case, the constant implied in the symbol 0 depends on r but not on z.
(Whittaker.)
■ Example 2. Shew that the values of a associated with (i) ce^ (z, q), (ii) cei {z, q),
(iii) sei {z, q), (iv) ce., {z, q) are respectively :
310 29
(i) -32(^H22V Q—q^' + Oiq^),
(ii) l-8q-8q' + 8q^-^q*+0iq%
(iii) l+8q-&q'^-8q'-^q* + 0iq'),
(iv) 4 + ^ j2 _ 6104 ^, _^ ^ ^^,^_ (Mathieu.)
Example 3. Shew that, if ?i be an integer,
Oe2n + 1 (2, ?) = ( - )" «<^2» + 1 (2 + i-T, - g).
19'31. The integral formulae for the Mathieu functions.
Since all the Mathieu functions satisfy a homogeneous integral equation
with a symmetrical nucleus (§ 19'22 example 3), it follows (§ 11-61) that
I ce„, {z, q) cen {z, q)dz=0 (m ^ w),
I 5e,„ (^, fy) 5e» (z, q) dz = 0 (m ^jt 71),
ce,n. (z, q) sen (2, q) dz = 0.
* The leading terms of these series, as given in example 4 at the end of the chapter (p. 420),
were obtained by Mathieu.
19-31, 19-4] MATHIEU FUNCTIONS 405
Example 1. Obtain expansions of the form :
(i) JC COS Z COS 6 ^ I ^^^^ (^^ ^) ^^^ (^^ ^^^
«=0
00
(ii) cos {k sin « sin 6)= 2 BnCCn (z, q) cen {0, q),
n=o
(iii) sin (^ sin 2 sin 6)= 2 CnSen {z, q) sen {6, q),
where Jl; = ^{32q).
Example 2, Obtain the expansion
as a confluent form of expansions (ii) and (iii) of example 1.
19'4. The nature of the solution of Mathieus general equation ; Floquet's
theory.
We shall now discuss the nature of the solution of Mathieu's equation
when the parameter a is no longer restricted so as to give rise to periodic
solutions ; this is the case which is of importance in astronomical (as opposed
to physical) applications of the theory.
The method is applicable to any linear equation with periodic coefficients
which are one-valued functions of the independent variable ; the nature of
the general solution of particular equations of this type has long been per-
ceived by astronomers, by inference from the circumstances in which the
equations arise. These inferences have been confirmed by the following
analytical investigation which was published in 1883 by Floquet*.
Let g (z), h (z) be a fundamental system of solutions of Mathieu's equation
(or, indeed, of any linear equation in which the coefficients have period 27r) ;
then, if i''(2;) be any other integral of such an equation, we must have
F(z) = Ag(z) + Bh(z),
where A and B are definite constants.
Since g(z + 2'7r), h(z + 27r) are obviously solutions of the equation f, they
can be expressed in terms of the continuations of g (z) and Ii (z) by equations
of the type
g{z + 277) = a,g (z) + a,h (z), h (z + 2it) = ^^g (^) + ^,/i (^),
where Wj, a^, /3i, ^^ are definite constants ; and then
F{z^ 2it) = (^Oi + B^^) g {z) + (^a, + i?/3,) h {z).
* Ann. dc VEcolc Normale (2), xiii. p. 47. Floquet's analysis is a natural sequel to Picard's
theory of differential equations with doubly-periodic (§ 'iO'l) coefficients, and to the theory
of the fundamental equation due to Fuchs and Hamburger.
t These solutions may not be identical with (j (z), h (z) respectively, as the solution of an
equation with periodic coefficients is not necessarily periodic. To take a simple case, u = t'^siu z
is a solution of — (l + cot^) u = 0.
406 THE TRANSCENDENTAL FUNCTIONS [CHAP. XIX
Consequently F (z -[■ 27r) = kF{z), where h is a constant*, if A and B are
chosen so that
Aa, + B^, = kA, Aa, + B^, = kB.
These equations will have a solution, other than A =B = 0, if, and
only if,
k-k, /3i 1=0;
0-2 , ^2—k\
and if k be taken to be either root of this equation, the function F (z) can be
constructed so as to be a solution of the differential equation such that
F(z+27r) = kF(z).
Defining /x by the equation k^e^''*^ and writing 4^{z) for e~>^^F{z), we see
that
0 (^ + 2-77) = e-" t^+^"' F{z + 27r) - </) {z).
Hence the differential equation has a particular solution of the form
e'^^ (p (z), where (f) (z) is a periodic function with period 27r.
We have seen, that in physical problems, the parameters involved in the
differential equation have to be so chosen that ^ = 1 is a root of the quadratic,
and a solution is periodic. In general, however, in astronomical problems, in
which the parameters are given, k^l and there is no periodic solution.
In the particular case of Mathieu's general equation or Hill's equation, a
fundamental system of solutionsf is then e'*^ (f> (z), e"*^^ <j> (— z), since the
equation is unaltered by writing — z for ^ ; so that the complete solution of
Mathieu's general equation is then
u = Cje'*^ ^ {z) + c^c'^ip (- z),
where Ci, Cg are arbitrary constants, and yu. is a definite function of a and q.
Example. Shew that the roots of the equation
a^-k, /3i =0
are independent of the particular pair of solutions, g {z) and h (z), chosen.
19'41. Hill's method of solution.
Now that the general functional character of the solution of equations
with periodic coefficients has been found by Floquet's theory, it might be
expected that the determination of an explicit expression for the solutions of
Mathieu's and Hill's equations would be a comparatively easy matter; this
however is not the case. For example, in the particular case of Mathieu's
general equation, a solution has to be obtained in the form
y^ei'^cf) (z),
* The symbol k is used in this particular sense only in this section. It must not be confused
with the constant ^ of § 19'21, which was associated with the parameter q of Mathieu's equation.
+ The ratio of these solutions is not even periodic ; still less is it a constant.
19-41] MATHIEU FUNCTIONS 407
where ^ (z) is periodic and ^t is a function of the parameters a and q. The
crux of the problem is to determine fi ; when this is done, the determination
of ^(^) presents comparatively little difficulty.
The first successful method of attacking the problem was published by
Hill in the memoir cited in § 19'12; as the method for Hill's equation is no
more difficult than for the special case of Mathieu's general equation, we shall
discuss the case of Hill's equation, viz.
^: + j-(.)«=o,
where J (z) is an even function of z with period tt. Two cases are of interest,
the analysis being the same in each :
(I) The astronomical case when z is real and, for real values of z, J{z)
can be expanded in the form
J(z) =$0 + 2^1 cos 2z + 202 cos 4!Z + 2^3 cos 6^ + . . . ;
CO
the coefficients 6n are known constants and X 6n converges absolutely.
(II) The case when ^ is a complex variable and J(z) is analjrtic in a
strip of the plane (containing the real axis), whose sides are parallel to the
real axis. The expansion of J{z) in the Fourier series 6o + 2 % On cos 2nz
is then valid (§ 9*1 1) throughout the interior of the strip, and, as before,
00
2 On converges absolutely.
Defining Q_n to be equal to On, we assume
00
ri,= - 00
as a solution of Hill's equation.
[In case (II) this is the solution analytic in the strip (!:$§ 10-2, 19-4); in case (I) it will
have to be shewn ultimately (see the note at the end of § 19-42) that the values of 6„
which will be determined are such as to make 2 u-6„ absolutely convergent, in order to
n— — 'x>
justify the processes which we shall now carry out.]
On substitution in the equation, we find
i (/^+2m)=6„e<'^+^»^'^ + ( ^ ^«e^'"^l ( ^ 6„e"^+-'»'^) = 0.
Multiplying out the absolutely convergent series and equating coefficients
of powers of e^^^ to zero (§§ 9-6-9-632), we obtain the system of equations
(/x + 2m)-^6„+ i ^m&«-m = 0 (n=..., -2, -1,0, 1, 2, ...).
408
THE TRANSCENDENTAL FUNCTIONS
[chap, XIX
If \ve eliminate the coefficients 6„ determinantally (after dividing the
typical equation by 6^ — 4<n- to secure convergence) we obtain * Hill's deter-
minantal equation :
(t/t + 4)2-
Oq -<9i
-02
4^^-00
-0z
4'' -00
-0i
•■• 42-^0
4^-^o
4'' -00 '"
-Si
(Z/X + 2)2-^0
22-^0
-01
-02
-0z
- 22-^0
2' -00
^'-00
2-^-e, -
-62
-01
{it^f-00
O-'-0o
-01
-02
'" 02-^0
o-'-Oo
o-'-e.
02-^0 ■■•
-^3
-02
-01
(i>-2)2-
22-^0
00 —01
- 2-^-^0
■P-80
2^-^o ■••
-^4
-03
4:-' -00
-02
-01
(2>- 4)2-^0
••• 4^-^o
4'' -00
42-^0
42-^0 •■■
We write A (^/u,) for the determinant, so the equation determining fi is
A (ifi) = 0.
19'42. The evaluation of Hill's determinant
We shall now obtain an extremely simple expression for Hill's deter-
minant, namely
A (ifji) = A (0) - sin'^ (h'""^/^) cosec- {^tt \/6o)-
Adopting the notation of § 2 "8, we write
A(*» = [^^,„],
(ifj,-2my~-e.
where A m, m —
The determinant [vl„,_,i] is only conditionally convergent, since the product
of the principal diagonal elements does not converge absolutely (§§ 2"81, 2'7).
We can, however, obtain an ahsolutely convergent determinant, Ai {ifi), by
dividing the linear equations of § 19"41 by 0Q-(i/u,— 2}if instead of dividing
by ^0 — 4h-. We write this determinant Aj {ifju) in the form [5„i,,i], where
^m.m — J-) J^n
{2m-iiJ.y-e,
(ni ^ n).
The absolute convergence of S On secures the convergence of the deter-
»=o
minant \_Bm,ii], except when fx has such a value that the denominator of one
of the expressions i^,„,,i vanishes.
* Since the coefficients 6„ are not all zero, we may obtain the infinite determinant as the
eliminant of the system of linear equations by multiplying these equations by suitably chosen
cofactors and adding up.
19-42] MATHIEU FUNCTIONS 409
From the definition of an infinite determinant (§ 2*8) it follows that
A(i^)=A,o»iim fi f'-/^:!"n,
p-».oo n=—p ( "q *»* )
and so A(z^)- A,0^) shi^O^^V^^ •
Now, if the determinant Aj (lyu,) be written out in full, it is easy to see
(i) that Ai (i/x) is an even periodic function of /x with period 2i, (ii) that Ai (ifi)
is an analytic function (cf §§ 2-81, 3*34, 5-3) of fi (except at its obvious simple
poles), which tends to unity as the real part of //. tends to + oo .
If now we choose the constant K so that the function D (/j,), defined by
the equation
D (fji) = Ai (ifl) - K {cot ^ TT (*> + V^o) - cot ^TT {ifi - ^/^,)],
has no pole at the point /i = iV^o, then, since i) (/u,) is an even periodic
function of //,, it follows that D (/x) has no pole at any of the points
2/11 ±i \/6o,
where n is any integer.
The function D {/j,) is therefore a periodic function of fj, (with period 2i)
which has no poles, and which is obviously bounded as R (/x.) -* + oo . The
conditions postulated in Liouville's theorem (§ 5"63) are satisfied, and so D (fju)
is a constant ; making /z. -^ + oo , we see that this constant is unity.
Therefore
Ai (i» = 1 + K [cot ^TT (^> + V^o) - cot I TT {{/M - s/e,)],
and so
A /,• \ sin|7r(tV-V6>n)sin|7rO> + V^o) , ^ ^^ , ,, ...
A M = sin-(i^v^o) + ^^ '"' ^*" ^^"^-
To determine K, put yu, = 0 ; then
A(0) = l + 2^cot(i7rV6'o).
Hence, on subtraction,
A(^^) = A(0)-4-^^^^*'^^■^>
sin2(i7r\/6'o)'
which is the result stated.
The roots of Hill's deterviinantal equation are therefore the roots of the
equation
sin^ i^irifM) = A (0) . sin- (i ir V^o)-
When fi, has thus been determined, the coefficients hn can be determined
in terms of &„ and cofactors of A (i/x) ; and the solution of Hill's differential
equation is complete.
410 THE TRANSCENDENTAL FUNCTIONS [CHAP. XIX
[In case (I) of § 19*41, the convergence of 2 | 6» | follows from the rearrangement theorem
00
of § 2"82 ; for 27^^ ] 6„ | is equal to | ftg I 2 | Cm,o I -^ | C'o.o I j where Cm,n is the cofactor of B^n
m = — <xi
in Ai {ijjL) ; and 2 | C,„_o I is the determinant obtained by replacing the elements of the row
through the origin by numbers whose moduli are bounded.]
It was shewn by Hill that, for the purposes of his astronomical problem, a remarkably
good approximation to the value of /x could be obtained by considering only the three
central rows and columns of his determinant.
19'5. The Lindemaym- Stieltjes theory of Mathieus general equation.
Up to the present, Mathieu's equation has been treated as a linear
differential equation with periodic coefficients. Some extremely interesting
properties of the equation have been obtained by Lindemann* by the sub-
stitution ^=cos^^, which transforms the equation into an equation with
rational coefficients, namely
4^(1-0^' + 2 (l-2O^ + (a-16^ + 325r)t^ = 0.
This equation, though it somewhat resembles the hypergeometric equation, is of higher
type than the equations dealt with in Chapters xiv and xvi, inasmuch as it has two
regular singularities at 0 and 1 and an irregular singularity at oo ; whereas the three
singularities of the hypergeometric equation are all regular, while the equation for W^^^ (z)
has one irregular singularity and only one regular singularity.
We shall now give a short account of Lindemann's analysis, with some
modifications due to Stieltjesf.
19'51. Lindemann's form of Floquet's theorem.
Since Mathieu's equation (in Lindemann's form) has singularities at ^=0
and ^=1, the exponents at each being 0, |, there exist solutions of the form
2/00= s a. r, 7/ol = r*S&n^^
2/,o = S «,/ (1 - r)^ vn = (1 - r)' i ^n (i - ^r ;
the first two series converge when ] ^| < 1, the last two when j 1 — ^[ < 1.
When the ^-plane is cut along the real axis from 1 to + oo and from
0 to — 00 , the four functions defined by these series are one-valued in the
cut plane ; and so relations of the form
2/io = ayoo + /S^/oi , 2/n = 72/00 + S?/oi
will exist throughout the cut plane.
Now suppose that ^ describes a closed circuit round the origin, so that the
circuit crosses the cut from — oo to 0 : the analytic continuation of y■^^ is
* Math. Ann. xxii. (1883), p. 117.
+ AAtr. Nach. cix. (1884). The analysis is very similar to that employed by Hermite
(Oeiiv7-es, III. p. 118) in connexion with Lame's equation.
19-5-19'52] MATHIEU FUNCTIONS 411
ayoo — ySyoi (since yw, is unaffected by the description of the circuit, but yoi
changes sign) and the continuation of yu is 73/00 — Syoi \ (^nd so Ay^^ + By^^ will
he unaffected by the description of the circuit if
A (ayoo + ySyoi)' + B (72/00 + Byo^Y = A {ay^ - ^y,,y + B (73/00 - %ol)^
i.e. if Aa^ + BjS = 0.
Also Ay-^o^+Byn obviously has not a branch-point at ^=1, and so, if
Aa/3 + ByB = 0, this function has no branch-points at 0 or 1, and, as it has no
other possible singularities in the finite part of the plane, it must he an
integral function of ^.
The two expressions
A^y.o + iB^-yn, A'^y^, - iB^y^
are consequently two solutions of Mathieu's equation whose product is an
integral function of f.
[This amounts to the fact (§ 19"4) that the product of e'^(f>{z) and
e~i^^<l>{—z) is a periodic integral function of zJ]
19'52. The determination of the integral function associated^ with Mathieu's
general equation.
The integral function F{z) ~ Ay^o'+ By^^ just introduced, can be deter-
mined without difficulty ; for if y^o and yn are any solutions of
J+p(?)|+«(r)»=o,
their squares (and consequently any linear combination of their squares)
satisfy the equation*
^ + 3P (0 ^ + [P' iO + 4Q (D + 2 {P iOY] I
+ 2[Q'(O+2P(OQ(O]2/ = 0;
in the case under consideration, this result reduces to
,a-r)™.ia-2r)-^)
+ (a - 1 - 16^ + 32^^) ^^^ + IQqF (^ = 0.
Let the Maclaurin series for P(^) be S CnC"} on substitution, we easily
n=0
obtain the recurrence formula for the coefficients c^, namely
where
{n + 1) {{n + If - a + 16q} n(n + l)(2n+l)
""" Wq{2n+1) ' ^"~ S2q{2n-1) '
* Appell, Cojuptes Eendiis, xci.
412
THE TRANSCENDENTAL FUNCTIONS
[chap. XIX
At first sight, it appears from the recurrence formula that Co and c^ can
be chosen arbitrarily, and the remaining coefficients Cg, Cg, ... calculated in
terms of them; but the third order equation has a singularity at ^= 1, and
the series thus obtained would have only unit radius of convergence. It is
necessary to choose the value of the ratio Ci/co so that the series may con-
verge for all values of ^.
The recurrence formula, when written in the form
/ / \ — J_ ^n+i
suggests the consideration of the infinite continued fraction
l("n +
+ 1*71+2 +
= lim \un +
^^Tl+l + . . . + W^
%)
The continued fraction on the right can be written^
UnK (n, n + m)IK {n+\, n + m),
where K (n, n + m) = 1
- K
0
— u'
0 ,
1 ,
— U
11+111)
The limit of this, as m^- ao , is a convergent determinant of von Koch s
type (by the example of § 2*82) ; and since
r=n
0 as 72. -^ CO ,
it is easily seen that K{n, oo ) ^- 1 as 7i ^- oo .
Therefore, if - ^'^ = ^'^/^i^L^"^ > ,
Cn+i K{n + 1, CO)
then Cn satisfies the recurrence formula and, since Cn+ijCn-^O as n -^ oo , the
resulting series for F(^) is an integral function. From the recurrence formula
it is obvious that all the coefficients c„ are finite, since they are finite when n
is sufficiently large. The construction of the integral function F(^) has
therefore been effected.
19'53. The solution of Mathieu's equation in terms of F (^).
If Wi and W.2 be two particular solutions of
then-f-
Wjw/ — iu^W2 = G exp \—\ P {^)d^\,
* Sylvester, Phil. Mag. (i), Vol. v. (1853), p. 446.
t Abel, Crelle, ii. p. 22. Dashes denote differentiations with regard to f.
19-53, 196] MATHIEU FUNCTIONS 413
where (7 is a definite constant. Taking w^ and W2 to be those two solutions of
Mathieu's general equation whose product is F {l^), we have
Wi w^ ^^{\-^)^F{^y w, W2 F{i)'
the latter following at once from the equation WiW2 = F{^).
Solving these equations for Wi/wi and w^jw^., and then integrating, we at
once get
where 71, 72 are constants of integration; obviously no real generality is lost
by taking Co = 71 = 72 = 1.
From the former result we have, for small values of | ^|,
^, = 1 + c/rU ^ (c, + G') r+o (^*),
while, in the notation of § 19 "51, we have aija^ = — ^a + 85.
Hence G^ = IQq — a — Ci.
This equation determines G in terms of a, q and Ci, the value of Cj being
K(l,oo)^{uoK{0, x)}.
Example 1. If the solutions of Mathieu's equation be e*'^^0(±2), where (f){z) is
periodic, shew that
Example 2. Shew that the zeros of F{() are all simple, unless (7=0.
(Stieltjes.)
[If F{() could have a repeated zero, Wj and ^V2 would then have an essential singularity.]
19'6. A second method of constructing the Mathieu function.
So far, it has been assumed that all the various series of § 19*3 involved
in the expressions for cej^ {z, q) and scj^ {z, q) are convergent. It will now he
shewn that cejff{z, q) and se]^{z, q) are integral functions of z and that the
coefficients in their expansions as Fourier series are power series in q which
converge absolutely when \q\ is sufficiently small*.
To obtain this result for the functions ce^{z, q), we shall shew how to
determine a particular integral of the equation
d^u
-r-j + (a + 16g cos 2z) u='y\r (a, q) cos Nz
* The essential part of this theorem is the convergence of the series which occur in the
coefficients ; it is already known {§§ 10-2, 10-21) that solutions of Mathieu's equation are integral
functions of z, and (in the case of periodic solutions) the existence of the Fourier expansion
follows from § 9*11.
414 THE TRANSCENDENTAL FUNCTIONS [CHAP. XIX
in the form of a Fourier series converging over the whole ^^-plane, where
^fr (a, q) is a function of the parameters a and q. The equation yjr (a, q) = 0
then determines a relation between a and q which gives rise to a Mathieu
function. The reader who is acquainted with the method of Frobenius* as
applied to the solution of linear differential equations in power series will
recognise the resemblance of the following analysis to his work.
Write a = N'^ + Sp, where iV is zero or a positive or negative integer.
Mathieu's equation becomes
-j-^ + N'^u = - S (p + 2q cos 2z) u.
If p and q are neglected, a solution of this equation is u = cos N'z = Uq (z),
say.
To obtain a closer approximation, write —8(p + 2q cos 2z) Uq (z) as a sum
of cosines, i.e. in the form
-H{qco8(N-2)z+p cos Nz + q cos {N +2)z\= V^ (z), say.
Then, instead of solving -^-^ -\- N^u=Vi (z), suppress the terms f in Fj (z)
which involve cosNz; i.e. consider the function Wi(z) where|
Fi (z) = Fi (z) + 8^ cos Nz,
A particular integral of
dz
is
"^''l + NHi^W.iz)
'' = ^ {lO^iT) ''''' (^ - 2> ^ + 1(1^^) cos {N+ 2) ^1 = U, (z), say.
Now express —8(p + 2q cos 2z) U^ {z) as a sum of cosines ; calling this
sum Fa {z), choose Og to be such a function of p and q that Fa (z) + o^ cos iV^2
contains no term in cos Nz ; and let Fa {z) + a^ cos iV"^; = TTg (^)-
Solve the equation -j-^ + iV^^w = W^. {z),
and continue the process. Three sets of functions U.,n{z), V^iz), TF'^(^)
are thus obtained, such that Z7,„ (z) and Wm (z) contain no term in cos JS'z
when m ^ 0, and
Tfm (z) = Vm (z) + a,„ cos Nz, F,^ (^) = - 8 (p + 2^ cos 2z) Um-i (z),
'^^^^ + N-^U,,{z)=W^{z),
CLz'
where a^„ is a function of jj and q but not of z.
* Crelle, lxxvi. pp. 214—224.
t The reason for this suppression is that the particular integral of —j + N'hi = cos Nz
contains non-periodic terms.
i Unless ^=1, in which case JVi {z) — V-i {z) + 8 {p + q) cos z.
19()1] MATHIEU FUNCTIONS 416
It follows that
(,»^ ) »»=0 »»=1
= S Vm{z) + [^ am) COS Nz
«-l / M \
= — 8 (j9 + 2^ COS 22) 2 C/^-i (^) + ( ^ Olm] COS iV>.
m=0 \w=l /
00
Therefore, if U (z)= S Um {2) be a uniformly convergent series of analytic
functions throughout a two-dimensional region in the 2-plane, we have
(§ 5-3)
(PU(z)
, \ + (a + 16g cos 2z) U{z) = '\\r {a, q) cos Nz,
00
where ■\lr(a,q)= S a,„.
w = l
It is obvious that, if a be so chosen that -»/r(a, q) = 0, then U (z) reduces
to cej^T^z).
A similar process can obviously be carried out for the function se^ {z, q)
by making use of sines of multiples of z.
19'61. The convergence of the series defining Mathieu functions.
We shall now examine the expansion of § 19'6 more closely, with a view to investigating
the convergence of the series involved.
When ?i ^ 1, we may obviously write
n n
Un{z)= 2 */3„,reos(iV-2r)2-f- 2 a„ ,. cos (iV^+ 2r) 2,
the asterisk denoting that the first summation ceases at the gi'eatest value of r for which
r^hN.
Since | ^ "*■ ^ ( ^n + i{z) = an + iGOii Nz -S{p + 2q cos 2z) U^ (2),
it follows on equating coefficients of cos (^± 2/-) z on each side of the equation t that
«n + l = 8^(n»,l+/3n,l),
r (r + JV) a,, + i^r = ^ {pan.r + q ian.r-l + a„^r + l)} (r=l, 2, ..,),
rir-N)^„^,,r = 2{p^n.r + q{^n.r-l + ^n.r + l)l- (r^^N).
These formulae hold universally with the following conventions | :
« "n.o = /3,,o = 0 (..= 1,2,...); V,=/3„„.=0 (r>«)
t When N=0 or 1 these equations must be modified by the suppression of all the coeflicieuts
J The conventions (ii) and (iii) are due to the fact that cosz = cos ( -s), cos 2^ = cos( - 2^).
416
THE TRANSCENDENTAL FUNCTIONS
[chap. XIX
The reader will easily obtain the following special formulae :
(I) «i = 8^, {lY^l); a, = 8(p + q), {JV^l),
dn^"- jV ' 9» + 1 nil
(III) a„_^ and ^n,r ^'I's homogeneous polynomials of degree n in p and q.
If i an,r=Ar, 2 ^n,r= B,.,
n=r n=r
we have yjr {a, q) = 8p + 8q{Ai+Bj) (-^^l),
r{r + N)Ar=2{pAr + q{A,_i + Ar + ,)} (A),
r(r-JV) B, = 2{pBr+q{B,._, + Br^^)} (B),
where Ao = B()=l and B^ is subject to conventions due to (ii) and (iii) above.
Now write w^ = -q {r {r + JV) — 2p} ~ i, w^ = —q{r{r — N) — %p} ~ ^
The result of eliminating Ji, A^i ... ^,--i) ^r + i? ••• from the set of equations (A) is
where A^ is the infinite determinant of von Koch's type (§ 2"82)
A^= 1 , w^ + i, 0 , 0 , ...
Wr + 2, 1 , W'r + 2) 0 , •••
0 , Wr + 3, 1 , Wr + 3, ■••
The determinant converges absolutely (§ 2'82 example) if no denominator vanishes;
and A,.-*-l as r-^cc (cf. § 19'52). If p and q be given such values that Aq^O,
2p^r{r + JV), where r=l, 2, 3, ..., the series
2 ( — )'■ W'l Wo • • • WrAr Ao ~ ^ COS {JV+ 2r) z
represents an integral function of z.
In like manner BrDQ = { — yw{w2 ... w/Z)^, where D^ is the finite determinant
1 , ?y',. + i, 0 , ...
the last row being 0, 0, ...0, 2w\ j^, 1 or 0, 0, ...0, w', ,y_-^,, l + w'i,y_j> as iV is even
or odd.
The series 2 Un [z) is therefore
CO
COS Nz + Ao~ 1 2 ( - )*■ Wj 1^2 . . . w,. A,, cos {N+ 2r) z
r = \
+ Do~ ^ 2' ( - ywi w./ . . . IV,: Br cos ( A^- 2r) z,
r=l
these series converging uniformly in any boiinded domain of values of z, so that term-by-
term difterentiations are permissible.
Further, the condition -v//- (a, q) = 0 is equivalent to
pAoD(t-q («'i Ai J)o+iViI)i A„) = 0.
If we multiply Ijy
n/i-
2p
r:^*.V
n U
r=i [ r{r + lY)j ,.1\ 1 r{r-N)j^
2p
19-7] MATHIEU FUNCTIONS 417
the expression on the left becomes an integral function of both p and q, NE' (a, q), say ; the
terms of ^ (a, q), which are of lowest degrees in p and g, are respectively p and
,2/_J LI
* \n-\ n+\]
Now expand - — •. j — ,„J^^ r — ^^ — -, — ^-^ dp
^ 27n J 'ir{If^ + 8p,q) dp ^
in ascending powers of q (cf. § 7'31), the contour being a small circle in the p-plane, with
centre at the origin, and | q \ being so small that * {JV^ + 8p, q) has only one zero inside the
contour. Then it follows, just as in § 7*31, that, for sufficiently small values of Ig-I,
we may expand p as a power series in q commencing* with a term in q^ ; and if | y |
be sufficiently small Bq and Ao will not vanish, since both are equal to 1 when q = 0.
On substituting for p in terms of q throughout the series for U{z), we see that the
series involved in cej^{z, q) are absolutely convergent when | y | is sufficiently small.
The series involved in se^{z, q) may obviously be investigated in a similar manner.
19'7. The method of change of parameter \.
The methods of Hill and of Lindemann-Stieltjes are effective in determining ^, but
only after elaborate analysis. Such analysis is inevitable, as fi is by no me^ns a simple
function of q ; this may be seen by giving q an assigned real value and making a vary
from — 00 to + 03 ; then /x alternates between real and complex values, the changes taking
place when, with the Hill-Mathieu notation, ^{0)sin^{\TT sja) passes through the values
0 and 1 ; the complicated nature of this condition is due to the fact that A (0) is an
elaborate expression involving both a and q.
It is, however, possible to express ^i. and a in terms of q and of a new parameter or,
and the results are very well adapted for purposes of numerical computation when | q \
is small J.
The introduction of the pai-ameter o- is suggested by the series for ce^ (z, q) and se^ {z, q)
given in § 19"3 example 1 ; a consideration of these series leads us to investigate the
potentialities of a solution of Mathieu's general equation in the form ^ = e'^^^(^), where
0 (2) = sin (2 — (r) + a3COs(3z— (r) + 63sin (32- a-) + «5COS (62 — o-) + 65 sin (52-a-) + ...,
the parameter o- being rendered definite by the fact that no term in cos (2 — <t) is to appear
in <^ (2) ; the special functions sei (2, q), ce^ (2, q) are the cases of this solution in which
o- is 0 or ^TT.
On substituting this expression in Mathieu's equation, the reader will have no difficulty
in obtaining the following approximations, valid for§ small values of q and real values of a- :
fj. =45- sin 2o- — 125'^sin 2(r — 12q*sin 4(r + 0{q^),
a =l + 8q cos 2o- + ( - 16+8 cos 4o-) q'^-8q^ cos 2(r + (^|^- 88 cos 4(r) q* + 0 {q%
a^ = 3^2 sin 2o- + 3^3 gin 4o- + ( - 2li sin 2(r + 9 sin 60-) q*-\-0 (q^),
bs=q + q^ cos 2(r + i-^^- + 5 cos 4a) q^ + {-^r^ cos 2a + 7 cos 6<r)q* + 0{q%
as = V- ?^ sin 2<r + |f <?* sin 4a- + f ( j^),
h = h<l'^ + ^9^ cos 2o- + ( - ^i£- + ^f cos 4(r) q^ + 0 {q%
«7 = t¥8?*sin2o- + 0(2'S), It = ^ q^ -i-^^q* cos 2(7 + 0{q% '
«9= 0 (j5), 69=3-1-5^1 + 0 (q-),
the constants involved in the various functions 0 (9") depending on a. •
* If iV=l this result has to be modified, since there is an additional term q on the right and
the term q'^l{N - 1) does not appear.
t Whittaker, Proc. Edinburgh Math. Soc. xxxn.
:;: They have been applied to Hill's problem by Ince, Monthhj Notices of the R. A. S. lxxv.
§ The parameters q and tr are to be regarded as fundamental in this analysis, instead of
a and q as hitherto.
W. M. A. 27
418 THE TRANSCENDENTAL FUNCTIONS [CHAP. XIX
The domains of values of q and a for which these series converge have not yet been
determined*.
If the solution thus obtained be called A {z, o-, q), then A {z, a; q) and A (z, — a, q) form
a fundamental system of solutions of Mathieu's general equation if /x=t=0.
Example 1. Shew that, if o- = ix0"5 and g'=0-01, then
a = 1-124,841,4 ..., /x, = i X 0-046,993,5 ... ;
shew also that, if o•=^ and §' = 0-01, then
a = l-321,169,3..., /i=:i x 0-145,027,6 ....
Example 2. Obtain the equations
joi = 4 J sin 2o- — 4g'a3 ,
■ a = 1 + Sg' cos 2(r — /i^ - Sqh^,
expressing fx and a in finite terms as functions of q, cr, a^ and 63.
Example 3. Obtain the recurrence formulae
{-4n{n + l) + 8qcos2(T-Sqb3±8qi(2n + l){a3-sin2(r)} Z2n + i + 8q{z2n-i + Z2n + 3)=0,
where «2n + i denotes 62n + i + ^<^2»t+i or b^n + i-iO'in + i, according as the upper or lower sign is
taken.
19 '8. The asymptotic solution of Mathieu's equation.
If in Mathieu's equation
d^u / 1 ,„ ^ .
we write ^ cos 2 = 1, we get
where M^=.a-\B.
This equation has an irregular singularity at infinity. From its resemblance to Bessel's
equation, we are led to write u = e^^^~^v, and substitute
v=\ + {a,l^) + {a2ie) + -
in the resulting equation for v ; we then find that
the general coefficient being given by the recurrence formula
2i(r + l)ar+i = {J-i/2 + p+;.(;.+ l)} + (2r-l)zX2ar_i-(r2-27- + |)Fa^_2.
The two series
«'V»(l + |+p4-...). «-V»(l-| + p-...
are formal solutions of Mathieu's equation, reducing to the well-known asymptotic
solutions of Bessel's equation (§ 17"5) when k-*~0. The formulae which connect them
with the solutions e '^~(^{±z) have not yet been published.
* It seems highly probable that, if | g' | is sufficiently small, the series converge for all real
values of tr, and also for complex values of <t for which j I (o-) | is sufficiently small. It may be
noticed that, when q is real, real and purely imaginary values of a correspond respectively
to real and purely imaginary values of ix.
198] MATHIEU FUNCTIONS 419
REFERENCES.
E. L. Mathieu, LiouvilUs Journal (2), xiii. (1868), pp. 137-203,
G. W. Hill, Acta Mathematical viiL (1886), pp. 1-36.
G. Floquet, Ann. de Vicole Normale (2), xil (1883), pp. 47-88.
C. L. F. LiNDEMANN, Math. Ann., xxil. pp. 117-123.
T. J. Stieltjes, Astronomische Nach., cix.
A. LiNDSTEDT, Astronomische Nach., cm, civ, cv.
R. C. Maclaurin, Trans. Gamb. Phil. Soc, xvii. pp. 41-108.
E. T. Whittaker, Proc. International Congress of Mathematicians, Cambridge, 1912,
L pp. 366-371.
E. T. Whittaker, Proc. Edinburgh Math. Soc, xxxii.
G. N. Watson, Proc. Edinburgh Math. Soc, xxxiii.
A. W, Young, Proc Edinburgh Math. Soc, xxxii.
E. Lindsay Ince, Pi-oc Edinburgh Math. Soc, xxxiii.
Miscellaneous Examples.
1. Shew that, if ^=^(325-),
27rceo (2, g) = ccq (0, q) I cos {k sin z sin d) ccq {6, q) dd.
2. Shew that the even Mathieu functions satisfy the integral equation
G{z) = \r Jo {ik {cos z + cos d)}G{e)d6.
3. Shew that the equation
{az^ + c) -,-^ +2az j_ +{X^cz^ + m) u=0
(where a, c, X, m are constants) is satisfied by
ii = ^e''^%{s)ds
taken round an appropriate contovir, provided that v (s) satisfies
(as2 ^ c) "^^ + 2as '^ + {X'-cs'' + m) v (s) = 0,
which is the same as the equation for u.
Derive the integral equations satisfied by the Mathieu functions as particular cases of
this result.
27—2
420 THE TRANSCENDENTAL FUNCTIONS [CHAP. XIX
4. Shew that, if powers of q above the fourth are neglected, then
cei iz^q) = cos z-\-q cos 2>z + (f- (^ cos hz — cos 3s)
+ ^ (tV '^^^ *^^~% ''OS 5^ + J cos Sz)
+ 2'* ( yIj cos 92 — y\ cos 72 + J cos hz + -y- cos Sz) ,
««! (2, y) = sin z + 2' sin 32 + 5"^ (^ sin 52 + sin 3z) ■ •
+ g:^ (jJg sin 72 + A sin 52+ J sin 32)
+ J* (y|(J sin 92 + yJg sin 72 + ^ sin 52 - \i- sin 32),
ce^ (2, g') = cos 22 + 2- (cos 42 — 2) -i- ^ y^. cos 62
+ ^3 ( J.^ cos 82 + If cos 42 + -if )
+?* (sitT cos IO2 + |fi§ cos 62).
(Mathieu.)
5. Shew that ..,..■.
ces (2, j) = cos 32 + 2' ( - cos 2+1 cos 52)
+ 2^ (cos 2 + j^j cos 'lz)-\-(f{-\ cos z-Vi^ cos hz-\--^ cos 92) + 0 (2*),
and that, in the case of this function
a = 9 + 422-823 + 0(2*). ''
(Mathieu.)
6. Shew that, if y (2) be a Mathieu function, then a second solution of the corresponding
differential equation is
y(2)j {y
Shew that a second solution* of the equation for ceo (2, 2) ^^
2 ceo (2, 2)~42''*i'^22 — 32^sin42 — ....
7. If 3/ (2) be a solution of Mathieu's general equation, shew that
{y(2 + 27r)+y(2-27x)}/y(2)
is constant.
8. Express the Mathieu functions as series of Bessel functions in which the coefficients
are multiples of the coefficients in the Fourier series for the Mathieu functions.
[Substitute the Fovirier series under the integral sign in the integral equations of
§ 19-22.]
9. Shew that the confluent form of the equations for ce„ (2, 2) *ind ?.en (2, q)^ when the
eccentricity of the fundamental ellipse tends to zero, is, in each case, the equation satisfied
by Jn {ik cos 2).
10. Obtain the parabolic cylinder functions of Chapter xvi as confluent forms of the
Mathieu functions, by making the eccentricity of the fundamental ellipse tend to unity.
11. Shew that ce,; (2, q) can be exjmnded in series of the form
2 J,„cos2'»2 or 2 5™cos2'» + i2,
m=0 jtt=0
according as n is even or odd ; and that these series converge when | cos 2 1 < 1.
* This solution is called ?«o {z, q) ; the second solutions of the equations satisfied by Mathieu
functions have been investigated by Ince, Proc. Edinburgh Math. Soc. xxxiii. See also § 19-2.
MATHIEU FUNCTIONS 421
12. With the notation of example 11, shew that if
ceniz, ?) = X„ J^^e*««»^°°«''ce„(d, q)de,
then X„ is given by one or other of the series
provided that these series converge,
13. Shew that the differential equation satisfied by the product of any two solutions
of Bessel's equation for functions of order n is
where S denotes a ^ .
az ' ■
Shew that one solution of this equation is an integral function of z ; and thence, by the
methods of §§ 19"5-19'5.3, obtain the Bessel functions, discussing particularly the case in
which n is an integer.
14. Shew that an approximate solution of the equation ;
^+(A+k^smh.^z)ii=0
az^
is u = C (cosech zf sin {k cosh z + «),
where C and t are constants of integi'ation ; it is to be assumed that k is large, A is not
very large and z is not very small.
CHAPTEK XX
ELLIPTIC FUNCTIONS. GENERAL THEOREMS AND THE
WEIERSTRASSIAN FUNCTIONS
20"1. Doubly -penodic functions,
A most important property of the circular functions sin 2^, cos 2^, tan 5;, ...
is that, i^ f{z) denote any one of them,
/(^ + 27r)=/(4
and hence f{z + 2n7r) =f{z), for all integer values of n. It is on account
of this property that the circular functions are frequently described as
periodic functions with period lir. To distinguish them from the functions
which will be discussed in this and the two following chapters, they are
called singly -periodic functions.
Let ft)i, &J2 bS any two numbers (real or complex) whose ratio is not purely
real. A function which satisfies the equations
f{z + 2a,,) =f{z), f{z + 2a),) =f{z\
for all values of z for which/ (2^) exists, is called a doubly -periodic function
of z, with periods 2a)i, 2a)2. A doubly-periodic function which is analytic
(except at poles), and which has no singularities other than poles in the
finite part of the plane, is called an elliptic function.
[Note. What is now known as an elliptic integral*' occurs in the researches of Jakob
BernoulH on the Elastica. Maclaurin, Fagnano, Legendre, and others considered such
integrals in connexion with the problem of rectifying an arc of an ellipse ; the idea of
'inverting' an elliptic integral (J5 21-7) to obtain an elliptic function is due to Abel,
Jacobi and Gauss.]
The periods 2a)i, 2a)2 play much the same part in the theory of elliptic
functions as is played by the single period in the case of the circular
functions.
Before actually constructing any elliptic functions, and, indeed, before
establishing the existence of such functions, it is convenient to prove some
* A brief discussion of elliptic integrals will be found in §§ 22"7-22'741.
20*1, 20'11] ELLIPTIC FUNCTIONS 423
general theorems (§§ 20"11 — 20*14) concerning properties common to all
elliptic functions ; this procedure, though not strictly logical, is convenient
because a large number of the properties of particular elliptic functions can
be obtained at once by an appeal to these theorems.
Example. The diflerential coefficient of an elliptic function is itself an elliptic
function.
20*11. Pemod-'parallelograirns.
The study of elliptic functions is much facilitated by the geometrical
representation afforded by the Argand diagram.
Suppose that in the plane of the variable z we mark the points 0, 2(o^,
2a>2, 2&)i + 2ft)2, and, generally, all the points whose complex coordinates are
of the form 2m&)i+ 2w&)2, where m and n are integers.
Join in succession consecutive points of the set 0, 2&>i, 2ft)] + 2g)2, ^w^, 0,
and we obtain a parallelogram. If there is no point w inside or on the
boundary of this parallelogram (the vertices excepted) such that
f{z + ^)=f{z)
for all values of 2^, this parallelogram is called a. fundamental period-parallelo-
gram for an elliptic function with periods 2a>i, ^co^.
It is clear that the ^-plane may be covered with a network of parallelo-
grams equal to the fundamental period-parallelogram and similarly situated,
each of the points 2m<o^ + ^noo^ being a vertex of four parallelograms.
These parallelograms are called period-parallelograms, or meshes ; for all
values of z, the points z, z+2q)i, ... z + 2m(0i + 2nco2, ... manifestly occupy
corresponding positions in the meshes ; any pair of such points are said to
be congruent to one another. The congruence of two points z, z' is expressed
by the notation / = 5 (mod. 2wi, 26)2).
From the fundamental property of elliptic functions, it follows that an
elliptic function assumes the same value at every one of a set of congruent
points ; and so its values in any mesh are a mere repetition of its values in
any other mesh.
For purposes of integration it is not convenient to deal with the actual
meshes if they have singularities of the integrand on their boundaries ; on
account of the periodic properties of elliptic functions nothing is lost by
taking as a contour, not an actual mesh, but a parallelogram obtained
by translating a mesh (without rotation) in such a way that none of the poles
of the integrands considered are on the sides of the parallelogram. Such a
parallelogram is called a cell. Obviously the values assumed by an elliptic
function in a cell are a mere repetition of its values in any mesh.
A set of poles (or zeros) of an elliptic function in any given cell is called
an irreducible set ; all other poles (or zeros) of the function are congruent to
one or other of them.
424 THE TRANSCENDENTAL FUNCTIONS [CHAP. XX
20'12. Simple properties of elliptic functions.
(I) The number of poles of an elliptic function in any cell is finite.
For, if not, the poles would have a limit point, by the two-dimensional
analogue of § 2 •21. This point is (§ 5'61) an essential singularity of the
function ; and so, by definition, the function is not an elliptic function.
(II) The number of zeros of an elliptic function in any cell is finite.
For, if not, the reciprocal of the function would have an infinite number
of poles in the cell, and would therefore have an essential singularity ; and
this point would be an essential singularity of the original function, which
would therefore not be an elliptic function. [This argument presupposes
that the function is not identically zero.]
(III) The sum of the residues of an elliptic function, f{z), at its poles in
any cell is zero.
Let G be the contour formed by the edges of the cell, and let the corners
of the cell be ^, ^ + 2&)i, t + 2a)i + 2a)2, t + 2a)2.
[Note. In future, the periods of an elliptic function will not be called 2coi, 2cd2
indifferently ; but that one will be called Stoi which makes the ratio 0)2/0)1 have a positive
imaginary part ; and then if C be described in the sense indicated by the order of the
corners given above, the description of C is counter-clockwise.
Throughout the chapter, we shall denote by the symbol C the contour formed by
the edges of a cell.]
The sum of the residues oi f{z) at its poles inside C is
If If /•<+2w, /•<+2w,+2<02 r<+2(02 n \
2^- /(^)^^ = 2^- + + + \f{^)dz.
^^^ J C ^TTt [J t J t+2oi^ J f+2a>,+2a)2 J t+2uij
In the second and third integrals write z + 2&)i, z + 2&)o respectively for
f(z), and the right-hand side becomes
2^- fj"^' \f{^) -/(^ + 2a).)} dz - 2^. J'^"""^ [f{z) -f{z + 20,,)} dz,
and each of these integrals vanishes in virtue of the periodic properties of
f{z) ; and so 1 f{z) dz = 0, and the theorem is established.
J c
(IV) Liouville's theorem*. An elliptic function, f{z), with no poles in a
cell is merely a constant.
For if f{z) has no poles inside the cell, it is analytic (and consequently
bounded) inside and on the boundary of the cell (§ 3'61 corollary ii); that is
to say, there is a number K such that \f{z) \ < K when z is inside or on the
boundary of the cell. From the periodic properties of f{z) it follows that
* This modification of the theorem of § 5-63 is the result on which Liouville based his
lectures on elUiDtic functions.
20-12, 20-13] ELLIPTIC FUNCTIONS 425
f{z) is analytic and \f{z)\ < K for all values of z; and so (§ 5-63) /(^) is a
constant.
It will be seen later that a very large number of theorems concerning
elliptic functions can be proved by the aid of this result.
20'13. The order of an elliptic function.
It will now be shewn that, if f{z) be an elliptic function and c be any
constant, the number of roots of the equation
fi^) = c
which lie in any cell depends only on f(z), and not on c ; this number is
called the order of the elliptic function, and is equal to the number of poles
oif{z) in the cell.
By § 6-31, the difference between the number of zeros and the number
of poles oi f{z) — c which lie in the cell C is
l.f
f'(^)
dz.
^'rriJcfi^)
Since /' (z + 2a)x) =/' (z + 2co2) =f' (z), by dividing the contour into four
parts, precisely as in § 20-12 (III), it follows that this integral is zero.
Therefore the number of zeros of f{z) — c is equal to the number of
poles of f(z) — c ; but any pole of f(z) — c is obviously a pole of f(z) and
conversely ; hence the number of zeros of f(z) — c is equal to the number
of poles of f(z), which is independent of c ; the required result is therefore
established.
[Note. In determining the order of an elliptic function by counting the number of
its irreducible poles, it is obvious, from § 6"31, that each pole has to be reckoned according
to its multiplicity.]
The order of an elliptic function is never less than 2 ; for an elliptic
function of order 1 would have a single irreducible pole ; and if this point
actually were a pole (and not an ordinary point) the residue there would
not be zero, which is contrary to the result of § 20*12 (III).
So far as singularities are concerned, the simplest elliptic functions are
those of order 2. Such functions may be divided into two classes, (i) those
which have a single irreducible double pole, at which the residue is zero in
accordance with § 2012 (III) ; (ii) those which have two simple poles at which
the residues are numerically equal but opposite in sign by § 20" 12 (III).
Functions belonging to these respective classes will be discussed in this
chapter and in Chapter xxii under the names of Weierstrassian and
Jacobian elliptic functions respectively ; and it will be shewn that any
elliptic function is expressible in terms of functions of either of these
types.
426 THE TRANSCENDENTAL FUNCTIONS [CHAP. XX
20*14. Relation between the zeros and poles of an elliptic function.
We shall now shew that the sum of the affixes of a set of irreducible
zeros of an elliptic function is congruent to the sum of the affixes of a set of
irreducible poles.
For, with the notation previously employed, it follows, from § 6'31, that
the difference between the sums in question is
TTlJc J{Z) ZTTl [Jt .^+2.0, ^ !^+2a,,+2a,2 J t+2oj,\ f{z)
1 f ^+2-' izf {z) (z + 2co,)f' (z + 2cc,)]
27ri
'^TriJt \f{z) f{z+2Q),)
_ J_ f*'-"'^ Wi^) _ (^+20)0/(^4-20,,)]
27rW. \f{z) f{z + 2c.,) r^
2o,,p/;i^)c^.+2o,.p-^>c^4
Jt f{z) Jt f{z) J
1
"liri
27ri \
_ 9
ZCO.,
log/(^)
«-|-2a
+ 2o,i
log/(^)
t
on making use of the substitutions used in § 2012 (III) and of the periodic
properties oi f{z) and/' (2).
Now /(2') has the same values at the points t+2oii, ^+2o,2as at t, so
the values of log/(^) at these points can only differ from the value oi f{z)
at t by integer multiples of 27rz, say — 2n7ri, 2rtnTi ; then we have
^-JriJc f{z)
and so the sum of the affixes of the zeros minus the sum of the affixes of
the poles is a period ; and this is the result which had to be established.
-Vr V dz = 2mo)i + 2?io,2,
■J c JKz)
20*2. The construction of an elliptic function. Definition of (^ {z).
It was seen in § 20*1 that elliptic functions may be expected to have
some properties analogous to those of the circular functions. It is therefore
natural to introduce elliptic functions into analysis by some definition
analogous to one of the definitions which may be made the foundation
of the theory of circular functions.
One mode of developing the theory of the circular functions is to start
CO
from the series 2 {z - ??i7r)~- ; calling this series (sin zy^, it is possible
m= - CO
to deduce all the known properties of sin z ; the method of doing so is briefly
indicated in ^ 20-222.
20'14-20'21] ELLIPTIC FUNCTIONS 427
The analogous method of founding the theory of elliptic functions is to
define the function p(z) by the equation*
/ wi+ V f 1 1 )
^ ^' m.n \(z - ^mm, - 2n(02y ( 2niQj, + 2n(o;)^\ '
where Wi, w^ satisfy the conditions laid down in §§ 20'1, 20*12 (III) ; the
summation extends over all integer values (positive, negative and zero) of
m and n, simultaneous zero values of m and n excepted.
For brevity, we write 12,^, n in place of 2m&)i + 2nco^, so that
^ (z) = z-^ + S' {(2 - n^, „)-^ - n-%].
m, n
When m and n are such that | H^^ „ | is large, the general term of the
series defining ^(z) is 0 (| n„,_ ,i j~'*), and so (§ 3*4) the series converges
absolutely and uniformly (with regard to z) except near its poles, namely
the points D,m,n'
Therefore (§5*3), ^(z) is analytic throughout the whole 2^-plane except
at the points n^_„, where it has double poles.
The introduction of this function ^ (z) is due to Weierstrass+ ; we now
proceed to discuss properties of ^(z), and in the course of the investigation
it will appear that ^(z) is an elliptic function with periods 2&)i, 2<b2-
For purposes of numerical computation the series for ^ (2) is useless on account of the
slowness of its convergence. Elliptic functions free from this defect will be obtained in
Chapter xxi.
Example. Prove that
20'21. Periodicity and other properties of <p{z).
Since the series for ^ {z) is a uniformly convergent series of analytic
functions, term -by-term differentiation is legitimate (§ 5 '3), and so
""* m, n V-^ ^'"in, n)
The function g)' {z) is an odd function of z ; for, from the definition of
^' {z), we at once get
<^o'{-z)==2 1 (z + a,,,n)-'.
m, n
* Throughout the chapter S will be written to denote a summation over all integer values
m, n
of VI and n, a prime being inserted (S') when the term for which m = 7^ = 0 has to be omitted
in,n
from the summation. It is also customary to write ^' (z) for the derivate of ^(z). The use of
the prime in two senses will not cause confusion.
t Werke, 11. pp. 245-255. The subject-matter of the greater part of this chapter is due to
Weierstrass, and is contained in his lectures, of which an account has been published by Schwarz.
See also Cayley, Liouville, x. (1845), and Eisenstein, Crelle, xxxv, (1847).
428 THE TRANSCENDENTAL FUNCTIONS [CHAP. XX
But the set of points — n,„_ „ is the same as the set D,rn, n and so the
terms of ^' {— z) are just the same as those of — ^j' (z), but in a different
order. But, the series for ^' (z) being absolutely convergent (§ 3'4), the
derangement of the terms does not affect its sum, and therefore
^'(-z)=-^o(z).
In like manner, the terms of the absolutely convergent series
m, n
are the terms of the series '•■'■.
m, n •> . ;.
in a different order, and hence v
i^ (- z) = ^ (z) ;
that is to say, ^(z) is an even function of z.
Further, ip' (z + 2(o,) = - 2 t (^ - fi^, „ + 2co^)-' ;
but the set of points Clm,n— 260i is the same as the set flm, n> so the series
for ^' (z 4- 2a)i) is a derangement of the series for g)' {z). The series being
absolutely convergent, we have
^y{z + 2ay,) = ^'{z);
that is to say, ^' (z) has the period 2&)i ; in like manner it has the period 2«d2.
Since ^/(^) is analytic except at its poles, it follows from this result that
g>' (z) is an elliptic function.
If now we integrate the equation ^' (z + 2coi) = ^' (z), we get
^{z-[-2a),) = ^o{z)+A,
where A is constant. Putting z = — Wi and using the fact that p (z) is an
even function, we get ^4 = 0, so that
^{z + 2&)i) = ^{z);
in like manner ^ (z + 2co2) = ^ (z).
Since p (z) has no singularities but poles, it follows from these two results
that p (z) is an elliptic function.
There are other methods of introducing both the circular and elliptic functions into
analysis ; for the circular functions the following may be noticed :
(1) The geometrical definition in which sin z is the ratio of the side opposite the angle
z to the hypotenuse in a right-angled triangle of which one angle is z. This is the definition
given in elementary text-books on Trigonometry ; from our point of view it has various
disadvantages, some of which are stated in the Appendix.
(2) The definition by the power series
z^ z°
sin. = .---f--....
2022] ELLIPTIC FUNCTIONS 429
(3) The definition by the product
(4) The definition by ' inversion ' of an integral
J 0
The periodicity properties follow easily from (4) by taking suitable paths of integration
(cf. Forsyth, Theory of Functions, pp. 213-222), but it is extremely difficult to prove that
sin z defined in this way is an analytic function.
The reader will see later (§§ 22-82, 22*1, 20-42, 20-22 and 5^ 20-53 example 4) that
elliptic functions may be defined by definitions analogous to each of these, with corre-
sponding disadvantages in the cases of the first and fourth.
Example. Deduce the periodicity of ^ (2) directly from its definition as a double series.
[It is not difficult to justify the necessary derangement.]
20*22. The differential equation satisfied by jp (2).
We shall now obtain an equation satisfied by ^ (z), which will prove to
be of great importance in the theory of the function.
The function ^ (z) — z~^, which is equal to S' [{z — n,„_ „)~2 — n~f„|, is
m, n
analytic in a region of which the origin is an internal point, and it is an
even function of z. Consequently, by Taylor's theorem, we have an expansion
of the form
^ (z) - z-'^ = i g,z^- + ^^g,z^ + 0 (^«)
valid for sufficiently small values of | ^ ] . It is easy to see that
^, = 60 2' n-\, gs = UO S' n^,.
m, n m, n
Thus (fj {z) = Z-' + ^^g.z-^ + yi,z* + 0 {z') ;
differentiating this result, we have
ip'{z)=-2z-' + ^^g,z+\g,z^+0{z%
Cubing and squaring these respectively, we get
f {z) = Z-' + l^g.z-- + l^g, -f 0 {z%
g)'=' {z) = 4^-" - I g,z-"- -^g,+ 0 (z').
Hence ^'^ (z) - 4>f (z) = - g.,z-^ -g,+ 0 {z%
and so <p'^ {z) - 4^j=' {z) + g. <p\z) -{-g-i^ 0 {z%
That is to say, the function y'''{z) - '^s^^ {z) -\- g.2,<^ {z) + g.,, which is
obviously an elliptic function, is analytic at the origin, and consequently
it is also analytic at all congruent points. But such points are the only
possible singularities of the function, and so it is an elliptic function ivith
no singularities ; it is therefore a constant (§ 20'12, IV).
On making ^-*0, we see that this constant is zero.
430 THE TRANSCENDENTAL FUNCTIONS [CHAP. XX
Thus, finally, the function <^{z) satisfies the differential equation
where g^ and g^ (called the invariants) are given by the equations
5r, = 60 2' n;;;f,„ g, = \^Q 2' n-^;,.
m, n m, n
Conversely, given the equation
{^£j = ^f-g.y-g.,
if numbers coj, eog can he determined* such that
m, n m,n
then the general solution of the differential equation is
where a is the constant of integration. This may be seen by taking a new
dependent variable u defined by the equation f y=^<^{u), when the dififerential
equation reduces to ( -i- ,
Since f {z) is an even function of z, we have y — ^(^z± a), and so the
solution of the equation can be written in the form
y=<p{z + a)
without loss of generality.
Example. Deduce from the differential equation that, if
then
02=5-2/2^5,
C4=.93/22.7, ^6=5-22/24. 3. 52,
^^"2^5. 7. 11'
i" 25.3.53.13 ' 24.72.13' '" ~ 2^. 3. 52. 7 . 11
20*221. The integral formula for ^ {z).
Consider the equation
/•OO
^=j {U'-gd-g,)-^dt,
determining z in terms of ^ ; the path of integration may be any curve Avhich
does not pass through a zero of H^ — g^t — g.^.
On differentiation, we get
[dz) =4^-^.r-^3,
and so ^ — ^j(^z + a),
where a is a constant.
* The difficult problem of determining Wj and wo when f/2 ^^^ 9z ^^'^ given is solved in
§ 21-73.
t This equation in u always has solutions, by § 20-13.
20-221, 20-222] elliptic functions 431
Make f ^ oo ; then z --0, since the integral converges, and so a is a pole
of the function g> ; Le., o is of the form 0.^, n. and so ^ = ^{z+ 0,^, «) = ^ (z).
The result that the equation 2 = I {^i^—git — gs) dt is equivalent to
the equation ^ = ^(z) is sometimes written in the form
{U'-g,t-g,) ^dt
20*222. An illustration from the theory of the circular functions.
The theorems obtained in §§ 20*2-20'221 may be illustrated by the corresponding
results in the theory of the circular functions. Thus we may deduce the properties
of the function cosec^z from the series 2 {z — mn)~^ in the following manner:
jn= — 00
Denote the series by/ (2) ; the series converges absolutely and uniformly* (with regard
to z) except near the points mn &t which it obviously has double pales. Except at these
points, f{z) is analytic. The effect of adding any multiple of tt to 2 is to give a series
whose terms are the same as those occurring in the original series ; since the series
converges absolutely, the sum of the series is unaffected, and so f{z) is a periodic function
of z with period re.
Now consider the behaviour oi f{z) in the strip for which —\t^R{z)^\iv. From
the periodicity of f{z\ the value of f{z) at any point in the plane is equal to its value at
the corresponding point of the strip. In the strip f{z) has one singularity, namely s = 0 ;
and f{z) is bounded as «-»-oo in the strip, because the terms of the series for /(s) are
small compared with the corresponding terms of the comparison series 2' m~'^.
m= — ai
In a domain including the point 2 = 0, f{z) — z~'^ is analytic, and is an even function ;
and consequently there is a Macl'aurin expansion
f{z)-z-^= I a2«2^
?l=0
valid when | s | < tt. It is easily seen that
a2„=27r-2»(2?i + l) 1 TO-2»-2,
)« = 1
and so «o=3> 02 = 677"* 2 m~'^ = ^g.
Hence, for small values of \z\,
f(z) = z-^ + i^ + ^z^ + Oiz*).
Differentiating this result twice, and also squaring it, we have
f"{z) = 6z-*+^^ + 0{z^),
/2(2) = 2-4 + ii2-2 + U+^(2')-
It follows that /" (z) - 6f^ (2) + 4/ (2) = 0 {z'-).
That is to say, the function /" (s) — 6/^^(2) + 4/(2) is analytic at the origin and it is
obviously periodic. Since its only possible singularities are at the points vitt, it follows
from the periodic property of the function that it is an integral function.
* By comparison with the series 2' m~^.
432 THE TRANSCENDENTAL FUNCTIONS [CHAP. XX
Further, it is bounded as 3-»-oo in the strip -^ir^E{z)^^7r, since f{z) is bounded
and so is* /" (2), Hence /" (z) - 6/2 (2) + 4/(2) is bounded in the strip, and therefore from
its periodicity it is bounded everywhere. By Liouville's theorem (§ 5-63) it is therefore
a constant. By making 2-»-0, we see that the constant is zero. Hence the function
cosec2 2 satisfies the equation
/"(2) = 6/2(2)-4/(2).
Multiplying by 2/' (2) and integrating, we get
/'2(2) = 4/2(2){/(2)-l}+C,
where c is a constant, which is easily seen to be zero on making use of the power series
for/'(2)and/(2).
We thence deduce that 2z= i t-^{t-l)~^dt,
Jf(z)
when an appropriate path of integration is chosen.
Example 1. li y = <^{z) and dashes denote differentiations with regard to 2, shew that
where ej, e^^ e^ are the roots of the equation 4:^—0^1—0^ = 0.
[We have y"^ = 4y^ — gi,y— g^
= 4(y-ei)(y-e2)(y-e3).
Differentiating logarithmically and dividing by y', we have
r=l
Differentiating again, we have *
Adding this equation multiplied by ^ to the square of the preceding equation,
multiplied by ^\, we readily obtain the desired result.
It should be noted that the left-hand side of the equation is half the Schwarzian
derivative t of 2 with respect to y ; and so 2 is the quotient of two solutions of the
equation
Example 2. Obtain the ' properties of homogeneity ' of the function ^ (2) ; namely that
^ (^' i 17) = ^"'^-^ i' I "') ' ^ ^^" ' ^~'32, X-65r3) = X-2p(,, g,, ^3),
\ ! ^0)2/ \ I 0)2/
where ^yz M denotes the function formed with periods 2co], 20)3 and ^(2; g,,, g^)
denotes the function formed with invariants g^, g^.
[The former is a direct conseqtience of the definition of ^ (2) by a double series ; the
latter may then be derived from the double series defining the g invariants.]
* The series for/" (2) may be compared with S' m~^.
m= — oo
t Cayley, Camb. Phil. Trans, xiii. p. 5.
20-3, 20-31] ELLIPTIC FUNCTIONS 438
20"3. The addition-theorem for the function ^ {z).
The function ^ {z) possesses what is known as an addition-theorem ; that
is to say, there exists a formula expressing ^ (^^ + 3/) as an algebraic function
of ip {z) and ^ {y) for general values * of 2^ and y.
Consider the equations
<p'{z) = Aip{z) + B, ^'{y) = Af{y) + B,
which determine A and B in terms of z and y unless ^{z) = f (y), i.e. unless f
z = ±y (mod. 2w^, 20)2).
Now consider ^' {^) — Aip {^) — B,
qua function of ^. It has a triple pole at ^ = 0 and consequently it has
three, and only three, irreducible zeros, by § 20'13 ; the sum of these is a
period, by § 20'14, and as ^=z, ^=y are two zeros, the third irreducible zero
must be congruent to —z — y. Hence —z — y isa, zero of ^' (^) — A^ (^) — B,
and so
^y(-z-y) = A^(-z-y) + B.
Eliminating A and B from this equation and the equations by which A
and B were defined, we have
iJiz) ^y{z) 1 =0.
p(y) f'iy) 1
Since the derived functions occurring in this result can be expressed
algebraically in terms of ^ {z), <p (?/), i^d (z + y) respectively (§ 20-22), this
result really expresses f{z-\-y) algebraically in terms of ^ {z) and <p (y).
It is therefore an addition-theorem.
Other methods of obtaining the addition-theorem are indicated in § 20-311
examples 1 and 2, and § 20-312.
A symmetrical form of the addition-theorem may be noticed, namely
that, if It -f- V + w = 0, then
I io{u) <p'{u) 1 =0.
f {w) iS (w) 1
20'31. Another form of the addition-theorem.
Retaining the notation of § 20*3, we see that the values of ^, which make
§>' (^) — A^ {^) — B vanish, are congruent to one of the points z, y, —z — y.
* It is, of course, unnecessary to consider the special cases when y, or z, or y + z is a period.
t The function <p [z) - ^ (y), qua function of z, has double poles at points congruent to z = 0,
and no other singularities ; it therefore (§ 20-13) has only two irreducible zeros ; and the points
congruent to 2= ±j/ therefore give all the zeros of ^ (2) - ip (y).
\V. M. A. 28
434 THE TRANSCENDENTAL FUNCTIONS [CHAP. XX
Hence ^'^(^) — {A^ (f) + B\^ vanishes when ^ is congruent to any of the
points z, y, —z — y. And so
^f (0 - -4^^^ iO - (2AB + g,) ^ (0 -(B^+ gs)
vanishes when ^(^) is equal to any one of ^i>(z), ^{y), fiz +y).
For general values of z and y, ^ {z), f (y) and ^(z + y) are unequal and
so they are all the roots of the equation
4>Z' - A'Z' - (2AB + g,)Z-(B' + g,) = 0.
Consequently, by the ordinary formula for the sum of the roots of a cubic
equation,
and so p (^ + 2/) = i {j^^^y)\ " ^ (^) " ^ (2/)'
on solving the equations by which A and B were defined.
This result expresses ^j (z + y) explicitly in terms of functions of z and
of y.
20*311. The duplication formula for ^ {z).
The forms of the addition-theorem which have been obtained are both
nugatory when y = z. But the result of § 20"81 is true in the case of any
given value of z, for general values of y. Taking the limiting form of the
result when y approaches z, we have
lim &(z + y) = \ lim \^ ,{~A^l\ - <^ (z) - Hm <a (y).
JFrom this equation, we see that, if 2z is not a period, we have
on applying Taylor's theorem to ^(z + h), ^' (z + h); and so
unless 2z is a period. This result is called the duplication formula.
Example 1. Prove that
l|F_(!)^E(y)|^ .,(,)_. ,(,+3,),
4l^(^)-^(.?/)/ ^^^ ^ ^ ^-^^^
qua function of z, has no singularities at points congruent with 2 = 0, ± y ; and, by naaking
use of Liouville's theorem, deduce the addition-theorem.
\
20-311, 20-312] ELLIPTIC FUNCTIONS 435
Example 2. Apply the process indicated in example 1 to the function
and deduce the addition-theorem.
Example 3. Shew that
P(^+y) + ^(':-y) = {&{z)-&{l/)}-n{^iP(z)ip(^)-y,}{ipiz) + ip(i/)}-g3l
[By the addition-theorem we have
Replacing ^2(2) and iP'H^) by 4^3(2)-5'2g>(2)-5'3 and 4.^^ (;!/)- 92 ^ (^) - ff3 respec-
tively, and reducing, we obtain the required result.]
Example 4. Shew, by Liouville's theorem, that
j^{iP(^-o^)P{z-b)}=^{a-b){^'{z-a) + ^'{z-b)}-^(a-h){^{z-a)-ip{z-b)}.
(Trinity, 1905.)
20*312. Abel's* method of proving the addition-theorem for ^(z).
The following outline of a method of establishing the addition-theorem for ^ (2) is
instructive, though a completely rigorous proof would be long and tedious.
Let the invariants of ^ (2) be g^, gz; take rectangular axes OX, OF in a plane, and
consider the intersections of the cubic curve
with a variable line y = mx-\-n.
If any point (xx , yi) be taken on the cubic, the equation in z
ip{z)-x, = Q
has two solutions -\-Zx, —Zi (§ 20-13) and all other solutions are congruent to these two.
Since ^'^{z) = 'i^^iz)—gi^{z)-gz, we have <^"^ {z)=yi ; choose z^ to be the solution for
which gJ' (2i)= +yi, not -y^.
A number Si thus chosen will be called the parameter of (^1 , yi) on the cubic.
Now the abscissae x^, x^, x^ of the intersections of the cubic with the variable line
are the roots of
0 {x) = 4.-^ —g-i^-g^- ("i-^ + ^0^ = '^>
and so ^ (x) = 4 (^ — .ri) (^ - x-i) {x - X3).
The variation S^,. in one of these abscissae due to the variation in position of the line
consequent on small changes bm, hi in the coefficients m, n is given by the equation
d)' (.^v) S^r + ^ 3'" + 5^ Sn = 0,
^ ^ cm on
and so ^' (o-v) bx^ = 2 {mx^. + n) {x^ hn 4- S?i),
3 g^. 3 Xrbm + bn
whence 2 — =22 ^-ttt — r— ,
,.=1 mXr-\-n ,.=1 <p [Xr)
provided that Xi, x^, x^ are unequal, so that (f)' (x^) =/=0.
* See Abel, Oeuvres, i. pp. 145-211.
28—2
436 THE TRANSCENDENTAL FUNCTIONS [CHAP. XX
Now, if we put x{x8m + 8n)l(j) {x\ qua function of x, into partial fractions, the result is
3
2 A^/{x — Xfj^
where Ar= lim x(x8m + 8n) ^
=Xr{Xr87)i + 8n) lim (x-Xr)/(l){x)
x-*-Xr
= Xr {Xj.8m + 8n)l(f>' {x^),
by Taylor's theorem.
3 3
Putting ^ = 0, we get 2 8xrl^r = ^, i-^- ^ §2^=0-
r=l r=l
That is to say, the sum of the parameters of the points of intersection is a constant
independent of the position of the line.
Vary the line so that all the points of intersection move off to infinity (no two points
coinciding during this process), and it is evident that Zi+z^ + Zz is equal to the sum of the
parameters when the line is the line at infinity ; but when the line is at infinity, each
parameter is a period of ^ (z) and therefore 2, +23 + 23 is a period of ip (z).
Hence the sum of the parameters of three collinear points on the cubic is congruent to
zero. This result having been obtained, the determinantal form of the addition-theorem
follows as in § 20'3.
20'32. The constants e^, e^, e^.
It will now be shewn that ^ (&)i), ^ {w-^, g> (ws), (where 0)3 = — &>i — twa), are
all unequal ; and if their values be gj, e<i, e-i, then gj, e^, e.^ are the roots of the
equation H^ — g2t — (^3 = 0.
First consider ip' (w^). Since ^/ (z) is an odd periodic function, we have
^y (coi) = - ^d' (- 6)1) = - ^/ (2a)i - ft)i) = - ^' (toi),
and so ^' (wj) = 0.
Similarly ^' (co^) = ^' (ws) = 0.
Since ^' (z) is an elliptic function whose only singularities are triple poles
at points congruent to the origin, ^./ (z) has three, and only three (§ 20'13),
irreducible zeros. Therefore the only zeros of ^' {z) are points congruent to
COj, &)2, COs.
Next consider ^ (z) — e-^. This vanishes at coi and, since ^y (&>i) = 0, it has
a double zero at cOi. Since p(z) has only two irreducible poles, it follows
from § 20"13 that the only zeros of ^){z)—ei are congruent to oji. In like
manner, the only zeros of ^j (z) - e^, ^J {z) — e-^ are double zeros at points con-
gruent to CO2, co;i respectively.
Hence Ci^e^i^ e-i. For if gj = e.., then tp {z) — e^ has a zero at tos^ which is
a point not congruent to coj.
Also, since ^'-{z) = A^<^d^{z) — g,^^{z) — g,, and since ^/(2') vanishes at tuj, CO2,
(03, it follows that 4gj» (z) - g^ip {z) — g.^ vanishes when |^i> {z) = e^, gg or 63.
That is to say, e^, e.,, e^ are the roots of the equation
4>t'-g,t-g, = 0.
20-32, 20*33] elliptic functions . 437
From the well-known formulae connecting roots of equations with their
coefficients, it follows that
61 + 6^ + 63 = 0,
_ 1
6263 + 6361 + 616.2 — ^9^'
_ 1
616263 — j^j.
* Example 1. When ^2 and gg are real and the discriminant ^2^ - ^^^ffs^ is positive, shew
that ei, 62, 63 are all real ; choosing them so that Cj >e2>e3, shew that
/oo
and 0)3= - ^• [ ' {gs+g-it- ^t^) ~^dt,
so that 0)1 is real and 0)3 a pure imaginary.
'■ Example 2. Shew that, in the circumstances of example 1, ^{z) is real on the peri-
meter of the rectangle whose corners are 0, 6)3, (Ui + wg, a^.
20 'SS. The addition of a half-period to the argument of <^ (z).
From the form of the addition-theorem given in § 20'31, we have
3
and so, since ^J'2(3) = 4 n {^(z)-er},
r=l
we have j,(,+„,) = <-*''^^-^f/*|^^^ -«-«-«„
3
on using the result 2 6^ = 0 ;
r=l
this formula expresses ^{z + (Oi) in terms of p{z).
Example 1. Shew that
^(ia)i) = ei±{(ei-e2)(ei-e3)}i
Example 2. From the formula for ip{z + a)i) combined with the result of example 1,
shew that
i^ (ioi + C02) = 61 + {(ei - 62) («! - 63)}*.
(Math. Trip. 1913.)
Example 3. Shew that the value of i^' {z)^' (z + <oi) ^' (z + ati) ^' {z+oi^) is equal to
the discriminant of the equation At^ — g2t—g3 = 0.
[Differentiating the result of § 20'33, we have
from this and analogous results, we have
P' (2) ^y (2 -t- coi) ^' {z + 0)2) f (2 -h o>3)
= {ei - e^2)' {^2 - <^^? («3 - e,f <^0'^ {z) n^ {^ (z) - e,} - ^
= lG{e,-e2)He2-e,f{es-ei)%
which is the discriminant g/ — 27g% in question.]
438 THE TRANSCENDENTAL FUNCTIONS [CHAP. XX
Example 4. Shew that
^' (|a)i) = - 2 {(ei - 62) (^1 - «3)}* {(ei - e^)^ + (ei - 63)^}-
(Math. Trip. 1913.)
Example 5. Shew that, with appropriate interpretations of the radicals,
{^ {2z) - e.,}i {^ (22) - es}i + {^ {2z) - e^}^ {g> {2z) - ^i}*
+ {^ {2z) - e,]^ {^ (22) - e,}i = ^ (2) - g> (22).
20-4. Quasi-periodic functions. The function* ^ (z).
We shall next introduce the function ^(z) defined by the equation
coupled with the condition lim {^(z) — z-'^] = 0.
z-*0
Since the series for ^ (z) - z'^- converges uniformly throughout any
domain from which the neighbourhoods of the pointsf Q>'rn,n are excluded, we
may integrate term-by- term (§ 4*7 ) and get
^{z)-z-^ = -r{&{^)-^-']dz
J 0
= - S' f{{z-nrr,,n)-'-^;:n]dz,
m,n J 0
1 ^, f 1 1 2
and so ^(z) = - + f ] 7^ 1- j^ — + ^2
^ m,n 1-2' ^^m,n ^^m,n ^^m,n)
The reader will easily see that the general term of this series is
Odn^.nl"') as |n^,n|-*oo;
and hence (cf § 20"2), ^(z) is an analytic function of z over the whole ^-plane
except at simple poles (the residue at each pole being + 1) at all the points
of the set n„i_„.
It is evident that
-^(-z) = l+ t' l-^^^- - J- + ^1 ,
Z m,n (Z -r i-'"m,n ^^m,n ^^m,n)
and as this series consists of the terms of the series for ^(z), deranged in the
same way as in the corresponding series of § 20"21, we have, by § 2'52,
!:(-z)=-^(z),
that is to say, ^(z) is an odd function of z.
* This function should not, of course, be confused with the Zeta-function of Eiemann,
discussed in Chapter xiii.
t The symbol fl',„,„ is used to denote all the points fl„,.„ with the exception of the origin
(cf. § 20-2).
20-4-20-4]l] ELLIPTIC FUNCTIONS 439
Following up the analogy of § 20-222, we may compare ({z) with the function cot 2
°° d
defined by the series z~'^+ 2' {(2-m7r)~^ + (m7r)~'}, the equation -j- cot z= - cos&c^ z
corresponding to -r- C {^) = ~ & i^)-
20'41. The quasi-periodicity of the function ^ (z).
The heading of § 20*4 was an anticipation of the result, which will now be
proved, that ^(z) is not a doubly-periodic function of z ; and the effect on
^(z) of increasing z by 2ci)i or by 2(02 will be considered. It is evident from
§ 2012 (III) that ^{z) cannot be an elliptic function, in view of the fact that
the residue of ^{z) at every pole is + 1.
If now we integrate the equation
^(z + 2ft)i) = ^ (z),
we get ^{z+2(o,)=^(z)+2v^,
where 27;i is the constant introduced by integration; putting z = — (Oi, and
taking account of the fact that ^{z) is an odd function, we have
In like manner, ?(^ + Scoj) = f (z) + 2r}2,
where V2 = ^((^d-
Example 1. Prove by Liouville's theorem that, if x+y-\-z=0, then
{c(^)+f(3/)+a^)F+r(^)+r(y)+r(2)=o.
(Frobenius u. Stickelberger, Crelle, lxxxviii.)
[This result is a pseudo-addition theorem. It is not a true addition-theorem since
C'{x), f'(y), C {z) are not algebraic functions of ({x\ ^(y), ({z).]
Example 2. Prove by Liouville's theorem that
2i 1 iO{x) i^^{x)
-r
1 ^{x) i^'{x)
= C{^^+y+z)-C{x)-({y)-az).
1 ^{y) r(^)
1 <P{y) f'iy)
1 i^{z) ia2(2)
1 <^{z) r(^)
Obtain a generalisation of this theorem involving n variables.
(Math. Trip. 1894.)
20"411. The relation between rji and rj^.
We shall now shew that
1 .
^i&>2 - V2<Oi =^7rt.
To obtain this result consider ^(z)dz taken round the boundary of a
J c
cell. There is one pole of ^(z) inside the cell, the residue there being -f- 1.
Hence ^(z)dz = 27ri.
i c
440 THE TRANSCENDENTAL FUNCTIONS [CHAP. XX
Modifying the contour integral in the manner of § 20-12, we get
27ri= I {^{z) - ^{z+2o),)] dz - {^(z) -^{z + 2co,)] dz
.' t -It
= - 2r), dt + 277i dt,
and so 2Tri = — ^rj^w^ + 47;ia)2,
which is the required result.
20-42. The function a {z).
We shall next introduce the function a {z), defined by the equation
j^\oga{z)=^{z)
coupled with the condition lim {a{z)/z} = 1.
On account of the uniformity of convergence of the series for ^(z), except
near the poles of ^(z), we may integrate the series term-by- terra. Doing so,
and taking the exponential of each side of the resulting equation, we get
. .(.) = . n'|(i-^)expr^ +?^)['
the constant of integration has been adjusted in accordance with the condition
stated.
By the methods employed in §§ 20-2, 20-21, 20-4, the reader will easily
obtain the following results :
(I) The product for a {z) converges absolutely and uniformly in any
bounded domain of values of z.
(II) The function a {z) is an odd integral function of z with simple zeros
at all the points n,„_„.
The function a {z) may be compared with the function sin z defined by
the product
z n' jfl-— ^e^/u«-)l
m=-aoiV tniTJ J
d . d
the relation -y- log sin z = cot z corresponding to y- log a {z) = ^{z).
20-421. The quasi-periodicity of the function a {z).
If we integrate the equation
C(^ + 2a,0=C(^)+^'7i,
we get or(2 + 2&)i)=- ce-'''-o-(^),
where c is the constant of integration; to determine c, we put z = — w^, and
then
a (coj) = — ce~-''i"'' a (wi).
20-42-2051] ELLIPTIC FUNCTIONS 441
Consequently c = — e-'''"',
and a-(z + 2a)i) = - e^^ (^+"') a- (z).
In like manner (t{z + 2co2) = — e2^s(«+"2) o- (z).
These results exhibit the behaviour of (t{z) when z is increased by a
period of ^ (z).
If, as in § 20-32, j^ve write 0)3-= — (o^- a)^, then three other Sigraa-functions
are defined by the equations
ar (z) = e-'"-^ a{z + Q)r)/<T {(Or) (v = 1, 2, 3).
The four Sigma-functions are analogous to the four Theta-functions dis-
cussed in Chapter xxi (see § 2 19).
* Example 1. Shew that, if m and n are any integers,
(r(z + 2ma>i + 2na2) = (-)"'*" o- (z) exp {(2OTr;i + 2nT}2) 2 + 2m^riia>i + 4m7i?/i<B2 + 2%2,^^ 0)2},
and deduce that rjico^ — ^201 is an integer multiple of ^ni.
• Example 2. Shew that, if 5' = exp(7ria)2/<»i)) so that |5'|<1, and if
then F{z) is an integral function with the same zeros as a{z) and also F{z)I(t{z) is a
doubly-periodic function of z with periods 2o)i, 2w2.
« Example 3. Deduce from example 2, by using Liouville's theorem, that
Example 4. Obtain the result of example 3 by expressing each foctor on the right as
a singly infinite product.
20"5. Formulae expressing any elliptic function in terms of Weiersti'assian
functions with the same periods.
There are various formulae analogous to the expression of any rational
fraction as (I) a quotient of two sets of products of linear factors, (II) a sum
of partial fractions ; of the first type there are two formulae involving Sigma-
functions and Weierstrassian elliptic functions respectively ; of the second
type there is a formula involving derivates of Zeta-functions. These formulae
will now be obtained.
20-51. The expression of any elliptic function in ter-ms of ^ (z) and f' {z).
'Lat f{z) be any elliptic function, and, let <p{z) be the Weierstrassian
elliptic function formed with the same periods 2&)3, 2&)2.
We first write
442 THE TRANSCENDENTAL FUNCTIONS [CHAP. XX
The functions
are both even functions, and they are obviously elliptic functions ■wheny"(^^) is
an elliptic function.
The solution of the problem before us is therefore effected if we can
express any even elliptic function <f) (z), say, in terms of g) {z).
Let a be a zero of <^ {z) in any cell ; then the point in the cell congruent
to — a will also be a zero. The irreducible zeros of <^ {z) may therefore be
arranged in two sets, say a^, a^, ... «„ and certain points congruent to — aj,
— (Xqj ... — Ctjl-
In like manner, the irreducible poles may be arranged in two sets, say
61, 62, ... hn, and certain points congruent io —h^, —h.,, ... — bn.
Consider now the function*
1 fj \iJ(z)-^(ar)
(fi(z),.^-,\^j{z)-^j(br)
It is an elliptic function of z, and clearly it has no poles ; for the zeros of
(f) (z) are zeros f of the numerator of the product, and the zeros of the
denominator of the product are polesf of (f){z). Consequently by Liouville's
theorem it is a constant, A^, say.
Therefore </> (.) = A. 11 I^Mnf M ,
that is to say, (f> (z) has been expressed as a rational function of g? (z).
Carrying out this process with each of the functions
f(z) +/(- z), {f{z) -/(- z)} {^y {z)]-\
we obtain the theorem that any elliptic function f{z) can he expressed in terms
of the Weierstrassian elliptic functions ^j (z) and p' (z) with the same periods,
the expression being rational in ^(z) and linear in ^' (z).
2052. llie expression of any elliptic function as a linear combination of
Zeta-f unctions and their derivates.
Let f(z) be any elliptic function with periods 2mi, 2&)2. Let a set of
irreducible poles of f{z) be a^, tta, ... a„, and let the principal part (§ 5'61)
of f{z) near the pole a^ be
(^k, 1 , (^k, 2 , , ^k, rjc
z - a^ {z- ajcf ' ' {2- akY"
* If any one of the points a,, or b^ is congruent to the origin, we omit the corresponding
factor ^ (2) - g? (a;} or ^J (z) - ^ (6^). The zero (or pole) of the product and the zero (or pole)
of 4> {z) at the origin are then of the same order of multiplicity. In this product, and in that of
§ 20-.53, factors corresponding to multiple zeros and poles have to be repeated the appropriate
number of times.
t Of the same order of multiplicity.
20-52, 20-53] elliptic functions 443
Then we can shew that
d'
where A^ is a constant, and f<*' (z) denotes -y-^ ^(z)-
Denoting the summation on the right by F (z), we see that
F(z + 2a),)-F{z)= t 2viCk,u
k = l
by § 20-41, since all the derivates of the Zeta-functions are periodic.
n
But S Ca; 1 is the sum of the residues of f(z) at all of its poles in a cell,
and is consequently (§ 20-12) zero.
Therefore F(z) has period 2(Oj, and similarly it has period 2&)2; and so
f(z) — F{z) is an elliptic function.
Moreover F (z) has been so constructed that f(z) — F (z) has no poles at
the points a^, a2, ...a„; and hence it has no poles in a certain cell. It is
consequently a constant, -4 2, by Liouville's theorem.
Thus the function f (z) can be expanded in the form
A,^X t ^-~^,Ck,,K''-''{^-(ik\
k = \s=\ \S— \.)\
This result is of importance in the problem of integrating an elliptic
function /(^) when the principal part of its expansion at each of its poles is
known ; for we obviously have
/
f{z)dz=A^z-V S
A;=l
CA,il0g0r(2:-a;(.)
« = 2 \P ~ ^) ■
where (7 is a constant of integration.
» Example. Shew by the method of this article that
r(^)=*r(^)+T^5'2,
and deduce that
where C is a constant of integration.
20*53. The expression of any elliptic function as a quotient of Sigma-
f unctions.
Let f{z) be any elliptic function, with periods 2&), and 2&)o, and let a set
of irreducible zeros oi f{z) be a^, a.,, ... a«. Then Q 20-14) we can choose a
444 THE TRANSCENDENTAL FUNCTIONS [CHAP. XX
set of poles b^, b^, ... ba such that all poles oi f{z) are congruent to one or
other of them andf
tti + C/a + . . . + ttn = &i + 62 + . • • + ^w
Consider now the function
n °"^^~^^\
r^l<T{z-br)'
This product obviously has the same poles and zeros as f{z)\ also the
effect of increasing z by 2w^ is to multiply the function by
^ exp [Itj, (z - ar)} _ ^
r=l exp {277i {z - br)}
The function therefore has period 2fWi (and in like manner it has period
2(1)2), and so the quotient
/{z}-:- n -~ r-:
r=l O- (^Z — Or)
is an elliptic function with no zeros or poles. By Liouville's theorem, it must
be a constant, A^ say.
Thus the function f{z) can be expressed in the form
r = \ (T {Z — Or)
An elliptic function is consequently determinate (save for a multiplicative
constant) when its periods and a set of irreducible zeros and poles are
known.
Example 1. Shew that
Example 2. Deduce by differentiation, from example 1, that
and by further differentiation obtain the addition-theorem for ^ (2).
Example .3. If 2 a,.= 2 b„ shew that
r=l
I (r{a^ - hi) o- {ar - h^) ...a^ar-bj ^ ^
r=i o" (% — «!) a-icir-a^ ... ^... (r{ar — an) '
the * denoting that the vanishing factor a{ar-ar) is to be omitted.
Example 4. Shew that
^J(,r)-e,= tr,.2(2)V(2) (^-=1,2,3).
[It is customary to define {^J(z)-e,]^ to mean o-,. (2)/o- (2), not - (Tr (z) I o- (z).]
t Multiple zeros or poles are, of course, to be reckoned according to their degree of multi-
plicity ; to determine h^ ,h,, ... 6„ , we choose hi,h.,, ...h^^i, h.^' to be the set of poles in the cell in
which rti, 02. ••• <in lie, and then choose h,^, congruent to V> i" such a way that the required
equation is satisfied.
20-54, 20-6] ELLIPTIC FUNCTIONS 445
Example 5. Establish the * three-term equation,' namely,
o- (2 + a) o- (2- a) o- (5 + c)(r (6- c) + o- (2 + 6) or (2- 6) o-(c + «)<r(c- a)
+ (r{z + c)(T{z-c)(T{a-\-h) (r{a-b)-=0.
20*54. The connexion between any two elliptic functions with the same
periods.
We shall now prove the important result that aa algebraic relation exists
between any two elliptic functions, f{z) and (f> (z), with the same periods.
For, by § 20*51, we can express /(^) and (f)(z) as rational functions of the
Weierstrassian functions ^ (2:) and ^' (z) with the same periods, so that
f(z) = R, {p (z), ^y {z)], 4> {z) = R, [<p (z), gy (z)},
where R^ and R^ denote rational functions of two variables.
Eliminating gJ (z) and ^' (z) algebraically from these two equations and
^" (^) = 4^' (^) - 92 p (^) - gs,
we obtain an algebraic relation connecting /(2^) and (f> (z); and the theorem
is proved.
A particular case of the proposition is that every elliptic function is con-
nected with its derivate by an algebraic relation.
If now we take the orders of the elliptic functions/ (2') and 0 (z) to be m
and n respectively, then, corresponding to any given value of f{z) there is
(§ 20'13) a set of m irreducible values of 2^, and consequently there are m
values (in general distinct) of <^ (z). So, corresponding to each value of/, there
are m values of (f) and, similarly, to each value of <^ correspond n values of /.
The relation between /(2) and ^(2) is therefore (in general) of degree m
in (p and n in/.
The relation may be of lower degree. Thus, iff{z) = ^0 (2), of order 2, and
(f) (2:) = ^0- (z), of order 4, the relation is/^ = cj).
As an illustration of the general result takef{z) = 'p(z), of order 2, and
(f) (^z) = (/ (z), of order 3. The relation should be of degree 2 in cf) and of
degree 3 in/; this is, in ftict, the case, for the relation is (f)- = 4f^ " Q^f— g^-
Example. If u, v, w are three elliptic functions of their argument of the second order
with the same periods, shew that, in general, there exist two distinct relations which are
linear in each of m, v, w, namely
A UVW + B vw + C ^m + D uv->r E u + F V + 0 w + H =0,
A'uvw + B'vw+C'ivu + D'uv-\-E'ii. + F'v+G\u-{-H' = 0,
where A, B, ..., H' are constants.
20*6. On the integration of [uoX^+A^a^x^ + Qa.,x- + 4<a-iX + ttj} ~K
It will now be shewn that certain problems of integration, which are
insoluble by means of elementary functions only, can be solved by the intro-
duction of the function ^J (z).
446 THE TRANSCENDENTAL FUNCTIONS [CHAP. XX
Let ttoos* + 4iaia^ + Qa^af + 4ia3X + Ui = f (x) be any quartic polynomial
which has no repeated factors ; and let its invariants* be
^2 = aotti — 4aia3 + Sag^ •
^3 = a^a^a^ + ^a^a^a^ — Uz^ — aotts^ — ai^a^.
Let z = I {/(t)} ~ ^ dt, where cc^ is any root of the equation /(x) = 0 ; then,
J x„
if the function p (z) be constructed f with the invariants g2 and g^, it is possible
to express x as a rational function of^{z;g^, g^.
[Note. The reason for assuming that f{x) has no repeated factors is that, when f{x)
has a repeated factor, the integration can be effected with the aid of circular or logarithmic
functions only. For the same reason, the case in which ao=ai=0 need not be considered.]
By Taylor's theorem, we have
f{t) = ^A, (t - X,) + 6^2 (t - ooo)' + 4A, (t - ^o)' + Aoit- x,y,
(since /(a?o) = 0), where
Ao = ao, A^ = aoXo + ai,
A2 = aoXfi^ + 2aiiro + ao,
Ag = a^x^ + SajiCo^ + Saa^o + a^.
On writing {t — Xq)-^ = t, (a; - x„)-^ = |, we have
2=1 {4>AsT'+6A,T' + 4A,T+A,}-^dT.
To remove the second term in the cubic involved, write|
and we get
The reader will verify, without difficulty, that
SA^^-^A^As and 2A,A,A,- A.f - AoA,'
are respectively equal to g., and g„ the invariants of the original quartic
and so
s=iJ(2;g.„gs).
Now x = Xo + A,{s-^A,}-\
and hence ^ = ^o + if'{xo){^(z;g„g,)-^^^f"(x,)]-\
so that X has been expressed as a rational function of ^0 (z; g^, g^).
* Burnside and Panton, Theory of Equations, u. p. 113.
t See §21-73.
+ This substitution is legitimate siuce J3 + O; for the equation ^3 = 0 involves f{x) = 0
having x^^^x^ as a repeated root.
20-7] • ELLIPTIC FUNCTIONS 447
This formula for x is to be regarded as the integral equivalent of the
relation
Example 1. With the notation of this article, shew that
{
' Example 2. Shew that if
J a
where a is ajiy constant, not necessarily a zero of /(a?), and f{x) is a quartic polynomial
with no repeated factors, then
^-^ I {/(«#r(^)+i/(«)«i?(^)-^vr(«)}+^/(«).r («)
2{ii)(2)-^/"(a)}2-3V/(a)/-(a)
the function ^(z) being formed with the invariants of the quartic f{x).
(Weierstrass.)
[This result was first published in 1865, in an Inaugural-dissertation at Berlin by
Biermann, who asgribed it to Weierstrass.]
Example 3. Shew that, with the notation of example 2,
gPM {/(•^)/(«^)}^+/(a) , /'(«) .f"{a)
"'^^'^ 2{x-af '^4:{x-a)^ 24 '
and
20'7. The uniformisation* of curves of genus unity.
The theorem of § 20'6 may be stated somewhat differently thus :
If the variables a; and y are connected by an equation of the form
y^ = aooc* + 4>aiaf -f 6a2^" + ^0,3^ + «45
then they can be expressed as one-valued functions of a variable z by the
equations
^ = ^0 + If (^0) [i^ (^) - i-J" (^0)}-^ \
y = if M ^' (z) W (^) - iV/" (^0)}-^ J '
where f{x) = aoX* + 4aia?^ + Gag^^ + ^ago; + a^, Xq is any zero of f{x), and the
function <^ {z) is formed with the invariants of the quartic ; and z is such that
^=r [f{t)]-ut.
■1 a;o
It is obvious that y is a two- valued function of x and a; is a four-valued
function of y; and the fact, that x and y can be expressed as one-valued
* This term employs the word uniform iu the sense one-valued. To prevent confusion with
the idea of uniformity as explained in Chapter in, throughout the present work we Lave used the
phrase ' one-valued function ' as being preferable to ' uniform function.'
448 THE TRANSCENDENTAL FUNCTIONS ' [CHAP. XX
functions of the variable z, makes this variable z of considerable importance
in the theory of algebraic equations of the type considered ; z is called the
uniformising variable of the equation
y^ = aocc* + 4!aiX^ + Ga^x^ + 4iasa; + a^.
The reader who is acquainted with the theory of algebraic plane curves will be aware
that they are classified according to their deficiency or genus*, a luimber whose geometrical
interpretation is that it is the difference between the number of double points possessed
by the curve and the maximum number of double points which can be possessed by a
curve of the same degree as the given curve.
Curves whose deficiency is zero are called unicursal curves. 1i f{a\ y) = 0 is the equation
of a unicursal curve, it is well known t that x and y can be expressed as rational functions
of a parameter. Since rational functions are one- valued, this parameter is a uniformising
variable for the curve in question.
Next consider curves of genus unity; let f{x,y) — 0 be such a curve; then it has
been shewn by Clebsch | that x and y can be expressed as rational functions of ^ and rj
where t;^ is a polynomial in | of degree three or four. Then, by § 20"6, ^ and rj can be
expressed as rational functions of if) (z) and ^' (z) (these functions being formed with
suitable invariants), and so x and ?/ can be expressed as one-valued (elliptic) functions of 2,
which is therefore a uniformising variable for the equation under consideration.
When the genus of the algebraic curve f{x,9/) = 0 is greater than unity, the uniformi-
sation can be effected by means of what are known as autoinorphic functions. Two classes
of such functions of genus greatei- than unity have been constructed, the first by Weber
[GiHtinger Nach., 1886), the other by Whittaker {Phil. Trans, cxcii. pp. 1-32, 1898).
The analogue of the period-parallelogram is known as the 'fundamental polygon.' In the
case of Weber's functions this polygon is 'multiply-connected,' i.e. it consists of a region
containing islands which have to be regarded as not belonging to it ; whereqiS in the case
of the second class of functions, the polygon is ' simplj'-connected,' i.e. it contains no
such islands. The latter class of functions may therefore be regarded as a more
immediate generalisation of elliptic functions. Cf. Ford, Introduction to theory of Auto-
morphic Functions (Edinburgh Math. Tracts, No. 6).
REFEEENCES.
K. Weierstrass, Werke, Bd. i. pp. 1-49, Bd. 11. pp. 245-255, 257-309.
C. Briot et J. C. Bouquet, Theorie des fonctions clUpticjues.
H. A. ScHWARZ, Formeln und Lehrsatze zum Gehrauche der elliptischen Funktionen,
Nach Vorhsimgen und Aufzeichmingen des Herrn Prof. K. Weierstrass.
A. L. Daniels, 'Notes on Weierstrass' methods,' American Journal, vi. pp. 177-182,
253-269 : vii. pp. 82-99.
J. LiouviLLE (Lectures pulilished by C. W. Borchardt), Crelle, Lxxx. pp. 277-311.
A. Enneper, Elliptische Funktionen. (Zweite Aiiflage, von F. Muller, Halle 1890.)
J. Tannery et J. Molk, Fonctions EUiptiques.
* French genre, German Gcuchlecht.
t See Salmon, Higher Plane Curves, Chapter 11.
+ Crelle, lxiv. pp. 210-270. A proof of the result of Clebsch is given by Forsyth, Theory of
Functions, pp. 536-538. See also Cayley, Proc. London Math. Soc. iv. pp. 347-352.
ELLIPTIC FUNCTIONS
449
1, Shew that
2. Prove that
Miscellaneous Examples.
^{'^+y)-p{z-y)=-f{z)^{y)W{z)-ipi^)}-\
where, on the right-hand side, the subject of differentiation is symmetrical in «, y, and w.
(Math. Trip. 1897.)
3. Shew that
r'{z-y) r'ky-y^) r"(«'-^) =\9'i
r(^-y) rky-y^) r(«'-^)
^(2-y) ^(^-w) <e{w-z)
f"{z-y) 'i"'^-y^) f"{w-z)
&{z-y) ^(^-w) f{io-z)
1 1 1
(Trinity, 1898.)
shew that y is one of the values of
f(^ "4^2 logy') +(ei-e2)(ei-e3)| .
(Math. Trip. 1897.)
5. Prove that
2{^(2)-e}{g>(y)-g>(ic;)P{^(y + t^;)-e}*{^(y-«.)-e}* = 0,
where the sign of summation refers to the three arguments z, y, w, and e is any one of the
(Math. Trip. 1896.)
roots gj, 62, 63.
6. Shew that
F(^) I iP(2)-iP(«i) J '
(Math. Trip. 1894.)
7. Prove that
^ (22) - ^ (o^i) = {^' (2)} -2 {^ (,) - ^ (|coi)}2 {g) (2) - ^ (a)2+^o,i)}2.
(Math. Trip. 1894.)
8. Shew that
g>(, + ,)g>(^-,)^(g^(^^)g>W + i^2}H.73{g>(^) + ^W} .
"^ ^ ^"^ ^ ^ • U-^W-^(^)}^
(Trinity, 1908.)
9. If ^{u) have primitive periods 2a)i, 2a)2 and
/(«)={^(«)-^(«2)}*,
while ipi (w) and/i (m) are similarly constructed with periods 2&)i/« and 2o}a, prove that
^i{u) = ^{u)+ '2' {^.>(w+2ma)i/«)-^(2mV;i)},
and
m=l
n f(u + 2mcii/7i)
M-)=r\
n f{2m<oi/n)
W. M. A.
(Math. Trip. 1914.)
29
where c=^ (2a).
11. Shew that
450 THE TRANSCENDENTAL FUNCTIONS [CHAP. XX
10. If x=^{u + a), 7/ = p{u-a\
where a is constant, shew that the curve on which (x, y) lies is
{xy + ex + cy + ig^f = 4:{x+y^c) {cxy - ^g^),
(Burnside, Messenger, xxi.)
2r^ {u) - Zg,r' («) +9i = 27 (P {u) +g,]\
(Trinity, 1909.)
12. If z=\ {x^ + Qcx'^ + e^y^dx,
- &' (z)
verify that ^^I^Tc'
the eUiptic function being formed with the roots -c, ^{c+e), ^{c-e).
(Trinity, 1905.)
13. If m be any constant, prove that
r^)j" ^•>(^)-^(y) " +^^'^j ' iO{z)-iJ{yf
where the summation refers to the values of ^ (z) for which ^' (z) is zero ; and the integrals
are indefinite.
(Math. Trip. 1897.)
14. Let R {x) = J.^ + Bx"^ + Cx''' + Dx + E,
and let ^ = 0 {x) be the function defined by the equation
where the lower limit of the integral is arbitrary. Shew that
2(f)' (a) _ (f)'(a+y) + cf)' (a) 4^' {ci-y) + (jj' (a) _^(p' (a+y)- cp' (x)
0(-«+y)-^(«)~' 0(«+y)-'^(«) 0(«-y)-9!>(«i 0(«+y)-0(^)
_ 0'(«-j/)^0' i^)
«^(«-y)-0(^') '
(Hermite, Proe. Math. Congress, Chicago, 1896, p. 105.)
15. Shew that, when the change of variables
is applied to the equations
they transform into the similar equations
ri'^ + V (1 +pr) + P=0, du^ -,^f-^ =0.
Shew that the result of performing this change of variables three times in succession
is a return to the original variables ^, rj ; and hence prove that, if | and ;; be denoted as
functions of w by JS (u) and F (w) respectively, then
where A is one- third of a period of the functions E (ti) and F{;u).
Shew that E{u) = ^^-iO (u ■ g.„ g,),
where g., = 9pj, L^^i^ ^3= _1 _ 1 p3_^l^ ^,o.
(De Brun, Ofversigt af K. Vet. Akad., Stockholm, Liv.)
ELLIPTIC FUNCTIONS
451
16. Shew that
and
where
2a- (z + ft)i) (r(z + co^) a- (z — coi — a^)
<T^ (2) cr{u>i)<r (0)2) a- (o)i + wa) '
60- (z + a) a- (z-a) a {z+g)(t{z-c)
(Math. Trip. 1895.)
0-* (2) 0-2 (a) 0-2 (c)
(Math. Trip. 1913.)
17. Prove that
^{z-a)^{z-h) = ip{a-h){i^{z-a) + (p{z-h)-^{a)-^{b)}
+ &'ia-b){C{z-a)-C{z-b) + Cia)-Cib)}
+ &icc)&{b).
18. Shew that
(Math. Trip. 1910.)
19. Shew that
2 {^ (u^) - ^ (M2)} W (%) - & (%)} {^ (%) - ^ («i)}
r («i) {g> w - ^^ («3)} + r («2) {&^ (%) - &> (%)} + r (%) {&> («i) - &^ K)}
(Math. Trip. 1912.)
20. Shew that
1 ^{x) iJ'ix)
1 g-^(y) r(y)
(T{x+y+z) (r{x-y)ar{y-z)<T{z—x) _ 1
(r3 (.r) a-^J^Hz) ~ 2
1 ^{z) ^'{z)
Obtain the addition-theorem for the function ^ {z) from this result.
21. Shew by induction, or otherwise, that
= (-)i'*('^-i>l!2!...'rt!
o- (^0 +2i + . • . + 2«) no- (2^ - Z^)
O-''^l(20)...O-" + l(2„)
1 ^•^(-"o) F(^o)...^(»-i)(2o)
1 ^(^i) r(^i)-r''-^>(^i)
where the product is taken for pairs of all integral values of X and fx from 0 to n, such
that X < /x.
(Frobenius u. Stickelberger* Crelle, Lxxxiii. p. 179.)
22. Express
1 ^(.^) ^\^) ^'{x)
1 ^(i/) P(y) F(.y)
1 ^(2) ^2(,) ^j'(,)
1 iJ{u) ^^u) <p'{n)
as a fraction whose numerator and denominator are products of Sigma-functions.
* See also Hermite, Crelle, lxxxii. p. 346.
29—2
452 THE TRANSCENDENTAL FUNCTIONS [CHAP. XX
Deduce that if a = ^(^), i3 = ^(?/), 7 = ^.>(4 8 = ^{u), where ^+y + z+« = 0, then
{€2 - es) {{a -e,)i^-e,){y-e,){8- e,)}* '
+ (eg - ei) {(a - 62) O - 62) (y - ^2) (S - eg)}*
+ (ei - ^2) {(a - 63) {8 - eg) (y - eg) (S - eg)}^ = (eg - eg) (eg - e,) (ei - e^).
(Math. Trip. 1911.)
23. Shew that
2f(2„)-4n«> = |;||,
(Math. Trip. 1905.)
24. Shew that
^= - r (-), ^5=3g> («) r (-) - w' («)>
(Math. Trip. 1912.)
and prove that a- {nu)l{a- {u))^^ is a doubly-periodic function of u
25. Prove that
^(,-,)-^(,-6)-C(a-6) + C(2a-2Z,)=."/;-':t^l"^'";'M-
''^ ^ ''^ ^ ^^ ^ ''^ ^ o-(26-2a)or(3-a)o-(2-o)
(Math. Trip. 1895.)
26. Shew that, if ^^-f ^2+zg+S4=0, then
{2C (2.)}^ = 3 {2f (.,)} {2^ (z,)} + 2r (^r),
the summations being taken for r = l, 2, 3, 4. (Math. Trip. 1897.)
27. Shew that every elliptic function of order n can be expressed as the quotient of
two expressions of the form
where 6, aj, 02, ... an are constants. (Painleve, Bulletin de la Soc. Math, xxvii.)
28. Taking ei>e2>eg, ^(a)) = e,, g>(a)') = eg,
consider the values assumed by
C{ti)-uCi(o')l(o'
as u passes along the perimeter of the rectangle whose corners are —a>, a, co + a, — w + o)'.
(Math. Trip. 1914.)
29. Obtain an integral of the equation
If-=6^(.) + 36
w dz^ °
in the form
d
dz
.cr(.-).(e)""l^U-2^(e) '^^'^/J'
where c is defined by the equation
Also, obtain another integral in the form
o-(2;-fa,)cr(3 + a2) < > / \ > / ni
-^r^ exp { - zC (ai) - zC (ag)},
where S-^ («i) + &> («2) = &, F («i) + ^ («2)=0,
and neither ai + a2 "oi" «i-a2 i« congruent to a period. (Math. Trip. 1912.)
ELLIPTIC FUNCTIONS
453
30, Prove that
S'W^
(T{z+Zi)<r(z-\-Z2) <T (z+Z:i)ar(z + Zi)
V{2« + ^(«l + 22 + 23 + 24)}
is a doubly -periodic function of z, such that
ff {z)-\-g {z+<,ii)+g (z + (02)+ ff (z + (01 + 0)2)
= - 2(T{^{Z2+Z3-Zi-Zi)}cr{^ (23+21 -22- •^4)} o- {i (^1+^2 -23-24)}.
(Math. Trip. 1893.)
31. It f{z) be a doubly-periodic function of the third order, with poles at 2=Cx, z=C2,
2=63, and if cf) (2) be a doubly -periodic function of the second order with the same periods
and poles at 0=a, 2=/3, its value in the neighbourhood of 2 = a being
<^(2)=^-fXi(2-a)+X2(2-a)2+...,
prove-that
^X2 {/" (a) -/" m - X {/' (a) + f m 2 <^ (c,) + {/(a) -/O)} {sXX^ + 2 (^ (c^) 0 (03)} = 0.
(Math. Trip. 1894.)
32. If X (2) be an elliptic function with two poles aj, a2, and if Zi, 22, ...22^ be 2n
constants subject only to the condition
Zl + Z2,+ ...+Z2n = n{ai+a2),
shew that the determinant whose nth row is
1, \{Zi), X^Zi), ... X»(2i), Xi(20, X(2i)Xl(2i), X2(2i)Xi(2i), . . . X» " 2 (2^) Xi (2f )
[where Xj (2j) denotes the result of writing Zi for 0 in the derivate of X (z)], vanishes
identically. (Math. Trip. 1893.)
33. Deduce from example 21 by a limiting process, or otherwise prove, that
^(2) ^'(z) ...'g?("-i)(2) =(-)»-i{l! 2! ... (?i-l)!Po-(?m)/{o- (%)}»'
f'{z) f"(2) ...^(")(2)
g>("-l)(2) ^(")(2) ... p»-3)(2)
(Kiepert, Crelle, Lxxvi.)
34. Shew that, provided certain conditions of inequality are satisfied,
^i^+y) . <o. ^J^(eot^-hCOt
2ci)i \ 2(»i
, H 20''^""* sin — (mz + ny),
2coi/ (oi loi
or (2) o- (?/)
where the summation applies to all positive integer values of m and 71, and 5' = exp (ttiooo/coi).
(Math. Trip. 1895.)
35. Assuming the formula
a- {z) = e
prove that
•Jif' ^ l-2o2»cos — +o<»
2o)i 2(Bi . 7r2 if 0)1
. — - Sm V— n rr-, ,
' (2)= --- + U— co-'^ec^ ^ 2 - 2 ^-— cos —
^^ 0)1 \2o)i/ 2o)i Vwi/ il-q^' 0)1
when 2 satisfies the inequalities
\l(oJ \lcoJ \iCOl
(Math. Trip. 1896.)
454 THE TRANSCENDENTAL FUNCTIONS [CHAP. XX
36. Shew that if 2ot be any expression of the form 2m(Oi + 2n(02 and if
then a; is a root of the sextic
jfi-5giX*-^g3X^ - 5g2^^^-Sg2Sf3^-5g3^=0,
and obtain all the roots of the sextic. (Trinity, 1898.)
37. Shew that
•^' = « + LA2V,^L.^2/,_^' 5-2=5:77:7-^' 5-3=0, ^Hz,) =
where
(Dolbnia, Darbouaf Bulletin (2), xix.)
CHAPTER XXI
THE THETA FUNCTIONS
21 "1. The definition of a Theta-function.
When it is desired to obtain definite numerical results in problems
involving Elliptic functions, the calculations are most simply performed
with the aid of certain auxiliary functions known as Theta-f unctions. These
functions are of considerable intrinsic interest, apart from their connexion
with Elliptic functions, and we shall now give an account of their funda-
mental properties.
The Theta- functions were first systematically studied by Jacobi*, who
obtained their properties by purely algebraical methods ; and his analysis
was so complete that practically all the results contained in this chapter
(with the exception of the discussion of the problem of inversion in §§ 21*7
et seq.) are to be found in his works. In accordance with the general scheme
of this book, we shall not employ the methods of Jacobi, but the more
powerful methods based on the use of Cauchy's theorem. These methods
were first employed in the theory of Elliptic and allied functions by Liouville
in his lectures and have since been given in several treatises on Elliptic
functions, the earliest of these works being that by Briot and Bouquet.
[Note. The first function of the Theta-function type to appear in Analysis was the
00
Partition function f H (l—x'^z)~''- of Euler, Introductio in Analysin Infinitorum, 1748,
I. § 304 ; by means of the results given in § 21*3, it is easy to express Theta-functions in
terms of Partition functions. Euler also obtained properties of products of the type
n (i±^'0) n (i±^2»i)^ n (i±^2n-i-)_
n=l n=l ii=\
The associated series 2 m^^^^^^'i 2 7?i'"*"^ ' and 2 »i" had previously occurred in the
n.=0 «=0 n=0
posthumous work of Jakob Bernoulli {A^-s Conjectandi, 1713, p. 55).
* Fnndamenta Nova Theoriae Functionum Ellipticarum, and Ges. Werke, i. pp. 497-538.
t The Partition function and associated functions have been studied by Gauss (Werke, ii.
pp. 16-21 and iii. pp. 433-480) and Cauchy (Comptes Renchts, x. p. 179). For a discussion of
properties of various functions involving what are known as Basic numbers (which are closely
connected with Partition functions) see Jackson, Proc. }{oyal Sac. lxxiv. , Proc. London Math.
Soc. (1) xxviii. and (2) i., ii. ; and Watson, Camb. Phil. Trans, xxi. A fundamental formula in
the theory of Basic numbers was given by Heine, KugelJ'unktionen, i. p. 107.
456 THE TRANSCENDENTAL FUNCTIONS [CHAP. XXI
Theta-functions also occur in Fourier's La Theorie Analytiqtte de la Chaleur, cf. p. 265
of Freeman's translation.
The theory of Theta-functions was developed from the theory of elliptic functions
by Jacobi in his Fundamenta Nova Theoriae Functionum Ellipticarum (1829), reprinted
in his Oes. Werke, i. pp. 49-239 ; the notation there employed is explained in § 21-62.
In his subsequent lectures, he introduced the functions discussed in this chapter ; an
account of these lectures (1838) is given by Borchardt in Jacobi's Ges. Werke, i. pp. 497-538.
The most important results contained in them seem to have been discovered in 1835,
cf. Kronecker, Sitzungsherichte der K. Preussischen Ahad. zu Berlin (1891), pp. 653-659.]
Let T be a (constant) complex number whose imaginary part is positive ;
and M^rite q = e^'^, so that \q\ < 1.
Consider the function ^(^, g), defined by the series
qua function of the variable z.
If A be any positive constant, then, when | 2; | < J., we have
n being a positive integer.
Now d'Alembert's ratio (§ 2-36) for the series t | g- j"^ e'^*'^ is | q\^''+^ e^^,
n=-Qo
which tends to zero as ?i -* 00 . The series for ^ {z, q) is therefore a series
of analytic functions, uniformly convergent (§ 3"34) in any bounded domain
of values of z, and so it is an integral function (§§ 5"3, 5*64).
It is evident that
^ (z, q) = I + 2 S ( -)" g»' cos 2nz,
and that ^ (z + tt, g) = ^ (z, q) ;
further ^ (^ + ttt, g) = ^ ( - )»* g"' r/^™ e^nfe
w= - X
and so '^ (^ + ttt, q) - - q-' er^^ ^ {z, q).
In consequence of these results, '^ (z, q) is called a quasi doubly -periodic
function of z. The effect of increasing 2^ by tt or ttt is the same as the effect
of multiplying ^ {z, q) by 1 or - q-^ e-^*^, and accordingly 1 and - q-'^ e'^^^ are
called the multipliers or periodicity factors associated with the periods tt and
TTT respectively.
2111. The four types of Th eta functions.
It is customary to write '^4 (z, q) in place of ^ {z, q) ; the other three
types of Theta-functions are then defined as follows :
21 11] THE THETA FUNCTIONS 457
The function ^s (z, q) is defined by the equation
^3 (z, q) = %(z + lTr,q) = i + 2 i ^"'^ cos 2nz.
Next, ^1 (z, q) is defined in terms of ^4 (z, q) by the equation
M= — 00
and hence* ^1 (^,q) = ^ 2 (-f q('' + ^)^ sin (2w + 1) z.
Lastly, ^2 {^, q) is defined by the equation
^, (z, q) = %(z + l7r,q] = 2% ^(« + ^)' cos {2n + 1) z.
Writing down the series at length, we have
^1 (-s^j q) = 2qi sin z — 2g^ sin 3^ + 2^- t" sin 5^; — . . . ,
^2 (■2', g") = 2q^ cos ^ + 2^-^ cos Sz + 2q^' cos oz + ...,
^3 (2:, ^) = 1 4- 2^- cos 2z + 2q* cos 4^^ + 2q^ cos 62^ + . . . ,
^4 (-2^, q) = \ — 2q cos 22^ + 2(/'* cos 4^ — 25^^ cos 6^ +
It is obvious that ^1 {z, q) is an odd function of z and that the other
Theta-functions are even functions of z.
The notation which has now been introduced is a modified form of
that employed in the treatise of Tannery and Molk ; the only difference
between it and Jacobi's notation is that ^4 {z, q) is written where Jacobi
would have written '^ {z, q). There are, unfortunately, several notations in
use; a scheme, giving the connexions between them, will be found in § 21"9.
For brevity, the parameter q will usually not be specified, so that ^1 {z), . . .
will be written for ^1 (z, q), When it is desired to exhibit the dependence
of a Theta-function on the parameter r, it will be written ^ {z \ r). Also
^2(0), ^3(0), ^4(0) will be replaced by ^2, ^3 > ^4 respectively; and ^1' will
denote the result of making z equal to zero in the derivate of ^i {z).
• Example 1. Shew that
S,{z,q) = h{-2z,q*)-9,{2z,q*).
* Example 2. Obtain the results
5i(2)= -S^{z + ^7r) =-0/53(2 + U+W)=-0/»4(2 + ^7rr),
'h{z)= Bi{2 + h^) = M$^(z + ^7r + UT)= M3.i{z + ^1TT\
Si{z)=-aiSi{z + ^7rr)= ^■J/92(^ + |7r + W)= Ssiz + i^n),
where M=q'e^'.
* Throughout the chapter, the many-vahied function q^ is to be interpreted to mean
exp (Xttit).
458
THE TRANSCENDENTAL FUNCTIONS
[chap. XXI
' Example 3. Shew that the multipliers of the Theta-functions associated with the
periods tt, itt are given by the scheme
^l(^) ^2(«)
^3(2)
^4{^)
TT
-1 , -1
1
1
7TT
-N ' N
iT
-N
where iV=2'"^e"2*^
* Example 4. If 5 (z) be any one of the four Theta-functious and ^' (2) its derivate with
respect to z, shew that
5 (2 + 7r) S(2)' 5 (2 + 7rr) •» («) '
2112. T/?e ^•ero* q/" the Theta-f unctions.
From the quasi-periodic properties of the Theta-functions it is obvious
that if ^ {z) be any one of them, and if z^ be any zero of ^ {z), then
Zq + imr + nirr
is also a zero of ^ {z), for all integral values of m and n.
It will now be shewn that if C be a cell with comers <, i + tt, i + tt -I- ttt,
t + ITT, then ^ {z) has one and only one zero inside C.
Since '^ {z) is analytic throughout the finite part of the 2^-plane, it follows,
from § 6"31, that the number of its zeros inside C is
^'^dz.
Treating the contour after the manner of § 20'12, we see that
1 ^ ^'Wd.
l-irijc'^iz)
J_ r'+'^ (^^(£) _ ^^(^_+7rT)| . _ J_ {'"^-^ {"^'J^ _ ^^(^ + 7r)]
27rtJ, h(^) ^(^-Httt)] 27rij, l^(^) ^(^-f7r)|
t + TT
„ . , 2i(^^',
by § 21 '11 example 4. Therefore
27ri ./ c ^ (^)
rf^ = l,
that is to say, ^{z) has one simple zero only inside (7; this is the theorem
stated.
21 12, 2 12] THE THETA FUNCTIONS 459
Since one zero of %(z) is obviously z = 0, it follows that the zeros of
^i(^), ^2(-s^), ^3('2^)> ^4(^) are the points congruent respectively to 0, 2"^,
^TT + 2 TTT, 2''"r- The reader will observe that these four points form the
comers of a parallelogram described counter-clockwise,
21 "2. The relations between the squares of the Theta-functions.
It is evident that, if the Theta-functions be regarded as functions of a
single variable z, this variable can be eliminated from the equations defining
any pair of Theta-functions, the result being a relation * between the functions
which might be expected, on general grounds, to be non-algebraic; there
are, however, extremely simple relations connecting any three of the Theta-
functions ; these relations will now be obtained.
Each of the four functions ^j^ (z), V (2), V (■2), V (z) is analytic for all
values of z and has periodicity factors 1, q~'^e~*^'' associated with the periods
TT, TTT ; and each has a double zero (and no other zeros) in any cell.
From these considerations it is obvious that, if a, b, a' and b' are suitably
chosen constants, each of the functions
ftV(^)-h6V(^)- a"^,Hz) + b'X^{z)
V(^) ' V(^)
is a doubly -periodic function (with periods tt, ttt) having at most only a
simple pole in each cell. By § 20"13, such a function is merely a constant;
and obviously we can adjust a, b, a', b' so as to make the constants, in each
of the cases under consideration, equal to unity.
There exist, therefore, relations of the form
%" (z) = a%' (z) + 6 V (z), %' (z) = a'%' (z) + 6' V (z).
To determine a, b, a', b', give z the special values 2 ttt and 0 ; since
we have V = - « V, V = i^/ ; %' = - «' V, %"" = 6' V-
Consequently, we have obtained the relations
%' (Z) V = ^4^ (z) %' - ^1'^ (z) %', %' (z) V = V (z) V - V (z) V.
If we write ^ + g tt for 2', we get the additional relations
%^ (z) V - V (^) V - ^2^ (^) %', V (z) V = %' (z) V - %' (z) V.
By means of these results it is possible to express any Theta-function in
terms of any other pair of Theta-functions.
* The analogous relation for the functions sin 2 and cos z is, of course, (sin2)2 + (cosz)2=l.
460 THE TRANSCENDENTAL FUNCTIONS [CHAP. XXI
Corollary. Writing 2 = 0 in the last relation, we have
that is to say
21*21. The addition-formulae for the Theta functions.
The results just obtained are particular cases of formulae containing two
variables ; these formulae are not addition-theorems in the strict sense, as
they do not express Theta-functions oi z -^-y algebraically in terms of Theta-
functions of z and y, but all involve Theta-functions of ^ — 3/ as well as of
z + y, z and y.
To obtain one of these formulae, consider ^3 (z + y) ^3 (z — y) qua function
of z. The periodicity factors of this function associated with the periods tt
and TTT are 1 and q-^ e'^^ '^+2^' . q-^ e'^^ <^-?/> = q-^ e-*'\
But the function a^3^ (z) + b%^ (z) has the same periodicity factors, and
we can obviously choose the ratio a:b so that the doubly -periodic function
a%Hz) + b%'(z)
%{z + y)'^,{z~y)
has no poles at the zeros of % (z — y); it then has, at most, a single simple
pole in any cell, namely the zero of ^3 (z + y) in that cell, and consequently
(§ 20*13) it is a constant, i.e. independent of z ; and, as only the ratio a : 6 is
so far fixed, we may choose a and h so that the constant is unity.
We then have to determine a and b from the identity in z,
a V {z) + h%-^ {z) = %{z^ y) % (z - y).
To do this, put z in turn equal to 0 and 2 '"" + 2 '""''"' ^^*^ ^® 8'®^
a V = V {y), b%^ [I'TT + l TTT j = ^3 (1 TT + ^ TTT -t- y) ^3 (^TT + ^ TTT - 2/) ;
and so a = V (y)/%\ b = %' (y)/%\
We have therefore obtained an addition-formula, namely
^3 (^ + y) % (z - y) %^ = V (2/) %' (^) + V (y) %' {z).
The set of formulae, of which this is typical, will be found in examples 1
and 2 at the end of this chapter.
21*22. Jacohi's fundamental formvlae*.
The addition -formulae just obtained are particular cases of a set of identities first given
by Jacobi, who obtained them by purely algebraical methods ; each identity involves as
many as four independent variables, w, ^, y, z.
Let w', x\ y', z' be defined in terms of w, x, y, z, by the set of equations
2w' := —w + x-\-y + 2,
1x' = tv — x+y + z^
2y' = tv + x-y + z,
2z' = w-{-x+y — z.
* Ges. Werke, i. p. 505.
21-21, 21-22]
THE THETA FUNCTIONS
461
The reader will easily verify that the connexion between w, x, y, z and w', of, y\ / is a
reciprocal one*.
For brevity t, write [r] for X (w) ^r {x) 3r (y) ^r («) and [r]' for 5^ (v/) Sr C-^) ^r i^') ^r («').
Consider [3], [1]', [2]', [3]', [4]' qua functions of z. The effect of increasing « by tt or wr
is to transform the functions in the first row of the following table into those in the second
or third row respectively.
[3]
[1]'
[2]'
[3]'
[4]'
(rr)
[3]
-[2]' .
-[1]'
[4]'
[3]'
(tt)
#[3] -iV[4]' i iV^[3]' ■
i
JV[2]'
-^[1]'
For brevity, N has been written in place of j"ie~2»«.
Hence both -[l]'+[2]' + [3]' + [4]' and [3] have periodicity factors 1 and JV, and so
their quotient is a doubly-periodic function with, at most, a single simple pole in any cell,
namely the zero of ^3 (z) in that cell.
By § 20*13, this quotient is merely a constant, i.e. independent oi z; and considerations
of symmetry shew that it is also independent of iv, x and y.
We have thus obtained the result
J[3]=-[l]' + [2]' + [3]' + [4]',
where A is independent of w, x, y^, z; to determine A put w=x=y = z=0, and we get
and so, by § 21 "2 corollary, we see that A =2.
Therefore 2 [3]= -[1]' + [2]' + [3]' + [4]' (i).
This is one of Jacobi's formulae ; to obtain another, increase w, .r, y, z (and therefore
also w', x', y', z) by ^n ; and we get
2[4] = [l]'-[2]' + [3]' + [4]' (ii).
Increasing all the variables in (i) and (ii) by ^n-r, we obtain the further results
2[2] = [l]' + [2]' + [3]'-[4]' (iii),
2[l] = [l]' + [2]'-[3]'+[4]' (iv).
[Note. There are 256 expressions of the form Sp{w) Sq{x) S^i^) ^s{^) which can be
obtained from ^3 (w) ^3 (x) B3 (y) S^ {z) by increasing w, x, y, z by suitable half-periods, but
only those in which the suffixes jo, j, r, s are either equal in pairs or all different give rise
to formulae not containing quarter-periods on the right-hand side.]
Example 1. Shew that
[l] + [2] = [l]' + [2]', [2] + [3] = [2]' + [3]', [l] + [4]=[l]' + [4]', [3] -h [4] = [3]' -f [4]',
[l] + [3] = [2]'-H[4]', [2]-i-[4] = [l]' + [3]'.
* In Jacobi's work the signs of ic, x', y', z' are changed throughout so that the complete
symmetry of the relations is destroyed ; the symmetrical forms just given are due to H. J. S.
Smith, Proc. London Math. Soc. i. (1865).
t The idea of this abridged notation is to be traced in H. J. S. Smith's memoir. It seems,
however, not to have been used before Kronecker, Crelle, cii. (1887).
462 THE TRANSCENDENTAL FUNCTIONS [CHAP. XXI
Example 2. By writing w-\-\ni x + \ir for w, x (and consequently y' + ^rr, z' + ^ir
for y, z'), shew that
[3344] + [22 1 1 ] = [4433]' + [11 22]',
where [3344] means ^3 (w) S3 (x) ^4 (y) ^4 (2), etc.
jExample 3. Shew that
2 [1234] = [3412]' + [2143]' - [1234]' + [4321]'.
Example 4. Shew that
S,^{z) + S3'(z)=S,''{z) + S,^z).
21*3. Jacobi's expressions for the Theta-functions as infinite products*.
We shall now establish the result
'^i{z) = G n (1 - 2^2n-i cog 2z + ^«-2),
(where G is independent of z), and three similar formulae.
Let f{z) = n (1 - f/'^-i e2i^) n (1 _ q^n-i g-2iz^ .
each of the two products converges absolutely and uniformly in any bounded
domain of values of z, by § 3"341, on account of the absolute convergence of
2 q^^~^ ; hence / (z) is analytic throughout the finite part of the ^-plane,
w=l
and so it is an integral function.
The zeros of f(z) are simple zeros at the points where
g2iz = g(2»+l)«V ^^^ =...,- 2, - 1, 0, 1, 2, ...)
I.e. where 2iz = (2n + 1) irir + 2ni'7ri ; so that f{z) and ^i{z) have the same
zeros ; consequently the quotient ^4 (z)/f{z) has neither zeros nor poles in
the finite part of the plane.
Now, obviously /(^r + tt) =f{z) ;
and f(z + TTt) = n (1 - 52«+l g2iz^ n (1 _ ^2n-3g-2w-)
= /(^) (1 - fy-1 e-2''^)/(l - ge2/^-)
= - q-'^ e-"'\f(z).
That is to say f{z) and ^^{z) have the same 2)enodicity factors (§ 21-11
example 3). Therefore %{z)/f(z) is a doubly-periodic function with no
zeros or poles, and so (§ 20-12) it is a constant G, say; consequently
00
^4(^) = G U (1- 2q'''-^ cos 2z + q'""-"-).
CO
[It will appear in § 21-42 that G= FI (1 - q^").]
n = l
Write 2; + ^ TT for ^ in this result, and we get
^3 (2) = (r n (1 + 2ry-«-i cos 2z + g^'^-2).
H = l
* Of. Fnndanienta Nova, p. 145.
21-3-21'41] THE THETA FUNCTIONS 463
Also ^1 {z) = - iq^ e^ X (z + \ TTT^
= - iq^ e^' G Tl {I - f"" e^'^) fi (1 - ^2»-2 g-2tz)
M=l »=1
= 2^5* sin z n (1 - 92n e^^^^) fi (1 - g2« ^-2^)^
w=l n-\
and so ^i (^) = 2Gq^ sin ^r 11 (1 - 252n cos 2z + ^")
while ^2 {z) = ^Az-\-\'rr\
= 2Gq^ cos ^ n (1 + 2^2" cos 2z + 3*").
M = l
Example. Shew that*
00 18 r « 1 8 c oc 18
n {\-qin-i)i +\^\ n (i+?2™)l =] n (i+?2»-i)i
(Jacobi.)
21*4. TAe differential equation satisfied by the Theta-functions.
We may regard ^3(5|t) as a function of two independent variables z
.1*- and t; and it is permissible to differentiate the series for ^^{z\r) any
\ ?" number of times with regard to z or t, on account of the uniformity of
• convergence of the resulting series (§ 4" 7 corollary) ; in particular
y ^ ' ^ = — 4 S n'^ exp (n^iriT + 2niz)
OZ" « = _ 00
^1
4 d%,{z\r)
iri dr
f > Consequently, the function ^3 (s j r) satisfies the partial differential equation
\i
1 V^'2/^^^-0
The reader will readily prove that the other three Theta-functions also
satisfy this equation.
21'41. A relation between Theta-functions of zero argument.
The remarkable result that
V(0) = ^,(0)^3(0)^4(0)
will now be established + It is first necessary to obtain some formulae for
differential coefficients of all the Theta-functions.
* Jacobi describes this result {Fund. Nova, p. 90) as 'aequatio identica satis abstrusa.'
t Several proofs of this important proposition have been given, but none are simple.
Jacobi's original proof {Ges. Werke, i. pp. 51.5-517), though somewhat more difficult than the
proof given here, is well worth study.
464
THE TRANSCENDENTAL FUNCTIONS
[chap. XXI
Since the resulting series converge uniformly, except near the zeros of
the respective Theta- functions, we may differentiate the formulae for the
logarithms of Theta-functions, obtainable from § 21 '3, as many times as we
please.
Denoting differentiations with regard to z by primes, we thus get
V(^) = ^3(^) 2 ,3
liq^
n^\ 1 + g'
2n-l c2iz
" 2ig2»-i e-2'>
"i 1 + i
2n— 1 a— 2iz
"Liq
Iriln—X o^iz
- t
2iq
2n— 1 ^— 2tz
+ i
_«=! (1 + 2'^-' e^^'f n=i (1 + 9'"-' e-''^'y_
Making ^ -* 0, we get
V(0) = 0, V(0) = - 8^3(0) S^^^^^^,,_,^,
In like manner,
V(0) = 0, V(0) = 8^,(0) 2 Tj^-^,,^^.'
,=1 (1-9^-7^
V(0) = o, V(0) = ^.(0)
-1-8 2
n=i{i^rjy
and, if we write ^i (z) = sin z . (^ (z), we get
GO ^,2
<^'(o) = o, f'(0) = 8(^(0) S -r-^-
we get
^=1 (1 - q^^y
If, however, we diflferentiate the equation ^1(2^) = sin^^. (}){z) three times,
Therefore :^i^v"/ _ 01 ^ '-/
and
V(0) -^ .rx(i-g-)-^
-1 ;
V(0) V(0) V(0)
"^ ^,(0) "^ ^3(0) "^ ^4(0)
L n = i (1 + g-^*^)^ «=i (1 + q'^'-J ^ nti (1 - q^'-')\
= 8
_ V
+ t
- S
«=i (1 + ^'0' «=i (1 - ry n=i (1 - q'^'T,
on combining the first two series and writing the third as the difference of
two series. If we add corresponding terms of the first two series in the last
line, we get at once
V' (0) ^3" (0) X' (0) _ 5 g- -^Z" (0)
^ ^.(0) + ^^3(0) + vo) ~ "■*«=! (T^r)^ - ^ + VTO) •
2142] THE THETA B^UNCTIONS 465
Utilising the differential equations of § 21*4, this may be written
1 d%' (0 1 t)
V(0|t) dr
1 d%{0\T) 1 <^^3(0iT) 1 d%(0\r)
^2(01 r) dr ^3(0 Jt) dr ^4 (Ok) dr
Integrating with regard to r, we get
V (0, q) = G% (0, q) % (0, q) % (0, q),
where 0 is a constant (independent of q). To determine G, make q-*(); since
limg~*V=2, Iim9-H2=2, lim^3 = l, lim^4 = l,
9-»-0 g-*0 3-*-0 g-*-0
we see that C = 1 ; and so
which is the result stated.
21*42. The value of the constant 0.
From the result just obtained, we can at once deduce the value of the
constant 0 which was introduced in § 21*3.
For, by the formulae of that section,
X = </> (0) = 2^^ Gil {1- q''% %=2qiG U {1 + q'^'f,
W = l Ji = l
00 • 00
^3 = G' n (1 + q'^^-'y, ^, = G u(i- f^-%
n=l n=l
and so, by § 21 •41, we have
GO QO 00 00
n (1 - q-""-)^ = G^ n (1 + q^^f U (1 + g2/i-i)2 n (1 - q^-')\
M=l w=l n-\ n=\
Now all the products converge absolutely, since |5'|<1, and so the
following rearrangements are permissible :
n (1 - f^-^) n (1 - f'^)\ . I n (1 + if-') n (i + q^'')\
= n (1 - q^) n (1 + 5")
w=l «=1
00
= n (1 - g^"),
the first step following from the consideration that all positive integers are
comprised under the forms 2n — 1 and 2?i.
Hence the equation determining G is
n {l-(f'f=G\
n = \
and SO G = ± IT (1 - g^").
W. M. A. 30
466 THE TRANSCENDENTAL FUNCTIONS [CHAP. XXI
To determine the ambiguity in sign, we observe that G is an analytic
function of q (and consequently one-valued) throughout the domain \q\<\]
and from the product for ^3 {z), we see that (r -* 1 as q^Q. Hence the
plus sign must always be taken ; and so we have established the result
00
G= n (1 - q^'%
• Example 1. Shew that Bi=2q^G^.
« Example 2. Shew that
54= n {(1 -?2"-^)(i -?")}•
I Example 3. Shew that
1+2 2 q'''= n {{l-f-')(l+f''-^f}.
21*43. Connexion of the Sigma-function with the Theta-fitnctions.
It has been seen (§ 20'421 example 3) that the function cr{z\(x>i, 0)2), formed with
the periods Swi, 2co2, is exj^ressible in the form
where q = e%.^{Tvi(i)2l<>>i)-
If we compare this result with the product of § 2r4 for 5i (s | r), we see at once that
TT \2cOj/ 2"- n=\ \^Wl <»1/
To express r^i in terms of Theta-functions, take logarithms and differentiate twice,
so that
'{^)='^-(~) cosec^ GP- +
^1« _ \<i>li^Y']
,0'
where v^^nzlcoi and the function 0 is that defined in v^ 2r41.
Expanding in ascending powers of z and equating the terms independent of z in this
result, we get
o^ni^^ fjL\\f ^y^l^''^^)
" 0)1 3 \2coJ \2o}J (p (0) '
and so rji = -7 .
Consequently cr (2 1 coi , to.^) can be expressed in terms of Theta-functions by the
formula
a(.ia.x,co,3)=^,exp(^---J5i^.|-j,
where i/ = |7r2/a)i.
' Example. Prove that
_ /tt'^cOoS/" Trl\
21"5. TAe expression of elliptic functions hy means of Theta functions.
It has just been seen that Theta-functions are substantially equivalent
to Sigma-functions, and so, corresponding to the formulae of §§ 20-5-20-58,
there will exist expressions for elliptic functions in terms of Theta-functions.
21-43-2r51] THE THETA FUNCTIONS 467
From the theoretical point of view, the formulae of §§ 20-5-20-53 are the
more important on account of their symmetry in the periods, but in practice
the Theta-function formulae have two advantages, (i) that Theta-functions
are more readily computed than Sigma-functions, (ii) that the Theta-
functions have a specially simple behaviour with respect to the real period,
which is generally the significant period in applications of elliptic functions
in Applied Mathematics.
hetfiz) be an elliptic function with periods 2&)i, 2&).2; let a fundamental
set of zeros («!, ota, ... «„) and poles (^i, ^^y ••• AO be chosen, so that
r=l
as in § 20-53.
Then, by the methods of § 20-53, the reader will at once verify that
TTZ — vra*
coj ' H 2a)i
V 2&)i
where J. 3 is a constant ; and if
t Ar,m{2-I3ry
m = l
be the principal part o^ f{z) at its pole /3r, then, by the methods of § 20-52,
where A^. is a constant.
This formula is important in connexion with the integration of elliptic
functions. An example of an application of the formula to a dynamical
problem will be found in § 22-741.
* Example. Shew that ,
and deduce that
21*51. Jacohi's imaginary transformation.
If an elliptic function be constructed with periods 2(Ui, 2co2, such that
1 {(0.2/00^) > 0,
it might be convenient to regard the periods as being 202, — 2(Wi; for these
numbers are periods and, if / (w./wi) > 0, then also / (— cojcoo) > 0. In the
case of the elliptic functions which have been considered up to this puint,
the periods have appeared in a symmetrical manner and nothing is gained
by this point of view. But in the case of the Theta-functions, which are
only quasi-periodic, the behaviour of the function with respect to the real
30—2
468 THE TRANSCENDENTAL FUNCTIONS [CHAP. XXI
period tt is quite different from its behaviour with respect to the complex
period ttt. Consequently, in view of the result of § 21 "43, we may expect to
obtain transformations of Theta-functions in which the period-ratios of the
two Theta-functions involved are respectively t and — 1/t.
These transformations were first obtained by Jacobi*, who obtained them
from the theory of elliptic functions. A direct proof of the transformations
is due to Landsberg, who used the methods of contour integration f. The
investigation of Jacobi's formulae, which we shall now give, is based on
Liouville's theorem ; the precise formula which we shall establish is
%(z\r) = {-ir)-Uxp(X~).%(-\--),
\7nTj \t\ tJ
where (— ir)~^ is to be interpreted by the convention | arg {—it) | < o^'''-
For brevity, we shall write — l/r = r', q' = exp (tti't').
The only zeros of ^3 (z \ r) and ^3 {r'z | t') are simple zeros at the points
at which
11 / / ,,11,
z = omr 4- wttt + 2 t^ + 2 ''"^' t z = m tt + n ttt + ^tt + 2'^'^
respectively, where m, n, m', n take all integer values; taking m' = — n — \,
n' = m, we see that the quotient
is an integral function with no zeros.
Also ^\r{z + ttt) -^ yjr (z) = ex-p ( — '""^ . "^ ''" ) -^ q-^ g-s'^ = 1 ,
while ^{r (z - tt) -^ yjr (z) = exp (^^—^ ~] X q'-'^ g-^iw/T = 1.
Consequently -v/r (2-) is a doubly-periodic function with no zeros or poles;
and so (§ 20-12) yjr {z) must be a constant, A (independent of z).
Thus ^^3 (z\r) = exp (iTz'/ir) % (zr' \ r') ;
and writing z + ^'rr, z + 2 ttt, z + ^tt -\- ^ttt in turn for z, we easily get
^"^4 (z\t)= exp (zW-'/tt) '^a (zT j t'),
A% (z \t)= exp (ir'z-jTT) '^4 (zr j t'),
A%{z\t)= - i exp {it'z'Jtt) \ (zt j t ).
We still have to prove that A = (- ir)^ ; to do so, differentiate the last
equation and then put ^ = 0 ; we get
AX{(y\r)= -zW(0|t').
* Gen. Werke, i. p. 2G4. A particular case of Jacobi's results had been previously given by
Poisson, Journal de V Kcole. polyteclmique, cahier xix.
+ This method is indicated in example 17 of Chapter vi, p. 124. See Landsberg, Crellc, cxi.
21 -52] THE THETA FUNCTIONS 469
But V (0 I t) = ^2 (0 I t) ^3 (0 I r) ^4 (0 ! r)
and X (0 I t') = %{0\ t') ^3 (0 | r) ^4 (0 | r) ;
on dividing these results and substituting, we at once get A~^ = — ir, and so
To determine the ambiguity in sign, we observe that
both the Theta-functions being analytic functions of t when Z(t)>0;
thus A is analytic and one-valued in the upper half r-plane. Since the
Theta-functions are both positive when t is a pure imaginary, the plus sign
must then be taken. Hence, by the theory of analytic continuation, we
always have
A = +(-ir)^;
this gives the transformation stated.
• Example 1. Shew that
when rr'= — 1.
• Example 2, Shew that
^4 (0 I r) _ So^ (0 I t')
53(0|r+l) ^4(0|r)'
• Example 3. Shew that
and shew that the plus sign should be taken.
21*52. Landens type of transformation.
A transformation of elliptic integrals (§ 22'7), which is of historical
interest, is due to Landen (§ 22*42) ; this transformation follows at once
from a transformation connecting Theta-functions with parameters r and 2t,
namely
^3(^|t)^4(^|t) ^ ^3(0|T)^4(0iT)
^4(2^ I 2t) ^4(0 I 2t) '
which we shall now prove.
The zeros of ^3 {z \ r) ^4 {z \ r) are simple zeros at the points where
z= [m + ^\'rT + {n-\--ATrr and where z = mir + ( ^ + o ) '^'^> where m and n
take all integral values ; these are the points where 2z = mir -f- in + -Att . 2t,
which are the zeros of ^4 {2z \ 2t). Hence the quotient
%{z\t)'^Mt)
^4 (2^ I 2t)
470 THE TRANSCENDENTAL FUNCTIONS [CHAP. XXI
has no zeros or poles. Moreover, associated with the periods ir and ttt, it
has multipliers 1 and {q-^ e'''') (- q-' 0-''') -r (- q-^ e''"') = 1 ; it is therefore
a doubly-periodic function, and is consequently (§ 20-12) a constant. The
value of this constant may be obtained by putting z = 0, and we then have
the result stated.
If we write z-^^tn for z, we get a corresponding result for the other
Theta-functions, namely
%{z\t)%{z 1 t) ^ ^3 (0 I t) ^4 (0 I t)
^i(2^j2t) ^4(0|2t)
21'6. The differential equations satisfied by quotients of Theta-functions.
From § 2ril example 3, it is obvious that the function
%(z)^Xiz)
has periodicity factors — 1, + 1 associated with the periods tt, ttt respectively ;
and consequently its derivative
{%' (z) % (z) - V (^) ^1 (^)] - V (^)
has the same periodicity factors.
But it is easy to verify that % (z) %(z)/^^^{z) has periodicity factors — 1,
+ 1 ; and consequently, if ^ (z) be defined as the quotient
{%' (z) % (z) - ^: (z) % (z)] - {% (z) % (z)},
then (f) (z) is doubly-periodic with periods tt and ttt ; and the only possible
poles of (f)(z) are simple poles at points congruent to ^tt and -tt + ^'^t.
Now consider cf) iz +~'7rT\; from the relations of § 21'11, namely
%(^z + l7rT^^iq-h-''%{z), %(^z + l7rT^=iq'ie'-''%(z),
% [z + Jttt) =q-^e-'' %(z), % (z + ^ttt) = q'ie-'' %{z),
we easily see that
(/> (s + I ttt) = { - X (^) ^: (^) + %' (^) ^4 (^)} ^ {% {Z) X {Z)].
Hence ^ {z) is doubly-periodic with periods tt and ^ ttt ; and, relative to
these periods, the only possible poles of (j) (z) are simple poles at points
congruent to g tt.
Therefore (| 20-12), <^{z) is a constant; and making z^O, we see that
the value of this constant is j^/ ^4] h- {^2 ^3} = V-
21-6, 21-61] THE THETA FUNCTIONS 471
We have therefore established the important result that
writing ^ = % (z)/'^i(z) and making use of the results of § 21 '2, we see that
(^y=(v-pv)(v-rv).
This differential equation possesses the solution ^i{z)l^i{z). It is not
difficult to see that the general solution is ± '^i (^ + a)/^4 (z + a) where a
is the constant of integration ; since this quotient changes sign when a is
increased by tt, the negative sign may be suppressed without affecting the
generality of the solution.
Example 1. Shew that
Example 2. Shew that
1 fV!)l _.2Mi)Mi)
dz\3,{z)j ""^ h{z)h{^y
21'61. The genesis of the Jacohian Elliptic function* sn u.
The differential equation
fy=(v-rv)(v-rv),
which was obtained in § 21 '6, may be brought to a canonical form by a slight
change of variable.
Writef ^%l% = y, z%^ = u;
then, if ^2 ijg written in place of B^o/^s, the equation determining y in terms
of u is
(iy=(i-/)<i-*:f>-
This differential equation has the particular solution
The function of u on the right has multipliers — 1, +1 associated with
the periods ir^^, ttt^s^; it is therefore a doubly-periodic function with
periods 27r^3^ irr^^. In any cell, it has two simple poles at the points
congruent to ^77x^3^ and 'rr^.^ + ^ttt^^ ; and, on account of the nature of the
quasi-periodicity of y, the residues at these points are equal and opposite in
sign ; the zeros of the function are the points congruent to 0 and tt^jI
* Jacobi and other early writers used the notation sin am in place of sn.
t Notice, from the formulae of § 21 "3, that ^2=1=0, Ss + O when j </ ] <; 1, except when ^ = 0, in
which case the Theta-functions degenerate ; the substitutions are therefore legitimate.
472 THE TRANSCENDENTAL FUNCTIONS [CHAP. XXI
It is customary to regard y as depending on k rather than on q ; and to
exhibit t/ as a function of u and k, we write
?/ == sn {u, k),
or simply y = sn.u.
It is now evident that sn {u, k) is an elliptic function of the second
of the types described in § 20-13 ; when g -* 0 (so that k -^ 0), it is easy to see
that sn {u, k) -^ sin u.
The constant k is called the modulus ; if k'^=%/%, so that k^ + k'^ = l,
k' is called the complementary modulus. The quasi -periods tt^s^ 7rT%^ are
usually written 2K, 2iK', so that sn (u, k) has periods 4/^, 2iK'.
From §21-51, we see that 2Z' = 77^3^(0 | r'), so that K' is the same
function of r' as K is of r, when tt' = — 1.
• Example 1. Shew that
(^2 \ (S) ' 54 (2) S4 (/) '
^ •& (z)
and deduce that, if y= r^ q^(> ^'^^'^ u=zB3\ then
■^2 '^4 (,'2j
Example 2. Shew that
rf^ ^4 (2) ^ ^4 (2) -34 (2) '
S S (z)
and deduce that, if i/=-^ ^^ , and ?f = 0^a2, then
^3 ^4 (s)
(IT =(i-f </-'">•
' Example 3. Obtain the following results :
2/''A''\-
-— -j =54=l-2^ + 2yH2y9-...,
/i" = A'7r-ilog(l/ry).
[These results are convenient for calculating k, k', K, K' when q is given.]
21-62. Jacobi's earlier notation*. The Theta- function 0 (m) and the
Eta-function H (u).
The presence of the factors ''^■f^ in the expression for sn {u, k) renders 'it
sometimes desirable to use the notation which Jacobi employed in the
Fundamenta Nova, and subsequently discarded. The function which is of
primary importance with this notation is (d (u), defined by the equation
(H) (u) = ^4 {u%-' 1 t),
so that the periods associated with 0 (u) are 2K and 2iK'.
* This is the notation employed throughout the Fundamenta Nova.
21-62, 21-7] THE THETA FUNCTIONS 473
The function S{u + K) then replaces ^3 (z) ; and in place of ^1 (z) we
have the function H (u) defined by the equation
H (w) = - iq ^ i e*""^^"'^> 0 (u + iK') = ^, (u%-^ I t),
and ^2 (-3") is replaced by H (u + ^).
The reader will have no difficulty in translating the analysis of this
chapter into Jacobi's earlier notation.
• Example 1. If Q' (u)——j^, shew that the singularities of — j-i- are simple poles
at the points congruent to iK' (mod 2K, 'iik') ; and the residue at each singularity is 1.
• Example 2. Shew that
H'(0)=|7rA'-iH (/i ) e (0) e (^).
21*7. The problem of Inversion.
Up to the present, the Jacobian elliptic function sn {u, k) has been
implicitly regarded as depending on the parameter q rather than on the
modulus k ; and it has been shewn that it satisfies the differential equation
/dsn uy , „ X /, 7„ „ N
V^^tTJ =a-sn^«0(l-^^sn^w),
where P = %* (0, q)/%' (0, q).
But, in those problems of Applied Mathematics in which elliptic functions
occur, we have to deal with the solution of the differential equation
in which the modulus k is given, and we have no a priori knowledge of the
value of q ; and, to prove the existence of an analytic function sn {u, k)
which satisfies this equation, we have to shew that a number r exists* such
that
k' = %*(0\T)/%^{O\T).
When this number t has been shewn to exist, the function sn{u, k) can
be constructed as a quotient of Theta-functions, satisfying the differential
equation and possessing the properties of being doubly-periodic and analytic
except at simple poles; and also
lim sn (u, k)lu = 1.
That is to say, we can invert the integral
so as to obtain the equation y = sn {u, k).
* The existence of a number r, for which / (r) > 0, involves the existence of a number q such
that I (/ I < 1. An alternative procedure would be to discuss the differential equation directly,
after the manner of Chapter x.
474 THE TRANSCENDENTAL FUNCTIONS [CHAP. XXr
The difficulty, of course, arises in shewing that the equation
c = ^,^(0 1t)/V(0|t),
(where c has been written for k^), has a solution.
When* 0<c<l, it is easy to shew that a solution exists. From the
identity given in § 21'2 corollary, it is evident that it is sufficient to prove
the existence of a solution of the equation
1_c = V(0|t)/V(01t),
which may be written 1 — c = 11 ,
w=i \i + q
Now, as q increases from 0 to 1, the product on the right is continuous
and steadily decreases from 1 to 0 ; and so (§ 3*63) it passes through the
value 1 — c once and only once. Consequently a solution of the equation
in T exists and the problem of inversion may be regarded as solved.
21*71. The problem of inversion for complex values of c. The modular functions
f{r\g(j\ h{r).
The problem of inversion may be regarded as a problem of Integral Calculus, and it
may be proved, by somewhat lengthy algebraical investigations involving a discussion of
the behaviour of I {\—fi) "^ {I- k'^fi) ^ dt^ when y lies on a ' Riemann surface,' that the
y 0
problem of inversion possesses a solution. For an exhaustive discussion of this aspect of
the problem, the reader is referred to Hancock, Elliptic Functions, Vol. i.
It is, however, more in accordance with the spirit of this work to prove by Cauchy's
method (§ 6-31) that the equation c = 52* (0 | r)/^^* (0 | r) has one root lying in a certain
domain of the r-plane and that (subject to certain limitations) this root is an analytic
function of c, when c is regarded as variable. It has been seen that the existence of this
root yields the solution of the inversion i:)roblem, so that the existence of the Jacobian
elliptic function with given modulus k will have been demonstrated.
The method just indicated has the advantage of exhibiting the potentialities of what
are known as modular functions. The general theory of these functions (which are of
great importance in connexion with the Theories of Transformation of Elliptic Functions)
has been considered in a treatise by Klein and Fricke +.
Let /(O^^IO.- n j^-tf"-^r = '^^^^^
- fl-e'-""^)"'^| 8 _ ^4^ (0 I r)
^^^' ," |,l+>-l)-/ ~53*(0|r)'
Hr)=-f{r)lg{T).
Then, if Tr'= — 1, the functions just introduced possess the following properties :
/(r + 2)=/(r), g{r + 2)==g{r\ f{r)+g{r)=\,
/ (r + 1 ) = /. (r ), / (r') =g (r), g (r') =/ (r),
by §§ 21-2 corollary, 21-51 example 1.
* This is the case whicli is of practical importance.
t F. Klein, Vorlesumicn ilhcr die Theorie der clUptischen Modulfunktionen, auspearbeitet
iind rervolhtiindigt von R. Fricke. (Leipzig, 1890.)
21-71, 21-711]
THE THETA FUNCTIONS
475
It is easy to see that as /(t) -»• +-oo , the functions Jj5^«~"'"/(r)=/, (t) and g (t) tend to
unity, uniformly with resj)ect to R (t), when —\^R{t)^\ ; and the derivates of these two
functions (with regard to r) tend uniformly to zero* in the same circumstances.
21*711. The principal solution of f(T) — c=^0.
It has been seen in § 6-31 that, if /(t) is analytic inside and on any contour, 2'iTi times
the number of roots of the equation /(r) -c = 0 inside the contour is equal to
1 clfir)
h
dr,
lf(T)-C dr
taken round the contour in question.
Take the contour ABCDEFE'D'C'B'A shewn in the figure, it being supposed
temporarily t that f{r) — c has no zero actually on the contour.
E' . F . E
-1 0 1
The contour is constructed in the following manner :
FE is drawn parallel to the real axis, at a large distance from it.
AB is the inverse of FE with respect to the circle | t | =1.
BC is the inverse of ED with respect to j r | = 1, i) being chosen so|that D\=AO.
By elementary geometry, it follows that, since G and D are inverse points and 1 is its
own inverse, the circle on Dl as diameter passes through C ; and so the arc CD of this
circle is the reflexion of the arc AB in the line R{t) = \.
The left-hand half of the figure is the reflexioji of the right-hand half in the line
R{r) = 0.
* This follows from the expressions for the Theta-f unctions as power series in q, it being
observed that | g | ■^- 0 as I (r) ^- 4- oo .
t The values of /(r) at points on the contour are discussed in § 21-712.
476 THE TRANSCENDENTAL FUNCTIONS [CHAP. XXI
It will now be shewji that, unless* c^l or c^O, the equation /(r) -c=0 has one, and
only one, root inside the contour, provided that FE is sufficiently distant from the real
axis. This root will be called the principal root of the equation.
To establish the existence of this root, consider I -r— r '-i dr taken along the
Jf{r)-c dr
various portions of the contour.
Since /(r + 2) = /(r), we have
[J BE ] E'D'] J{t)-C dr
Also, as T describes BC and B'C, r'(= — 1/r) describes E'D' and ED respectively;
and so
{] BG J C'B') f{r)-C dr [J BC J C'B') g{r)-G dr
[J E'D' J DE) g{r)-G dr
= 0,
because ^(r' + 2)=^ (r'), and consequently corresponding elements of the integrals cancel.
Since f (r±l) = /i (r), we have
D'C J CD) fir)- G dr jB'ABh{T)-C dr
but as / describes B'AB, r describes EE', and so the integral round the complete contour
reduces to
df{r), 1 dh jr') ^ 1 df jr')] ^^
jEE'\f{r) — c dr Ii{t')-c dr f{r') dr
= ( f 1 dfjr) J. dhjr) 1 dg(r)]
jEE'\f{r)-C dr A (r) {1 - C . /i (t)} dr '^ g {t) - C dr\ '
Now as EE' moves off to infinityt, /(■r)-c-»-- 0 4=0, ,9(7-)— c^»- 1 -c=}=0, and so the
limit of the integral is
] E'E 1 - c . A (r) [ dr dr J
But l-c./i(r)^l, /i(r)-^l, ^i(r)^l, ^^^0, ^^^ ^ 0, and so the limit of the
(XT dr
integral is
/
TTldT—^TTi.
E'E
Now, if we choose EE' to be initially so far from the real axis that/(r) - c, \-c.h (r),
g(T)-c have no zeros when r is above EE', then the contour will pass over no zeros
of /W-*^' ;^« ^^" moves off to infinity and the radii of the arcs CD, UC, B'AB diminish
to zero ; and then the integral will not change as the contour is modified, and so the
original contour integral will be 2ni, and the number of zeros of /'(r) -c inside the original
contour will be precisely one.
* It is sbewn in § 21-712 that, if c ^ 1 or c ^ 0, then /(r) - c has a zero on the contour.
t It has been supposed temporarily that c 4= 0 and c + 1.
<">=i^-/.
21-712-21-73J THE THETA FUNCTIONS 477
21*712. The values of the modular function fir) on the contour considered.
We now have to discuss the point mentioned at the beginning of § 21"711, concerning
the zeros of /(t) — c on the lines* joining +1 to ±H-xit and on the semicircles of
OjBCI, (-1)(7'5'0.
As T goes from 1 to 1 +<» * or from — 1 to - 1 + & i, /(t) goes from — oo to 0 through
real negative values. So, if c is negative, we make an indentation in DE and a corre-
sponding indentation in D'E' ; and the integrals along the indentations cancel in virtue of
the relation /(r + 2)=/(r).
As T describes the semicircle 05(71, t' goes from — 1 + oo i to — 1, aud/(r)=^(r') = 1 -/(t'),
and goes from 1 to -}- oo through real values ; it would be possible to make indentations in
BC and EC to avoid this difficulty, but we do not do so for the following reason : the
effect of changing the sign of the imaginary part of a number is to change the sign of the
real part of t. Now, if 0< iZ (c) < 1 and /(c) be small, this merely makes t cross QF by a
short path; if li{c)<0, t goes from BE to D'E' (or vice versa) and the value of q alters
only slightly ; but if E (c)>l, t goes from BC to B'C, and so q is not a one- valued function
of c so far as circuits round c= -|- 1 are concerned ; to make q a one- valued function of c,
we cut the c-plane from +\ to +oo ; and then for values of c in the cut plane, q is
determined as a one-valued analytic function of c, say q{c), by the formula q(c) = e"^'''^^'
where
dfir)
f{T)-c dr
as may be seen from § 6-3, by using the method of § 5'22.
If c describes a circuit not surrounding the point c=l, q{c) is one-valued, but t{c) is
one-valued only if, in addition, the circuit does not surround the point c = 0.
21 •72. The periods, regarded as functions of the modidus.
Since ^=^5x53^(0, q) we see from v:^ 21-712 that K is a one-valued analytic function of
c{ = k^) when a cut from 1 to -i-oo is made in the c-plane; but since K'= — irK, we see
that K' is not a one-valued function of c unless an additional cut is made from 0 to — oc ;
it will appear later (§ 22-32) that the cut from 1 to -|- oo which was necessary so far as
K is concerned is not necessary as regards K'.
21*73. The inversion-problem associated with Weierstrassian elliptic functions.
It will now be shewn that, when invariants g.^^ and g^ are given, such that g^^'ilg^, it
is possible to construct the Weierstrassian elliptic function with these invariants ; that is
to say, we shall shew that it is possible to construct periods 2a)i, 2aj2 such that the function
^ (z I 0)1 , 0*2) has invariants g2 and gs .
The problem is solved if we can obtain a solution of the differential equation
of the form ^^^ {^\<^iy (^2)-
We proceed to effect the solution of the equation with the aid of Theta-functions.
Let v = Az, where A is a constant to be determined presently.
* We have seen that EE' can be so chosen that/(T) -c has no zeros eitlier on EE' or on
the small circular arcs.
478 THE TRANSCENDENTAL FUNCTIONS [CHAP. XXI
By the methods of § 21-6, it is easily seen that
•82' (v) ^1 (v) - Si (") 52 (v) = - ^3 {v) ^4 (^) S2\
and hence, using the results of § 21-2, we have
Now let ei, 62, 63 be the roots of the equation Ay^-g^y-g^^O, chosen in such an order
that (ei-e3)/(ei-e2) is not* a real number greater than unity or negative.
In these circumstances the equation
ei - 63 ^ V (0 I r)
61-62 53*(0|r)
possesses a solution (§ 21-712) such that /(r)>0; this equation determines the parameter
T of the Theta-functions, which has, up till now, been at our disposal.
Choosing r in this manner, let A be next chosen so thatt
/12V = ei-e3.
Then the function
satisfies the equation
(j)''=4(2/-6i)(y-e2)Cy-e3).
The periods of y, qua function of z, are it A , ttt/A ; calling these 2a)i , 2a)2 we have
/((»2/cOl)>0.
The function ^(z\coi, ©2) niay be constructed with these periods, and it is easily
seen that ^{z)-A'^ Q-JfAA ^3^(0 I ^) ^t^i^ 1 r)-ei is an elliptic function with no pole at
the origin | ; it is therefore a constant, C, say.
If G-j, 6*3 be the invariants of p {z\ a>i, u^), we have
4Fi^)-(^2P{^)-Gs = r"{z) = 'iW{z)-C-e,} {p{z)-C-e^i {p{z)-C-e,},
and so, comparing coefficients of powers of ^ (z), we have
0 = 12C, G^2=.72-12C2, G'3=^3-^2C+4C3.
Hence C=0, G.,==g2, ^3=9^;
and so the function ^J (2 | wi , a^) '^^'ith the required invariants has been constructed.
21*8. lite numerical computation of elliptic functions.
The series proceeding in ascending powers of q are convenient for
calculating Theta-functions generally, even when | g' [ is as large as 0"9. But
it usually happens in practice that the modulus k is given and the calculation
* If ^^ > 1, then 0 < ^^ < 1 ; and if '^ < 0, then 1 - ^^ > 1, and
''k
-1
<1.
The values 0, 1, cc of (cj - e3)/((?i - e.^) are excluded since g'J^^^Tgz^-
+ The sign attached to J is a matter of indifference, since we deal exclusively with even
functions of v and z.
X The terms in z-^ cancel, and there is no term in s"! because the function is even.
21-8] THE THETA FUNCTIONS 479
of K, K' and q is necessary. It will be seen later (§§ 22*301, 22*32) that
K, K' are expressible in terms of hypergeometric functions, by the equations
but these series converge slowly except when | k \ and | k' \ respectively are
quite small ; so that the series are never simultaneously suitable for numerical
calculations.
To obtain more convenient series for numerical work, we first calculate q
as a root of the equation k = ^./(O, q)l^i (0, q), and then obtain K from the
formula ^ = ^7r^3'^(0, q) and K' from the formula
K' = 'rr--K\oge{llq).
The equation k = V (0, ^)/V (0, q)
is equivalent to* '^Jk' = ^i{0, q)l%{0, q).
Writing 2e = _ -,, , (so that 0 < e < - when 0 < A; < 1), we get
^ %{Q,q)-^, (0, q) ^ X {0,q^
^ %iO,q) + %{0,q) %{0,q^)-
We have seen (§§21*71-21*712) that this equation in q* possesses a
solution which is an analytic function of e* when | e j < ^ ; and so q will be
expansible in a Maclaurin series in powers of e in this domainf.
It remains to determine the coefficients in this expansion from the
equation
_ g + g^ + ^-° + . ■ .
^ ~ 1 + 2q*^ 2gi« + . . . '
which may be written
q=€ + 2q*6-q'> + 2q"'e- q-'+ ...;
the reader will easily verify by continually substituting e + 2q'^€ — q^ + ...
for q wherever q occurs on the right that the first few terms j are given by
q = e+2€' + 1563 + 1506^=' + 0 (e").
It has just been seen that this series converges when j e j < g.
[Note. The first two terms of this expansion usually suffice; thus, even if k be as
large as V(0-8704) = 0 933..., e = |, 2^^ = 0*0000609, 1569 = 0-0000002.]
Example. Given k = M = \lsJ2, calculate q, K, K' by means of the expansion just
obtained, and also by observing that r = i, so that q^e"".
[^ = 0-0432139, /i:= A" = 1-854075.]
* In numerical work 0 < A; < 1, and so q is positive and 0 < ^/A;' < 1.
t The Theta-functions do not vanish when | (/ j < 1 except at 5 = 0, so this gives the only
possible branch point.
X This expansion was given by Weierstrass, Werke, ii. p. 276.
480
THE TRANSCENDENTAL FUNCTIONS
[chap. XXI
21 '9. The notations employed for the Theta-ftmctions.
The following scheme indicates the principal systems of notation which have been
employed by various writers; the symbols in any one column all denote the same
function.
5i (,r2)
SA^z)
33 {m)
S{m)
Jacobi 1
1
5i(.)
h{z)
3s (z)
3i{z)
1
Tannery and Molk
6x (W2)
62 (tO-2)
63 {<oz)
6{a>z)
Briot and Bouquet '
6,{z)
^2(2)
e^iz)
e,{z)
Weierstrass, Halphen, Hancock
e{z)
^1(2)
6z{z)
02 (Z)
Jordan, Harkness and Morley
The notation employed by Hermite, H. J. S. Smith and some other mathematicians is
expressed by the equation
e^ ^{x)= 2 (_)«-2i(2«+^)^gi7r{2«+M)a;/a.
with this notation the results of § 2ril example 3 take the very concise form
Cayley employs Jacobi's earlier notation (§ 21-62). The advantage of the Weierstrassian
notation is that unity (instead of n) is the real period of ^3(2) and 6n{z).
Jordan's notation exhibits the analogy between the Theta-functions and the three
Sigma-functions defined in § 20'421. The reader will easily obtain relations, similar
to that of § 21*43, connecting ^,.(2) with o-,. (2a)i2) when r=l, 2, 3.
REFERENCES.
L. EuLER, Opera Omnia, ser. 1, Vol. xx.
C. G. J. Jacobi, Fundamenta Nova*; Oes. Math. Werke, i. pp. 497-538.
C. Hermite, Oeuvres Matheraatiques.
F. Klein, Vorlesungen iiber die Theorie der elliptischen Modulfiinktionen (Ausgear-
beitet und vervoUstandigt von R. Fricke). (Leipzig, 1890.)
H. Weber, Elliptische Funktionen tmd algehrai^chc Zahlen. (Brunswick, 1891.)
J. Tannery et J. Molk, Fonctions Elliptiques. (Paris, 1893-1902.)
Miscellaneous Examples.
Obtain the addition-formulae
3y (// ^z)3,{y-z) V = 53^' (2/) 3.^ (z) - 3,^ (y) S-J^ {z) = 3{^ (y) 3,^ (z) - 3^ {y) 3,^ (z),
S, {y + z) S, {y - z) ^^ = ^4^ (y ) ^./ {z) -3^^ {y) 3.^ (z) = 3,^ (y) S,^ (.*) - ^3^ (y) 3,^ (z),
3, {y + z) 3, iy - z) 54- = 3,^ iy) 3,'^ (z) - 3,^ (y) 3./ (z) = 3-,^ {y) V (2) - 3.J^ (y) 3,^ (z),
34 iu + z) 3, {y - z) 3i' = 3-/ (y) 3./ (z) - 3,^ (y) 5./ (z) = ^4'^ (y) 3,^ {z) - 3,^ (y) 3,^ (z).
(Jacobi.)
* Reprinted in his Ges. Math. Werke, i. pp. 49-239.
21-9]
THE THETA FUNCTIONS
481
• 2. Obtain the addition-formulae
^4 (y + 2) Si 0/-2) S2' = »i' (y) V {z) + h' (y) ^i^ iz) = S2' iy) S*^ iz) + 9^ (y) ^3^ (z),
S* (y + z) ^4 (y-^) ^3== = V (I/) V (2) + ^2=' (y) ^-^ (2) = V (i/) 5*2 (2) + S,2 (y) ^^^ (^) ;
and, by increasing 1/ by half periods, obtain the corresponding formulae for
5r(y + 2)-9rO/-2)V and 3r(^ + z)Sr(^-z)Ss\
where r=l, 2, 3. (Jacobi.)
, 3. Obtain the formulae
^1 (2/ ± z) S2 iv + 2) 5354 = Si (if) S, (.y) 53 (z) S, {z) ± ^3 (j/) ^4 (y) 5, (2) ^2 (2),
^x(y±^)53 0/+^)-a2-94 = -»,(2/)-»3(y)'92(2)54(«)±^2(3/)-94(3')^l(2)-»3(2),
^1 (y ±^) S, {y + z) 5,^3 = ^1 {y) S, (3/) S, (z) Ss (2) ± S, {y) S, {y) B, (z) S, (z), .
S,(^±z) 3, 0/ + z) S2S3 = S, (y) 53 (y) ^2 (z) S, {z) + &, {y) S, iy) 5i {z) 3, (z),
S2 LV ± z) S* iv + z) S,3, = S, (y) 3, (y ) 3, (z) 3^ (z) + 3, (y) ^3 (y) -^i (z) Ss (z),
S3(2/±z)Sdy + z)3,3, = 3s(y)3,{y)3^(z)3^{z) + 3,(y)3,(9/)3,{z)3,iz).
(Jacobi.)
4. Obtain the duplication-formulae
^2 (2y) 5,542=.9,2 cy) V (y) - ^1^ (y) ^3^ (y),
^3 (2y) -93542=53^ (.'/) V (y) - ^1^ (y) 52'-^ (y),
^4(2y) V =^3ny)-^2*(y)=V(y)-^i*(y).
5. Obtain the duplication-formula
3, {2y) 3,3^3,^23, (y) 5^ (y) S3 (y) ^4 (y).
6. Obtain duplication-formulae from the results indicated in example 2.
7. Shew that, with the notation of § 21 "22,
[l]-[2] = [4]'-[3]', [l]-[3] = [lJ'-[3]', [l]-[4] = [2]'-[3]',
[2]-[3] = [l]'-[4]', [2] -[4] = [2]' -[4]', [3]-[4]=[2]' - [1]'.
8. Shew that
2[1122] = [1122]'-j-[2211]'-[4433]'-f[3344]',
2 [1133] = [1133]'-!- [3311]' -[4422]'-!- [2244]',
2 [1144] = [1144]'-f [4411]' -[3322]' + [2233]',
2 [2233] = [2233]' + [3322]' - [441 1]' + [1 144]',
2 [2244] = [2244]' + [4422]' - [331 1 ]' + [ 1 133]',
2 [3344] = [3344]' + [4433]' - [22 1 1 ]' + [ 1 1 22]'.
9. Obtain the formulae
(Jacobi.)
(Jacobi.)
(Jacobi.
,2«-l^-2|
y2n-l\-2\
« = 1
k^k'-i=2q^ n {{l + q^"f{l-q''
10. Deduce the following results from example 9 :
n (i-^2»-i)o=2^*F/t"^ n (i+^2»-i)6^2^*(H-';
K=l «=1
-k
n (!-(/")« =27r-3fy-'/-X-'A'3,
n {l-qnf =4.7r-'q-hH'-^K-\
n (i-i-j-")"
H = l
n (1-hr)
■■%q
'kk'-
■^k^k-
W. M. A.
(Jacobi.)
31
482 THE TRANSCENDENTAL FUNCTIONS [CHAP. XXI
i'S ' (z)
11. By considering I ^^4- e^"*^ dz taken along the contour formed by the parallelogram
whose corners are — iw, ^tt, ^tt + ttt, —^tt + ttt, shew that
and deduce that, when [ I{z) | <|/(7rr),
Si{z)_. '^ 9'"sin2?i3
12. Obtain the following expansions :
&i (z) ^ , °° o^" sin 2?i2
^^ = -t. + 4^./-^-^,
^3'(z)_. "^ (-)"g"sin2?i2;
each expansion being valid in the strip of the 3-plane in which the series involved is
absolutely convergent.
(Jacobi.)
13. Shew that, if | /(y) | < /(ttt) and \I{z)\<I (ttt), then
S (V + Z) $ ' 00 00
n / Tn / N = cot w + cot 2 + 4 2 2 o2""» sin (2my + 2nz).
^1(2/) ^l(^) m=ln=l
(Math. Trip. 1908.)
14. Shew that, if | I{z)\<^I (ttt), then'
— - ~-r = i «o + 2 a„ cos 2?12,
IT ^i {Z) - n=i
where a„ = 2 2 ^("'■+i)(2»+"'+i).
(Math. Trip. 1903.)
[Obtain a reduction formula for a„ by considering j {Si{z)}-^e^"^^dz taken round the
contour of example 11.]
15. Shew that
-^1 (2) L ,i = 1 1 - 2y^" cos 22 + q*\
is a doubly-periodic function of s with no singularities, and deduce that it is zero.
Prove similarly that
■^2' (2) . , ^ o2nsin22
•^2 (^) M=i 1 + 2f" cos 22 + J''" '
■^3'(2)^_^^ 22n-lgijj22
^3 (2) ,,=1 1 1 2p" ^^cos 2i+q*''-^ '
^4 (•j) «=i 1 +2^'-"- 1 COS 22 + j^''-^ •
16. Obtain the values of k, I-', A', A" correct to six places of decimals when q = ^Q.
[/?,■ = 0-895769, /{•' = 0-4445 18,
A= 2-262700, A" = 1-658414.]
THE THETA FUNCTIONS 483
17. Shew that, ii w+x + i/+z=0, then, with the notation of § 21-22,
[3]+[l]=[2] + [4],
[1234]+[3412] + [2143] + [4321] = 0.
18. Shew that
19. By putting x=y=z, w«=3;» in Jacobi's fundamental formulae, obtain the following
results :
B^ {x) ^1 (3^)+^43 {x) Si {3x) = Si^ {2x) $t,
^3^ (^) ^3 (3.r) - .943 (x) 9i (3a;) = V (2^) K
^/ (x) ^2 (3^) + ^43 (x) Si {3x):=S3^ {2x) Ss.
20. Deduce from example 19 that
{Si^ (x) Si {3x) Si^ + Si^ {x) Si {3x) Si'f + {S^^ (x) S3 {3x) S^^ - S^ {x) Si {3x) S^^f
= {Si (x) S, {3x) Ss^ + Si^ (x) Si {3x) S,f.
(Trinity, 1882.)
31—2
CHAPTER XXII
THE JACOBIAN ELLIPTIC FUNCTIONS
22"1. Elliptic functions with two simple poles.
In the course of proving general theorems concerning elliptic functions
at the beginning of Chapter XX, it was shewn that two classes of elliptic
functions were simpler than any others so far as their singularities were
concerned, namely the elliptic functions of order 2. The first class consists
of those with a single double pole (with zero residue) in each cell, the second
consists of those with two simple poles in each cell, the sum of the residues
at these poles being zero.
An example of the first class, namely ^{z), was discussed at length in
Chapter xx ; in the present chapter we shall discuss various examples of
the second class, known as Jacohian elliptic functions* .
It will be seen (§22"122, note) that, in certain circumstances, the Jacobian
functions degenerate into the ordinary circular functions ; accordingly, a
notation (invented by Jacobi and modified by Gudermann and Glaisher) will
be employed which emphasizes an analogy between the Jacobian functions
and the circular functions.
From the theoretical aspect, it is most simple to regard the Jacobian
functions as quotients of Theta-functions (§ 21"61). But as many of their
fundamental properties can be obtained by quite elementary methods,
without appealing to the theor}^ of Theta-functions, we shall discuss the
functions without making use of Chapter xxi except when it is desirable
to do so for the sake of brevity or simplicity.
22"11. The Jacohian elliptic functions, sn u, en u, dnii.
It was shewn in § 21 -61 that if
^ ^, MW V)
^"^, M^/V)'
the Theta-functions being formed with parameter r, then
* These functions were introduced by Jacobi, but many of their properties were obtained
independently by Abel, who used a different notation. See the note on p. 505.
221, 22-11] THE JACOBIAN ELLIPTIC FUNCTIONS 485
where ^•^ = ^2 (0 i t)/^3 (0 1 t)- Conversely, if the constant k (called the
modulus*) be given, then, unless k^^\ or P ^ 0, a value of t can be found
(§§ 21-7-21-712) for which V(0| t)/V(0 | t) = A;^^ so that the solution
of the differential equation
subject to the condition \-r-] = 1 is
\duJu = y = t)
_%%(u/%')
the Theta-functions being formed with the parameter t which has been
determined.
The differential equation may be written
u= {l-t')-^(l-kH')-^dt,
J 0
and, by the methods of § 21 '7 3, it may be shewn that, if y and u are con-
nected by this integral formula, y may be expressed in terms of u as the
quotient of two Theta-functions, in the form already given.
Thus, if
fy
w= (l-t')-i(l-kH')-idt,
J 0
y may be regarded as the function of u defined by the quotient of the Theta-
functions, so that y is an analytic function of u except at its singularities,
which are all simple poles ; to denote this functional dependence, we write
y = sn (u, k),
or simply y = sn w, when it is unnecessary to emphasize the modulusf.
The function sn u is known as a Jacohian elliptic function of a, and
on w -— ^1 (^AsO /* .
'^"-^,^,(W) ^^>
[Unless the theory of the Theta-functions is assumed, it is exceedingly difficult to shew-
that the integral formula defines y as a function of u which is analytic except at simple
poles. Cf. Hancock, Elliptic Functions, Vol. i.]
Nowwite c„(»,i) = ||gM (B).
dnoa-)4:lw! <c)-
Then, from the relation of § 21*6, we have
^- snM = cnzidnw (I),
dii ^ '
* If 0 < fc < 1, and Q is the acute angle such that siu d = k, ^ is called the modular angle.
t The modulus will always be inserted when it is not k.
486 THE TRANSCENDENTAL FUNCTIONS [CHAP. XXII
and from the relations of § 21-2, we have
sn^ u + cn^u = l .(II),
¥sn^u + dn^u = l (Ill),
and, obviously, cnO = dnO=l (IV).
We shall now discuss the properties of the functions sn u, en ?«, dn to as defined by the
equations (A), (B), (C) by using the four relations (1), (II), (III), (IV) ; these four relations
are sufficient to make sn u, en «, dn u determinate functions of u. It will be assumed,
when necessary, that sn %, en u, dn ti are one- valued functions of u, analytic except at their
poles ; it will also be assumed that they are one-valued analytic functions of k^ when cuts
are made in the plane of the complex variable F from 1 to -f oo and from 0 to — oo .
22*12. Simple properties of sn u, cnu, dn u.
From the integral u= I (1 —t-)"^ (1 — kH^)'^ dt, it is evident, on writing
Jo
— t for t, that, if the sign of y be changed, the sign of u is also changed.
Hence sn u is an odd function of u.
Since sn (—u)= — sn u, it follows from (II) that en (— m) = + en m ; on
account of the one-valuedness of en u, by the theory of analytic continuation
it follows that either the upper sign, or else the lower sign, must always be
taken. In the special case u = 0, the upper sign has to be taken, and so it
has to be taken always ; hence en (— u) = en u, and en u is an even function
of u. In like manner, dn u is an even function of u.
These results are also obvious from the definitions (A), (B) and (C) of
§ 2211.
Next, let us differentiate the equation sn- «* -f cn^ m = 1 ; on using equation
(I), we get
d en H .
— -, — = — sn w dn M ;
du
in like manner, from equations (III) and (I) we have
ddnu , „
— J = — A;^ sn M en u.
du
22'121. The complementary modulus.
If A;^ -f- A:'- = 1 and k' ^ + 1 as k-*0, k' is known as the complementary
modulus. On account of the cut in the /..-^-plane from 1 to + oo , A;' is a one-
valued function of k.
[With the aid of the Theta-functions, we can make k'^ one-valued, by defining it to be
54(0|r)/53(0|r).]
- Example. Shew that, if
« = I \ 1 - ^2) - i (X''2 + ]cH^)~^ dt,
J u
the" y = en i^a^k).
22-12-22-2] THE JACOBIAN ELLIPTIC FUNCTIONS 487
Also, shew that, if «= f {l-t^y^ {t'^-if^r^dt,
J V
then y=dn(M, k).
[These results are sometimes written in the form
J cnu J dn Jt »
22*122. Glaisher's flotation* for quotients.
A short and convenient notation has been invented by Glaisher to express
reciprocals and quotients of the Jacobian elliptic functions ; the reciprocals
are denoted by reversing the order of the letters which express the function,
thus
nsu = l/snu, ncu^l/cnu, ndM = l/dnrf;
while quotients are denoted by writing in order the first letters of the
numerator and denominator functions, thus
sc w = sn u/cn u, sd m = sn u/dn u, cd w = en w/dn u,
cs w = en uJBXi u, dsu = dn w/sn u, dc w = dn u/cn u.
[Note. Jacobi's notation for the functions sn u, en u, dn u was sinam ?4, cosam u,
Aam M, the abbreviations now in use being due to Gudermannt, who also wrote tn m,
as an abbreviation for tanamw, in place of what is now written sew.
The reason for Jacobi's notation was that he regarded the inverse of the integral
u= ["^ (1-Jc^ singer ^dd
Jo
as fundamental, and wrote J <j) = a,vau; he also wrote Acji = {l—k^sin^(piy for ^ •]
* Example. Obtain the following results ;
Jo J csu
Jo J dSM
= r (i-«2)-i(i_p^2)-4^; =r- {fi-iy^{t^-py^dt
J cdu J (lc«
= r {fi-iy^(t^-k^y^dt = l"'"(<2_i)-i(^'2^2+^2)-i^^
J nsu J 1
(f^-iy^i-k'H'-y^dt.
T.
22'2. The addition-theorem for the function snu.
We shall now shew how to express sn (u + v) in terms of the Jacobian
elliptic functions of u and v ; the result will be the addition- theorem for the
function snw; it will be an addition-theorem in the strict sense, as it can
be written in the form of an algebraic relation connecting sn u, sn v, sn {u + v).
* Messenger of Mathematics, xi. p. 86.
t Crelle, xviii. pp. 12, 20.
X Fundamenta Nova, p. 30. As k-»-0, am «-»•(*.
488 THE TRANSCENDENTAL FUNCTIONS . [CHAP. XXII
[There are numerous methods of establishing the result; the one given is
essentially due to Euler*, who was the first to obtain (in 1756, 1757) the
integral of
dx dy _ ,\
in the form of an algebraic relation between x and y, when X denotes a
quartic function of x and Y is the same quartic function of y.
Three f other methods are given as examples, at the end of this section.]
Suppose that u and v vary while u + v remains constant and equal to a,
say, so that
dv _
du
Now introduce, as new variables, Sj and s^ defined by the equations
Si = sn u, So = sn V,
so that J .V = (1 - SiO (1 - k%%
and 42 ^ Q _ g2^ (^i _ ]^2g2-^^ sij^ce v" = 1.
Differentiating with regard to u and dividing by 2ii and 24 respectively,
we find that, for general values § of w and v,
s, = - (1 + k"") s^ + 2J<^s,\ s, = - (1 + I<f) s^ + 2kW-
Hence, by some easy algebra,
S1S2 ^s^l ^iC ^1^2 \^1 — ^2 )
s,'s^' - 4V ~ {s,'-Si'){l-k's,^s,') '
and so
d d
{s,s, - s,s,)-^ ^ (siSo - .%s,) = (1 - k's,%^)-' du^'^~ k%'s/) ;
on integrating this equation we have
i- /»/ Si'S^
where 0 is the constant of integration.
Replacing the expressions on the left by their values in terms of u and v
we get
en i< dn ?< sn v + en w dn V sn M _ „
1 — A;^ sn- u sn- v
* Acta Petropolitana, vi. (1761), pp. 37-57. Euler had obtained some special cases of
result a few years earlier.
t Another method is given by Legeudre, Fonctiom EUiptiqucs, i. p. 20.
X For brevity, we shall denote differential coefficients with regard to u by dots, thus
._dv .,_d^v
§ I.e. those values for which cnw dn ?( and cnv Anv do not vanish.
22-2] THE JACOBIAN ELLIPTIC FUNCTIONS 489
That is to say, we have two integrals of the equation du+ dv = 0, namely
(i) u + V = a and (ii)
sn M en i; dn v + sn V en w dn w
1 — A;'' sn^ u sn"^ v
G,
each integral involving an arbitrary constant. By the general theory of
differential equations of the first order, these integrals cannot be functionally
independent, and so
sn w en ■?; dn V + sn V en M dn w
1 — k^ sn^ u sn* v
is expressible as a function of w + v ; call this function /(m + v).
On putting V = 0, we see that f(u) = sn w ; and so the function / is the
sn function.
We have thus demonstrated the result that
, . sn w en V dn i; + sn i; en w dn w
sn {u + v) = -, ;„ — -, z — '
which is the addition-theorem.
Using an obvious notation*, we may write
an(„ + ,,) = -j--^,^,^^-.
' Example 1. Obtain the addition-theorem for sinw by using the results
• Example 2. Prove from first principles that
and deduce the addition -theorem for sn u. (Abel.)
* Example 3. Shew that
S^C^CL^ — S.^C^d] CjC2-f SjaiS2"2 »ia2 + »«i 32^1^2
(Cayley.)
• Example 4. Obtain the addition-theorem for sn u from the results
^1 (3/+^) -94 {y-z) ^2^3 = ^t {y) Si {y) 92 {z) h {z)+h {y) 9, {y) 9, (z) S^ (z\
^4 (y + z)Si(^- Z) Si' = V (y) 54^ (Z) - Si2 (y) 3^2 (,)^
given in Chapter xxi, IVfiscellaneous Examples 1 and 3 (pp. 480, 481). (Jacobi.)
. Example 5. Assuming that the coordinates of any point on the curve
/=(l-:*72)(i_p.^2)
can be expressed in the form (sn w, en u dn m), obtain the addition-theorem for sn u by
Abel's method (§ 20-312).
* This notation is due to Glaisher, Messenger, x. pp. 92, 124.
490 THE TRANSCENDENTAL FUNCTIONS [CHAP, XXII
[Consider the intersections of the given curve with the variable curve 2/--=l+mx+nx^ ;
one is (0, 1) ; let the others have parameters u^, u^, %, of which Ui, u^ may be chosen
arbitrarily by suitable choice of m and n. Shew that Ui + Ui+Us is constant, by the
method of § 20-312, and deduce that this constant is zero by taking
m = 0, %=-A(l+F).
Observe also that, by reason of the relations
{k^ - %2) x\ X2X3 = 2m, {B - nF) {X]^ + ^2 + ^3) = 2wi%,
we have
.^3 (1 -l?'X^x^)=X3-{\-\-j^^A 2mXj^X2=Xs-2mXj^X2 - nx^x^ {X1+X2+X3)
==[xi+X2 + Xs — nxi x^ ^3) — (^1 + ^2) - 2')nx-^ x^ - nx^ x^ix^+x^
= - *i 2^2 - ■^23/1- ]
22"21. The addition-theorems for an u and &nu.
We shall now establish the results
en M en y — sn u sn i; dn w dn w
en (u->rv)= = jz — I 1 ,
, , . dn M dn V — A;- sn w sn w en w en y
dn {u + v) = :. 7- — I 1 ;
1 — A;^ sn^ w sn^ v
the most simple method of obtaining them is from the formula for sn {a + v).
Using the notation introduced at the end of § 22*2, we have
(1 - k^s^^sif cn^ {u + v) = {\- ¥s^^siy {1 - sn^ {u + v)]
= (1 — k-s-cs^^y — {s^c^dQ + s^Cid-^)"
= 1 - W-s^^s.^ + k's.'s.^ - Si" (1 - si) (1 - k'^s,'')
- si (1 - sr) (1 - k'^si) - 2s,s.,c,c.Ad^
= (1 - si) (1 - si) + sisi (1 - k'si) (1 - k'si)
— Is-^s.^CiC^d^d^
= (Cia2 — s^Sod^d2y
and so en (u + v)=± -fzTj^sisi '
But both of these expressions are one-valued functions of u, analytic
except at isolated poles and zeros, and it is inconsistent with the theory
of analytic continuation that their ratio should be + 1 for some values of u,
and — 1 for other values, so the ambiguous sign is really definite ; putting
w = 0, we see that the plus sign has to be taken. The first formula is
consequently proved.
The formula for dn (11 + v) follows in like manner from the identity
(1 — k'^sisi)'^ — k- (siC.do + 6'oCirfi)-
= (1 - k-si) (1 - k-si) + k'sisi (1 - si) (1 - s.i) - 2k^SiSoCiC2d^d.i,
the proof of which is left to the reader.
22-21, 22-3] THE JACOBIAN ELLIPTIC FUNCTIONS 491
I Example 1. Shew that
dn (tt+ v) dn {u - «) = i_p^2g8 •
(Jacobi.)
[A set of 33 formulae of this nature connecting functions of w + v and of « — v is given
in the Fundamenta Nova, pp. 32-34.]
, Example 2, Shew that
3 cntt + cn» d cnM + cnv
du, snwdni; + sni)dn«~ dv snwdnv + sn ydnw'
so that (cntt + cnt;)/(snMdnv+sn vdnw) is a function of u-\-v only ; and deduce that it is
equal to {1 + en (« + v)}/sn {u + v).
Obtain a corresponding result for the function {siC2 + S2C^i{di + d^.
(Cayley.)
• Example 3. Shew that
1 -k'^^i\^{u + v)sn^{u — v) = {\ -Fsn*M) (1 -Fsn*v) {\—]c^sn'^u^n^v)~'^,
y?;'2 + F cn2 (m + v) cn2 (tt - v) = (^'2 + /{;2 en* «) (^'2 + F en* y) ( 1 - F sn2 M sn2 v) - 2.
(Jacobi and Glaisher.)
• Example 4. Obtain the addition-theorems for cn(M+i;), dn(M-l-v) by the method of
§ 22-2 example 4.
• Example 5. Using Glaisher's abridged notation {Messenger, x. p. 105), namely
s, c, d=an u, en u, dn u, and S, C, I)=sn 2u. en 2u, dn 2u,
prove that
2scd 1-232 + Fa* l-2Fg2+F3*
''^~l-/&2s*' ^~ l-Fs* ' r^¥s* '
(i+>S)^-(i->so^
^"(l+y?;-^)*-^!--?;,^)^*
• Example 6. With the notation of example 5, shew that
^ ~1+I>~k^l + C)~ k^{D-C) ~kf^+D-k'^C'
,_D^C _D + k'-C-k'^ ^k'^{\-D) _ k'^\ + C)
^~1+Z>~ P(l + (7) Ic^lD-C) k'^ + D-BC
k'^+I)+k^C_D+C_k'^l-C)_ k'\l + D)
I+jD l + C D-C k^+D-k'^C
(Glaisher.)
22-3. The constant K.
We have seen that, if
.'0
then y = sn {u, k).
If we take the upper limit to be unity (the path of integration being
a straight line) it is customary to denote the value of the integral by the
symbol K, so that sn (K, k)= 1.
[It will be seen in § 22*302 that this definition of K is equivalent to the definition as
i7r332 in §21-61.]
492 THE TRANSCENDENTAL FUNCTIONS [CHAP. XXII
It is obvious that cnK = 0 and dnK=±k'; to fix the ambiguity in sign,
suppose 0 < ^ < 1, and trace the change in (1 —k^t^)^ as t increases from 0 to 1;
since this expression is initially unity and as neither of its branch points (at
t = ± k~^) is encountered, the final value of the expression is positive, and so
it is + A;' ; and therefore, since dn ^ is a continuous function of k, its value is
always + k'.
The elliptic functions of K are thus given by the formulae
sn^=l, cn^ = 0, dnJr = ^'.
22*301. The expression of K in terms of k.
In the integral defining K, write t = sin <^, and we have at once
,47r
K= r {1 -k"" sin' 4>)-^d(f).
Jo
When I A; I < 1, the integrand may be expanded in a series of powers of k,
the series converging uniformly with regard to (j> (by § 3"34, since sin-^ ^ ^ 1) ;
integrating term-by-term (| 4'7), we at once get
K = ^'rrF{^^, -; 1; k-J=^7rF (^^, ^i 1; cj,
where c = ¥. By the theory of analytic continuation, this result holds for-
all values of c when a cut is made from 1 to + oo in the c-plane, since
both the integrand and the hypergeometric function are one-valued and
analytic in the cut plane.
• Example. Shew that
^(^^"S)=^^'- (Legendre.)
22*302. The eqtdvalence of the definitions of K.
Taking u = \iv^^ in §21-61, we see at once that ^w{\Tr^-^) = \ and so GVi{^\Tr^^) — (d.
Consequently, 1-snM has a doable zero at Jtt^s^. Therefore, since the number of poles
of sn« in the cell with corners 0, 'inB^^ it {j -{-l) ^■i; tt (t-1)^3^ is two, it follows from
§ 20'13 that the only zeros of l-snw are at the points ^l = \^^ {'^m + l + ^nr)^^, where
m and n are integers. Therefore, with the definition of § 22-3,
Now take r to be a pure imaginary, so that 0<^<1, and K is real ; and we have
?i=0, so that
^7r(4m + 1)^32= r''(l-Fsin2</))-^o?(^,
where m is a positive integer or zero ; it is obviously not a negative integer.
If m is a positive integer, since I (1 - F sin^ 0) ^dc^ is a continuous function of a and
J 0
so passes through all values between 0 and K as a increases from 0 to ^tt, we can find
a value of a less than W, such that
^/(4m + 1 ) =: 1 77 532 = I (1 - F sin2 0) - * t/0 ;
Jo
and so sn (1^77^32) = sin a <1,
which is untrue, since sn (iTr^32) = l.
22-301-22-31] THE JACOBIAN ELLIPTIC FUNCTIONS • 493
Therefore m must be zero, that is to say we have
But both K and ^nSs^ are analytic functions of k when the c- plane is cut from 1 to
+ 00, and so, by the theory of analytic continuation, this result, proved when 0<^<1,
persists throughout the cut plane.
The equivalence of the definitions of K has therefore been established.
• Example 1. By considering the integral
J 0
shew that sn 2K=0.
, Example 2. Prove that
sn^K={l+k'y^, en I A'= yf *(!+>?;')"*, du^K^k'K
[Notice that when u = ^K, cq2ic = 0. The sinrplest way of determining the signs to
be attached to the various radicals is to make ^"-^0, k'-*-l, and then snu, cnu, dnu
degenerate into sin ?<, cosm, 1.]
' Example 3. Prove, by means of the theory of Theta-functions, that
cs^A'=dn^A'=^'i
22*31. The periodic properties (associated with K) of the Jacobian
elliptic functions.
The intimate connexion of K with periodic properties of the functions
sn i(, en 11, dn u, which may be anticipated from the periodic properties of
Theta-functions associated with -tt, will now be demonstrated directly from
the addition-theorem.
By § 22'2, we have
„, snucnK dn K + snK cnudnu ,
'^(^ + K)= l-k^sn^usn^K = '^''-
In like manner, from § 22"21,
en (u + K) = — k' sd u, dn (u + -fiT) = k' nd u.
TT / r. rr\ ^n {u +K) k' sd u
Hence sn (u + 2K) = ,') j-/. = — r> — r- = — sn u,
^ ^ dn{u+K) kudu
and, similarly, en (w + 2K) = — en w, dn (w + 2K) = dn u.
Finally, sn(w + 4ir) = — sn(«. + 2/{') = snM, c\\{it-\-^K) = cwu.
Thus 4:K is a period of each of the functions sn il, en u, tuJiile dn it has
the smaller per-iod 2K.
• Example 1. Obtain the results
sn (?i + A'') = cd«, en (u + K)= —k'sdv, dn{u + K) = k'ndu,
directly from the definitions of sn u, cu u, dn u as quotients of Theta-functions.
» Example 2. Show that cs u cs (A -u) = k'.
494 • THE TRANSCENDENTAL FUNCTIONS [CHAP. XXII
22-32. The constant K'.
We shall denote the integral
Wl-t^y'^il-k'H'y^dt
Jo
by the symbol K', so that K' is the same function of k'^ (= c') as K is of
k^ (= c) ; and so
when the c'-plane is cut from 1 to + x , i.e. when the c-plane is cut from
0 to — 00 ,
To shew that this definition of K' is equivalent to the definition of § 21-61, we observe
that if tt'= — 1, -ff" is the one-valued function of k^, in the cut plane, defined by the equations
£:=inS,^ (0 I r),' F = V (0 I r)-^3'' (0 I r),
while, with the definition of § 21-51,
K' ^ |^^32 (0 I r'), k'-i = &2^ (0 I r') ^^3* (0 1 r'),
so that K' must be the same function of ^'^ as K is of k'^ ; and this is consistent with the
integral definition of K' as
J 0
It will now be shewn that if the c-plane be cut from 0 to — oo and from
1 to + 00 , then in the cut plane K' may be defined by the equation
K'=r (s'-l)-i{l-k's')-ids.
First suppose that 0<A;<1, so that 0<k'<l, and then the integrals
concerned are real. In the integral
f {l-t')-^{l-k'H')-^dt
Jo
make the substitution
s = {l-k'H')-i,
which gives
(s' - l)i = k't (1 - k'H') -K (1 - f^'s')i = k'(l- t')^ (1 - k'H') '- i,
ds _ ¥H
'dt'il-k'H')^"'-
it being understood that the positive value of each radical is to be taken.
On substitution, we at once get the result stated, namely that
fl/k
K' = (6*^ - 1) - 4 (1 _ k's') - 4 ds,
provided that 0<k<l; the result has next to be extended to complex values
of A;.
22-32, 22-33] the jacobian elliptic functions 495
Consider T'^l - f') " ^ (1 -k'^tT^dt,
J 0
the path of integration passing above the point 1, and not crossing the imaginary axis*.
The path may be taken to be the straight lines joining 0 to 1 - 8 and 1 +8 to ^~i together
with a semicircle of (small) radius 8 above the real axis. If (1-^^)* ^^(j (i_Pf2)i
reduce to +1 at <=0 the value of the former at 1 +8 is e'^'^S* (2 + 8)*= -i{fi - 1)*, where
each radical is positive ; while the value of the latter at t=l is 4-^ when k is real, and
hence by the theory of analytic continuation it is always + k'.
Make 8-^0, and the integral round the semicircle tends to zero like 8* ; and so
P'*(l-«2)-i(l-F^2)-i^^_j^+J^^*(^2_l)-i(l_p^2)-i^^^
Now P^*(l-f-')-*(l-F«2)-i^;= r(y{;2_^42)-i(l_M2)-i^^^
which t is analytic throughout the cut plane, while K is analytic throughout the cut plane.
Hence j^%t^-l)-i (l-kH^^dt
is analytic throughout the cut plane, and as it is equal to the analytic function K' when
0</(;<l, the equality persists throughout the cut plane ; that is to say
K'= r^^it^-iy^il-kH'T^dt,
when the c-plane is cut from 0 to — oo and from 1 to + co .
rVk
Since K-\-iK'= {l-t')-^^{l-kH')-^dt,
Jo
we have sn {K + iK')^ Ijk, dn {K + iK') = 0 ;
while the value of cn{K + iK') is the value of (1 — t^)^ when t has followed
the prescribed path to the point l/k, and so its value is — ik'/k, not + ik'Jk.
« Example 1. Shew that
lj\til-i){l-kH)}-Ut=lj^^^^Jt{t-l){&^t-l)}-idt=K,
' Example 2. Shew that A"' satisfies the same linear differential equation as K {^ 22'301
example).
22'33. The periodic propeHiesX {associated with K -¥ iK') of tlie Jacobian
elliptic functions.
If we make use of the three equations
sn (K + iK') = k-\ en {K + iK') = - ik'/k, dn {K + iK') = 0,
* li (k) > 0 because ] argc | < tt.
+ The path of integration passes above the poiut u — k.
X The double periodicity of sn u may be inferred from dynamical considerations. See
Whittaker, Analytical Dynamics, § 44.
496
THE TRANSCENDENTAL FUNCTIONS
[chap. XXII
we get at once, from the addition-theorems for sn u, en u, dn u, the following
results :
sn u en (K + iK') dn {K + iK') 4- sn (iT + iK') en u dn w
sn (w -\-K + iK') =
and similarly
1 - P sn'' u ^li' (K + iK')
= k~^ do a,
cn{u + K + iK') = — ik'k~^ no u,
dn {u + K + iK') = ik' sc u.
By repeated applications of these formulae we have
' sn (i* + ^K + UK') = sn u,
en (m + 4ir + 4iX') = en u,
dn (w + 4if + 4tX') = dn m.
sn (w + 2ir + 2iZ') = - sn it,
en (w + 2Z + 2ii5r') = en u,
dn (t< + 2ir + 2iK') = - dn w,
Hence the functions sn u and dn w have period 4iK + 4>iK', while en u has
the smaller period 2K + 2iK'.
22*34. The periodic properties (associated with iK') of the Jacobian
elliptic functions.
By the addition-theorem we have
sn (ii + iK') = sn (u - K+ K + iK')
= k-'dG(u-K)
= k~^ ns u.
Similarly we find the equations
en (a + iK') = — ik~^ ds u,
dn ( w -I- iK') = — ics u.
By repeated applications of these formulae we have
' sn (u + 2iK') = sn u, ( sn{u + UK') = sn %,
- en (w -f 2iK') = — en u, \ en (it -i- UK') = en u,
dn{u-\- 2iK') = - dn ii, [ dniu + UK') = dn u.
Hence the functions en u and dn w have period UK', while snu has the
smaller period 2iK'.
Example. Obtain the formulae
sn (M-f2;nA' + 2;w/i') = ( — )"'snM,
en {u + 2//1 A'+ 2?ii A'') =:(-)'" + « en u,
dn ( « + 2m /i + iniK') = ( - )" dn u.
22'341. The behaviour of the Jacobian elliptic functions oiear the origin
and near iK'.
We have
d d^
-.- sn u = en u dn u, -j-., sn u = Uc^ sn^ u en w dn u — Gwudnu (dn^ i< -H ^■- cn^ u).
an du' ^ ^
22-34-22-35] the jacobian elliptic functions 497
Hence, by Maclaurin's theorem, we have, for small vahies of | w | ,
snu = u-^(l +k^)u^ + 0 (u%
on using the fact that sn u is an odd function.
In like manner
cnw = l-|w2^.o (^^4)^
It follows that
sn (it + iK') = k~^ ns w "
— i 2}c^ — 1
and similarly en (u + iK') = -7^ H wr — *w + 0 (u^),
i 2 — A;^
dn (u + iK') = + — ^ — m + 0 (v?).
u b
It follows that at the point iK' the functions sn v, en v, dn v have simple
poles with residues k~^, — ik~^, — i respectively.
Example. Obtain the residues of su u, en u, dn u at iK' by the theory of Theta-
f unctions.
22'35. General description of the functions sn u, en 11, dn u.
The foregoing investigations of the functions sn u, en u and dn u may be
summarised in the following terms :
(I) The function snu is a doubly-periodic function of u with periods
4<K, 2iK'. It is analytic except at the points congruent to iK' or to 2K + iK'
(mod. 4iK, 2iK') ; these points are simple poles, the residues ab the first set all
being k~^ and the residues at the second set all being — k~^ ; and the function
has a simple zero at all points congruent to 0 (mod. 2K, 2iK').
It may be t)bserved that sn u is the only function of u satisfying this description ; for
if 0 {u) were another such function, su u - ^ («) would have no singularities and would be
a doubly-periodic function; hence (§ 20"12) it would be a constant, and this constant
vanishes, as may be seen by putting u=0 ; so that ^ («) = snM.
When 0 < A;^ < 1, it is obvious that K and K' are real, and sn u is real for
real values of u and is a pure imaginary when a is a pure imaginary.
(II) The function en u is a doubly -periodic function of u with periods
4/f and 2K + 2iK'. It is analytic except at points congruent to iK' or to
2K + iK' (mod. 4^, 2K -1- 2iK') ; these points are simple poles, the residues
w. M. A, .S2
498 THE TRANSCENDENTAL FUNCTIONS [CHAP. XXII
at the first set being — ik~^, and the residues at the second set being ik~^ ;
and the function has a simple zero at all points congruent to ^ (mod, 2K, 2iK').
(Ill) The function dn u is a doubly-periodic function of u with periods
2K and 4iiK'. It is analytic except at points congruent to iK^ or to SiK'
(mod. 2K, 4)iK') ; these points are simple poles, the residues at the first set
being — i, and the residues at the second set being i ; and the function has
a simple zero at all points congruent to K + iK' (mod. 2K, 2iK').
[To see that the functions have no zeros or poles other than those just specified,
recourse must be had to their definitions in terms of Theta-fuuctions.]
22*351. The connexion between Weierstrassian and Jacobian elliptic functions.
If gj, 62 > H be any three distinct numbers whose sum is zero, and if we write
e, -e.
i/ = e2 +
sn^ (X^, k) '
we have ( ;7 ) = ^ (cj - 62)^ X'^ ns^ \u cs^ \u ds- \u
= 4 (<3i - 62)^ ^^ ns^ \u (ns^ \u-l) (ns^ \u - P)
= 4X2 (^1 - 62) ~ Uy - ^2) iy-ei){y- k"- {e^ - e^) - e^.
Hence \i^ = e^ — e-i and ^"^ = (63 — 6'2)/(ej — 62), y satisfies the equation*
and so e^ + (ei - e^) ns^ ] it (ci - e.f, f'^m^l = ^J (?^+ a ; ^2, 5^3),
I V ^1 — 62J
where o is a constant. Making u ->- 0, we see that a is a period, and so
i^ {^^ ; 9-2 , ffs) = 62 + (ei - 62) ns^ {« (^i - e.^)^},
the Jacobian elliiitic function having its moduhis given by the equation
F=
63 — 62
61 - 62
22"4. Jacobi's irnaginai^y transformation^.
The result of § 21 "51, which gave a transformation from Theta- functions
with parameter t to Theta-functions with parameter r = — l/r, naturally
produces a transformation of Jacobian elliptic functions ; this transformation
is expressed by the equations
sn {iu, k) = i sc {u, k'), en {ia, k) = nc (u, k'), dn {iu, k) = dc {u, k').
Suppose, for simplicity, that 0 < c < 1 and y >0; let
{l-t'-)--^(l-kH-)-^^dt = iu,
J 0
so that {y = sn (iu, k) ;
take the path of integration to be a straight line, and we have
en (ill, k) = {l + y-f\ dn (iu, k) = (l + k^y^-)K
* The values of g-^ and ^3 are, as usual, -|St'2e3 aud ^eit'2e3.
t Fundameiita Nocu, pp. 34, 35. Abel {Crelle, 11. p. 104) derives the double periodicity of
elliptic functions from tliis result. Of. Jacobi, Werke, i. p. 402.
22-351-22-41] the jacobian elliptic functions 499
Now put y = 7^I{1 — rf)^, where 0 < ?; < 1, so that the range of values
of t is from 0 to i't)l{l — 7f^)^, and hence, if t=-it-^l{\ — t^^)^, the range of
values of ti is from 0 to 77,
Then dt = l{l-t^)-^idt„ {l-t"^)^ = {l-t,^)-^,
/ 1 - kH\ = (1 - k'H,^) ^ * (1 - t,') - i
and we have iu = [\l - 1^^) " ^ (1 - k'Hy^) " ^ idt^ ,
Jo
so that 17 = sn (w, k')
and therefore y = 8c {u, k').
We have thus obtained the result that
sn {iu, k) = i sc (u, k').
Also (in{iv,k) = {\-\-y'^)^ = {l-r]'^)~^ = nc{u,k'),
and dn (tw, A;) = (1 -/- ¥y'')^ = (1 - A;'^'?')* (1 - '^') " * = dc {u, k').
Now sn(m, ^) and isc {u, k') are one-valued functions of u and k (in the
cut c-plane) with isolated poles. Hence by the theory of analytic continuation
the results proved for real values of u and k hold for general complex values
of u and k.
22'41. Proof of Jacobi's imaginary transformation by the aid of Theta-
functions.
The results just obtained may be proved very simply by the aid of
Theta-functions. Thus, from § 21-61,
where ^ = m/V(0|t),
J 1, e oi ei / • 7 \ ^3 (0 I t') — ^"^1 {izT I r)
and so, by §21-61, sn(»„, ^) = ^ i- L ,^ . ^-^^^'
• = — I SC {v, k'),
where v = izT ^3- (0 | t) = izr . (— ir) '^3^ (0 j t) = — m,
so that, finally, sn {iu, k) = i sc (u, k').
• Example 1. Prove that en (w, X-) = nc (?<, k'), dn (m, ^)=dc (?*, J(f) by the aid of Theta-
functions.
Example 2. Shew that
sn i^iK', k) =i sc {\K', k') = tX-~*,
en {hiK\ k) = {\ ■{■kfk-^, dn {hJK', k) = {\+kf-.
[There is great difficulty in determining the signs of sn^zA'', cnii'/t', dnii'A'', if any
method other than Jacobi's transformation is used.]
32—2
500 THE TRANSCENDENTAL FUNCTIONS [CHAP. XXII
• Example 3. Shew that
dnf (A+zA )= j^ .
• Example 4. UO<k<l and if 6 be the modular angle, shew that
sn i (A>iA') = ei"*'"^'V(cosec ^), en ^ (^+iZ') = e"*"*' V(cot 6),
dn I (^+{^0 = 6-**'' V(coa^).
(Glaisher.)
22*42. Landens transformation*.
We shall now obtain the formula
f ' (1 - ytr sin^ 0,)-^de^ = {l+ k') I {1-t sin^ d)'^ dO,
Jo .0
where sin <^i = (1 + A;') sin (/> cos </> (1 — Aj^sin^^)"^
and k = (l-k')/(l + k').
This formula, of which Landen was the discoverer, may be expressed by
means of Jacobian elliptic functions in the form
sn {(1 + k') u, ki] = (1 + k') sn (u, k) cd (u, k),
on writing (f> = am u, ^i = am Ui .
To obtain this result, we make use of the equation of § 21 '52, namely
^3 (z\t)%(z\t) ^ %(z \ t)%(z\t) ^%(0\ t) V(0|t)
"^4(2^|2t) ^7(2^ I 2t) ' ^,(0|2t) ""
Write-f- Ti=2t, and let k^, A, A' be the modulus and quarter-periods
formed with parameter Tj ; then the equation
may obviously be written
k sn {2Kzl'Tr, k) cd {2Kz/'7r, k) = k,^ sn (4AW7r, k,) (A).
To determine k^ in terms of k, put ^ = t tt, and we immediately get
k/(l+k')^kK
which gives, on squaring, k, = (1 - k')/{l + k'), as stated above.
To determine A, divide equation (A) by z, and then make z^O; and
we get
2Kk=4>kM,
so that K=]^{l + k')K.
* Phil. Trans, i.xv. (1775), p. 285.
t It will be supposed that | li (r) ] < J, to avoid difficulties of sign which arise if R (ri) does
aot lie between ±1. This condition is satisfied when 0 < /c < 1, for r is then a pure imaginary.
22-42-22-5] the jacobian elliptic functions 501
Hence, writing u in place of IKzlnr, we at once get from (A)
(1 4- k') sn {u, k) cd {u, A;) = sn {(1 + k') u, k^],
since ^lAzjir = 2Au/K = (1 +k')u;
so that Landen's result has been completely proved.
• Example 1. Shew that A'I\ = 2K'/K, and thence that A' = (l + ^) K'.
• Example 2. Shew that
en {(1 +lf) u, k^} = {1 - (1 4 k') sn2 (u, k)} nd (u, k\
dn {(1 + k') u, ki) = (^' + (1 - k') cn^ {u, k)] nd (m, k).
' Example 3. Shew that
dn{u,k) = i\-y)cn{{\+k')u,ki} + (l+k')din{{\-\-k')u,ki],
where k = '2k^^l{l-{-ki).
22 •421. Transform/itions of elliptic functions.
The formula of Landen is a particular case of what is known as a transformation
of elliptic functions ; a transformation consists in the expression of elliptic functions with
parameter r in terms of those with parameter (a 4- 6t-)/(c + <ir) where a, b, c, d are integers.
We have had another transformation in which a= — 1, 6=0, c = 0, d—\, namely Jacobi's
imaginary transformation. For the general theory of transformations, which is out-
side the range of this book, the reader is referred to Jacobi, Fundamenta Nova, to Klein,
Vorlesungen uber die Theorie der Modulfunktionen (edited by Fricke), and to Cayley,
Elliptic Functions.
* Example. By considering the transformation T2 = r±l, shew, by the method of
§ 22-42, that
sn Qc'u, k^ = kf sd (m, k),
where /^2= ±ikjk', and the upper or lower sign is taken according as R {t) <0 or R {r)>0 ;
and obtain formulae for en {k'^l,, ^2) and dn {k'u, k2).
22"5. Infinite products for the Jacobian elliptic functions*.
The products for the Theta-functions, obtained in § 21%3, at once yield
products for the Jacobian elliptic functions ; writing ti, = ^Kxlir, we obviously
have, from § 22-11, formulae (A), (B) and (C),
^ 4. , _ i . -^ f 1 - 2o2« cos ^x 4- o"'* ]
^^^ = 2^*^^"*^^^\^ai-2r-cos2^4-r-r
en u = 2qik'ik~^cosxh |x ^^^^^2^+ g^^n-j '
4»1— 2
, ,1 ^ (14- 2o'2'»-i cos 2x 4- q
M=i (1 — ^ff cos Ix 4- q
From these results the products for the nine reciprocals and quotients can
be written down.
There are twenty-four other formulae which may be obtained in the following manner :
From the duplication-formulae (^ 22 '21 example 5) we have
1 — en ?< 1 ,1 1 4- dn ?i ,1 1 dn u + en ?« 1,1
= sn -21 dc - 11, = as - w nc - u, = en - « ds - u.
snw 2 2 ' sn?/. 2 2 ' su?t 2 2
* Fiiiidaiiienta Nova, pp. 84-115.
502 THE TRANSCENDENTAL FUNCTIONS [CHAP. XXII
Take the first of these, and use the products for an^u, cn^u, dn ^m ; we get
1 — cn« 1 — cos^ "^ Q — 2 ( -2')"cos^' + j2)i.
snu sin .17 ^"1 U + 2 ( — 3')"cos^+5'2"' '
on combining the various products.
Write u + K for u, x+^rr for x, and we have
dn M + ^'sn M_ ] +sin;?7 °" (l + 2{ — q)^sinx + q^"-''i
en u cos X n=i U "~ 2 ( — 3')" si
Writing u + iK' for u in these formulae we have
"sin .r + j-"'!
" sin X + g'2"J "
km\u + idin.u = i H
1 + 2t ( - f (f"-^ sin g; - g'^^ " ^
„=i ll-2i(-r^''"*sin.r-j-'*-V'
and the expression for kcdxi + ik' i\du is obtained by writing cos.^; for sin.r in this
product.
From the identities {\ — cnu){\+cnu)=.^n^u, {ksnu + iAi\u){ksnu-iA\i'u)=.\^ etc.,
we at once get four other formulae, making eight in all ; the other sixteen follow in the
same way from the expressions for ds\7ir\c\ii and cn^zids^M. The reader may obtain
these as an example, noting specially the following :
. . » r(l_o4n-3e2ta;)(X-o4»-le-2«\1
< Example 1. Shew that
»=i i(l-^V2'^-^)(l + ^^2'^-^))
*■ Example 2. Deduce from example 1 and from § 22-41 example 4, that, if 6 be the
modular angle, then
„=o ii + (_)'^-y«
and thence, by taking logarithms, obtain Jacobi's result
I ^ = 2 ( - )'* arc tan j'* "^ ^ = arc tan >^q — arc tan ^q^ + arc tan ^Jq^ — •••,
Jl=0
'quae inter formulas elegantissimas censeri debet.' {Fund. Nova, p. 108.)
Example 3. By expanding each term in the equation
logsn ?( = log(2g'*)-ilog^ + logsin.r+ 2 {log ( 1 - ^'^^ e^":)
n=l
+ log (1 - j2,ig-2.x) _ log (1 _ ^2,1-1 ^iix^ _ log (1 _ qin - lg-2ia;)|
in powers of e*" '^, and rearranging the resultant double series, shew that
1 1 , -. 4\ 1 1 7 1 ■ '^' 2(7'" cos 2mx
logsn« = log(2j^)-ilog^'+logsmA'+ 2 ^tj-,— ,. ,
»t=i ni\i.-^q )
when ! 7(i) I < \TrI{T).
Obtain similar series for log en ?j, logdnw.
(Jacobi, Ftmdamenta Nova, p. 99.)
■ Example 4. Deduce from example 3 that
fK
log sn u du = —^nK' - ^K log k.
(Glaisher, Proc. Royal Soc. xxix.)
2 26] THE JACOBIAN ELLIPTIC FUNCTIONS 503
226. Fourier series for the Jacohian elliptic functions*.
If M = ^Kxjir, sn u is an odd periodic function of x (with period 27r), which
obviously satisfies Dirichlet's conditions (§9*22) for real values of x; and
therefore (§ 9'31) we may expand sn w as a Fourier sine-series in sines of
multiples of x, thus
00
sn w = S hn sin nx,
n=l
the expansion being valid for all real values of x. It is easily seen that the
coefficients hn are given by the formula
irihn = \ sn w . exp (iiix) dx.
J —77
To evaluate this integral, consider jsnu . exp (nix) dx taken round the
parallelogram whose corners are — tt, tt, ttt, — 27r + ttt.
From the periodic properties of sn u and exp (nix), we see that I cancels
J n
; and so, since — tt + - ttt and ^ tt are the only poles of the integrand
/
— 2jr+TT
{qua function oi x) inside the contour, with residues f
— k~^ i-ir/Kj exp f — mV + ^ nTrir j
and k~^ I ^ tt/Kj exp ( - nTrir)
respectively, we have
- )■ snu.exx) (nix) dx^-yyjq^^ {I - (-)'"'].
Writing a; — tt + ttt for x in the second integral, we get
|1 + (_)n ^n| r gn u . exp (nix) dx = '^ g^" {1 - (-)«}.
Hence, when n is even, 6„ = 0 ; but when n is odd
2. q^
^" Kkl-q^^'
Consequently
27r iq^ sin « o^ sin 2,x q^ sin 5a; 1
Kk ( 1 — g 1 - (/■* 1- q^ )
when a; is real ; but the right-hand side of this equation is analytic when
5^^" exp (ma?) and q^^ eyi^ (— nix) both tend to zero as ti-^x, and the left-
hand side is analytic except at the poles of sn u.
* These results are substantially due to Jacobi, Werke, i. p. 157.
t The factor ^ttIK has to be inserted because we are dealing with sn {2Kxlv).
504 THE TRANSCENDENTAL FUNCTIONS [CHAP. XXII
Hence both sides are analytic in the strip (in the plane of the complex
variable x) which is defined by the inequality | / (a-) | < - wl (t).
And so, by the theory of analytic continuation, we have the result
sn w =
27r * g" + *sin(2/i + l)a;
Kk nZ, 1 - ?^+^
(where u = IKxIir), valid throughout the strip j / (a;) | < ^ tt/ (t).
Example 1. Shew that, if u = '2KxJit, then
/■« - - °° 2tf" sin 2n^
these results being valid when \I {x)\<\nl {t).
Example 2. By writing x + \n for x in results already obtained, shew that, if
u-=2Kxl7r and \I{x)\<^7rI(r),
27r "^ (-)'*o"+*cos(2« + l)J-; , 27r ^ (-)''o''+*sin(2?i+l)^
then cd.= ^.-^J^L^-L__ , sdu=^^,^l^y \^^r- ' ,
IT 27r ■* ( - )" o" cos 2/iA'
"^^=2AT + ZF„!, iT?^^— •
22*61. Fourier series for reciprocals of Jacobian elliptic functions.
In the result of § 22"6, write u + iK' for w and consequently x + ^ttt for «;
then we see that, if 0 > / (,r) > — irl (t),
, .„,, 27r ^ a'''^*sin(27i+ l)(a; + i7rT)
and so (§ 22-34)
74 = 0
=: (- z'tt/A") 2 {2if"+' sin (2?i + 1) ^- + (1 - g-^^-^) e-<-'"+^' ^^]/(l - f/"+i)
That is to say
TT 27r ^ 0-"-+^ sin (2n + 1) X
2A K ,,% 1 - r/»+i
But, apart from isolated poles at the points x = nir, each side of this
equation is an analytic function of x in the strip in which
ttI (t) > I (x) > - TT I (t) :
— a strip double the width of that in which the equation has been proved to
be true ; and so, by the theory of analytic continuation, this expansion for
ns u is valid throughout the wider strip, except at the points x = wtt.
22-61, 22-7] THE JACOBIAN ELLIPTIC FUNCTIONS 505
Example. Obtain the following expansions, valid throughout the strip | I{x) \ < rrI{T)
except at the poles of the first term on the right-hand sides of the respective expansions :
J TT 27r * o2» + isin(2« + l)^
ds«= -^ cosec^- ^ ^J/— T+^2r.-r--.
TT , 27r * g^''iim2nx
,77 27r » (-)" o2" + 1 COS (2n+ 1)07
dcu= ^ secx + ^ „!/- \_^Ji ^.
IT 27r * (-)"o2» + icos(2n-f-l)a;'
ncM=--^^secA' -7^772 ^ — ^ 1 .- 9n+i >
IT ^ 27r « ( - )" o^™ sin 2n^
sc w= — ^i^ tan ^ + -ftt; 2 ^^ — —,— ,v •
22*7. Elliptic integrals.
An integral of the form IR(w, x) dx, where R denotes a rational function
of w and x, and w"^ is a quartic, or cubic function of x (without repeated
factors), is called an elliptic integral*.
[Note. Elliptic integrals are of considerable historical importance, owing to the fact
that a very large number of important properties of such integrals were discovered by
Euler and Legendre before it was realised that the inverses of certain standard types of
such integrals, rather than the integrals themselves, should be regarded as fundamental
functions of analysis.
The first mathematician to deal with elliptic functions as opjiosed to elliptic integrals
was Gauss (see § 22-8), but the first results published were by Abelt and Jacobi J.
The results obtained by Abel were brought to the notice of Legendre by Jacobi
immediately after the publication by Legendre of the Traite des fonctions elliptiques. In
the supplement (tome in. p. 1), Legendre comments on their discoveries in the following
terms : "A peine mon ouvrage avait-il vu le jour, a peine son titre pouvait-il gtre connu
des savans etrangers, que j'appris, avec autant d'etonneinent que de satisfaction, que deux
jeunes geomfetres, MM. Jacobi (C.-G.-J.) de Koenigsberg et Ahel de Christiania, avaient
r^ussi, par leurs travaux particuliers, k perfectionner considerablement la theorie des
fonctions elli^Jtiques dans ses points les plus elev^s."
An interesting correspondence between Legendre and Jacobi was printed in Grelle, Lxxx.
(1875), pp. 205-279 ; in one of the letters Legendre refers to the claim of Gauss to have
made in 1809 many of the discoveries published by Jacobi and Abel. The validity of
this claim was established by Sobering (see Gauss, Werke, in. pp. 493, 494), though the
researches of Gauss ( Werke, in. pp. 404-460) remained unpublished until after his death.]
We shall now give a brief outline of the important theorem that every
elliptic integral can be evaluated by the aid of Theta-functions, combined
* Strictly speaking, it is ouly called an elliptic integral when it cannot be integrated by
means of the elementary functions, and consequently involves one of the three kinds of elliptic
integrals introduced in § 22-72.
t Crelle, n. pp. 101-19G.
:J: Jacobi announced his discovery in two letters (dated June 13, 1827 and August 2, 1827)
to Schumacher, who published extracts from them in Astronomische Nach. vi. (No. 123) in
September 1827— the month in which Abel's memoir appeared.
506 THE TRANSCENDENTAL FUNCTIONS [CHAP. XXII
with the elementary functions of analysis ; it has already been seen (§ 20*6)
that this process can be carried out in the special case of jw~^dcc, since
the Weierstrassian elliptic functions can easily be expressed in terms of
Theta-functions and their derivates (§ 21 '73).
[The most important case practically is that in which ^ is a real function of x and w,
which are themselves real on the path of integration ; it will be shewn how, in such
circumstances, the integral may be expressed in a real form.]
Since R (w, x) is a rational function of w and x we may write
R (w, x) = F (w, x)IQ {w, x),
where P and Q denote polynomials in w and x ; then we have
w(4 {W, X) (j/ (— W, X)
Now Q (w, x) Q (— w, x) is a rational function of w^ and x, since it is
unaffected by changing the sign of iv ; it is therefore expressible as a
rational function of x.
If now we multiply out wP (tu, x) Q (— w, x) and substitute for w^ in terms
of X wherever it occurs in the expression, we ultimately reduce it to a poly-
nomial in X and w, the pol3momial being linear in w. We thus have an
identity of the form
R {w, x) = {Ri (x) + wR^ (x)]/w,
by reason of the expression for lu^ as a quartic in x ; where R^ and R^ denote
rational functions of x.
Now lR2{x)dx can be evaluated by means of elementary functions onl}^* ;
so the problem is reduced to that of evaluating I w~^ R^ (x) dx. To carry out
this process it is necessary to obtain a canonical expression for w^, which we
now proceed to do.
22'71. 27te expression of a quartic as the product of sums of squares.
It will now be shewn that any quartic (or cubicf) in x (with no repeated
factors) can he expressed in the form
[A,(x- ay + B, (x - (3f} [A, (x - af + B, (x - /3)^|,
where, if the coefficients in the quartic are real, A-^, B^, A^, B.2, oc, ^ are all
real.
* The integration of rational functions of one variable is discussed in text-books on Integral
CalcuUis.
t In the following analysis, a cubic may be regarded as a quartic in which the coefficient of
.r'' vanishes.
2271, 2272] the jacobian elliptic functions 507
To obtain this result, we observe that any quartic can be expressed in
the form SiS^ where ^i, S<i are quadratic in x, say*
/Si = OriX^ + 2biOC + Ci, 82= a.2*"^ + '^h^ + ^2-
Now, \ being a constant, S^ — XS^ will be a perfect square in x if
(tti - XOa) (Ci - XC2) - (bi - Xh-if = 0.
Let the roots of this equation be Xj, X^; then, by hypothesis, numbers
a, y9 exist such that
Si — XiSi = (o-i — Xi tta) (x — ay, Si — Xg'S'a = («! — ^2 a^) (x — I3f ;
on solving these as equations in S-^, S.2, we obviously get results of the form
Si = Ai(x-ay + B,{x-0r, S,~ A,(x-ay+B,(x-^y,
and the required reduction of the quartic has been effected.
[Note. If the quartic is real and has two or four complex factors, let Si have com-
plex factors ; then X^ and Xo are real and distinct since
(«! - Xa2) (ci - XC2) - (5i - Xb^y
is positive when X = 0 and negative t when X — aija^.
When S\ and S2 have real factors, say {x — ^\) {x - ^{), (^"i? — 12) (■^ - |2')j the condition
that Xi and X2 should be real is easily found to be
(^l-^2)(^l'-^2)(6-|2')(^l'-6')>0,
a condition which is satisfied when the zeros of ^1 and those of *S^2 do not interlace ; this
was, of course, the reason for choosing the factors *S^i and S^ of the quartic in such a way
that their zeros do not interlace.]
2212. The three kinds of elliptic integrals.
Let a, yS be determined by the rule just obtained in § 22'7l, and, in the
integral \w~^Ri (x)dx, take a new variable t defined by the equation J
t = (x — a)/{x — y8) ;
,, , dx (a—0)~'^dt
we then have — = +
w [(Aif+Bi)(A,t' + B,)]i'
* If the coefficients in the quartic are real, the factorisation can be carried out so that the
coefficients in Si and S2 are real. In the special case of the quartic having four real linear
factors, these factors should be associated in pairs (to give Si and S^) in such a way that the
roots of one pair do not interlace the roots of the other pair ; the reason for this will be seen in
the note at the end of the section.
t Unless ai : a2=bi : 1)2, in which case
Si = ai {x-a)^ + Bi, S2 = a2(x- a)'^ + B2.
J It is rather remarkable that Jacobi did not realise the existence of this homograpliic
substitution ; in his reduction he employed a quadratic substitution, equivalent to the result of
applying a Landen transformation to the elliptic functions which we shall introduce.
508 THE TRANSCENDENTAL FUNCTIONS [CHAP. XXII
If we write R^ (x) in the form + (a — /3) R3 (t), where R3 is rational, we get
rR,(x)dx_ r R3(t)dt
. •'' w ~ J {{A,t' + B,)iA,f~ + B,p'
Now R, (t) + i^3 (- 0 = 2 R, (t% R3{t)-R3{-t) = 2tR, (t%
where JK4 and Rg are rational functions of t^, and so
R3{t)=R,(t') + tR,(t').
But l{(A^t' + B,)(A,t' + B,)}-^tR,{t'')dt
can be evaluated in terms of elementary functions by taking t^ as a new
variable*; so that, if we put R^it^) into partial fractions, the problem of
integrating \R(w, x)dx has been reduced to the integration of integrals of
the following types
{f^ [{A,t' + B,) (A,t' + B,)] - * dt,
/^
in the former of these m is an integer, in the latter m is a positive integer
and i\^^0.
By differentiating expressions of the form
it is easy to obtain reduction formulae by means of which the above
integrals can be expressed in terms of one of the three canonical forms :
(i) [K^i^2 + 5,)(^.f^ + 5,)}-*c;^,
(ii) \t''[{A,t'' + B,)(^AJ?^B.X^^^^
(iii) |(1 + Ni?)-^ K^it^ + B,) {A,t' + B,)] -^t.
These integrals were called by Legendref elliptic integrals of the first,
second and third kinds, respectively.
The elliptic integral of the first kind presents no difficulty, as it can be
integrated at once by a substitution based on the integral formulae of
§§ 22121, 22-122; thus, if A„ B„ A^, B^ are all positive and A^B,>A,B^,
we write
A,h = B,^ cs (u, k). [k'-' = (A,B,)/(A,B,).]
* See, e.g.. Hardy, Integration of Functions of a ungle Variable (Camb. Math. Tracts, No. 2).
t Exercices de Calcul Integral, i. p. 19.
22-72]
THE JACOBIAN ELLIPTIC FUNCTIONS
509
'Example 1, Verify that, in the case of real integrals, the following scheme gives
all possible essentially different arrangements of sign, and determine the appropriate
substitutions necessary to evaluate the corresponding integrals.
^
+
+
+
+
-
Bi
+
+
-
-
+
^2
+
+
+
+
+
A
+
+
+
-
+
Example 2. Shew that
'sn udu =
1—kcdu
\snudu= zrr log ^ — '-, , ,
J 2k ^1+kcd^i'
I en udu = k 1
arc tan {k sd u),
dnudu = iixn u,
I ds udu
f , 1 , dnu + kf
jscudu=-^,log^^^^^,,
1 , 1 - en M
2 ''i + cn%'
ldcudu=- log —
1 + snw
anu'
and obtain six similar formulae by writing u + K for u.
(Glaisher.)
t Example 3. Prove, by differentiation, the equivalence of the following twelve
expressions :
k"^u-\-k'^ lev? udu,
u — dnucsu — \ ns^ udu,
P sn ucdu-\- k'^ Jnd^ udu,
k'^ u ■{- k^ s>n u ad u + k^kf^\s,d^ udu,
— dnwcs u — \c&^udu,
M -f dn M sc ?( - Jdc^ ttc?M.
u—k^la\\^itdu,
Idn^udu,
k"^u + dnuficu — ^'- jnc^ udu,
duuacu — k"^ Jsc^ u du,
u+k'^smi cd u — k^ I cd^ u du,
k'^u — dnucau — Jds^ u du.
Example 4. Shew that
d^ sn" ^(,
du^
—n{n — \) sn"~2 u — ii^ {\-\-k'^) sn"M + /i (n + 1) ^^sn""''^ ^^^
and obtain eleven similar formulae for the second differential coefficients of en" u,
dn" u, ... nd" u. What is the connexion between these formulae and the reduction
formula for lt'<'{{Ait- + B{) {A2t^ + D^)}'^ dt^i
(Jacobi ; and Glaisher, Messenger, xi.)
> Exam,ple 5. By means of § 20"6 shew that, if a and ^ are positive,
{\j,{d'-x'^){x^+^^)}-Ux=r{As^-g2S-g^)~Us,
where ej is the real root of the cubic and
and prove that, if 5*2 = 0, then a and /3 are given by the equations
a2-^2= - 3 (2^3)*, a2 + i32 = 2 V3 • | 25-3 |*.
510 THE TRANSCENDENTAL FUNCTIONS [CHAP. XXII
Example 6. Deduce from example 5, combined with the integral formula for en u,
that, if ^3 is positive,
f {A^-g^)'^ds = ^{a^->r0^)-^K, f (A^+g^y^ ds = 2 {a^+^^y^ K',
J ^1 J —ei
where a- = (v/3-f) (2^3)*, ^2=,(^3 + |)(2^3)*, and the modulus is a{a^+^'^y^.
22"73. The elliptic integral of the second kind. The function* E (u).
To reduce an integral of the type
ft^ {(A,t' + B,) (A.t' + B,)} -^dt,
>
we employ the same elliptic function substitution as in the case of that
elliptic integral of the first kind which has the same expression under the
radical. We are thus led to one of the twelve integrals
1 sn^wdw, I cn^ udu, ... 1 nd^ udu.
By § 22'72 example 3, these are all expressible in terms of u, elliptic
functions of u and jdn-udu; it is convenient to regard
ru
E (u) = / dn^ 11 du
Jo
as the fundamental elliptic integral of the second kind, in terms of which all
others can be expressed ; when the modulus has to be emphasized, we write
E (u, k) in place of E{u).
We observe that
dE(u) , ., „,-, ^
— ^-^ = dn- u, E(0) = 0.
du
Further, since dn- u is an even function with double poles at the points
ImK + (2n + 1) iK, the residue at each pole being zero, it is easy to see that
E (ii) is an odd one-valued f function of u with simple poles at the poles
of dnu.
It will now be shewn that E {u) may be expressed in terms of Theta-
functions ; the most convenient type to employ is the function 0 {u).
Consider -^ 1r^/ tI"
du (B(u)j
du \^{u)] '
it is a doubly-periodic function of n with double poles at the zeros of 0 {u),
i.e. at the poles of dn u, and so, if J. be a suitably chosen constant,
, . , d m'{u))
an- u — A -^ \ -rrr\ \
du [0(w)j
* This notation was introduced by Jacobi, Crelle, iv. p. 373. In the Fiindavienta Nova, he
writes E (am 11) where we write E (u).
t Since the residues of dn^ u are zero, the integral defining E (u) is independent of the path
chosen (§ 6-1).
22-73-22-732] the jacobian elliptic functions 511
is a doubly-periodic function of u, with periods 2K, 2iK', with only a single
simple pole in any cell. It is therefore a constant ; this constant is usually
written in the form E/K. To determine the constant A, we observe that
the principal part of dn^u at iK' is —{u — iK')~% by § 22-341; and the
residue of 0' (u,)/^ (u) at this pole is unity, so the principal part of
^ \^rl\ is - (« - iK'r\ Hence A = l, so
du [©(w)j K
Integrating and observing that ©' (0) = 0, we get
E{u) = 0' {u)l% (u) + uE/K.
Since @'{K) = 0, we have E{K) = E\ hence
E = \^ dn^udu = [^"(l - k^ sin^ <^)^ d(^ = \'Tr F {-\,\; 1; k^Y
It is usual (cf § 22'3) to call K and E the complete elliptic integrals of the
first and second kinds. Tables of them qua functions of the modular angle
are given by Legendre, Fonctions Elliptiques, t. ii.
• Example 1. Shew that E{u + 2nK) = £{u) + 2nE, where n is any integer.
• Example 2. By expressing 6 (u) in terras of the function .^4 {^ttu/K), and expanding
about the point ^c=^iK', shew that
E=i{2-k^—srKh'^i')}£:-
22-731. The Zeta-function Z {u).
The function E (u) is not periodic in either 2K or in 2iK', but, associated
with these periods, it has additive constants 2E, [2iK'E — 7ri\/K ; it is
convenient to have a function of the same general type as E{u) which is
singly-periodic, and such a function is
Z (u) = 0' (u)/@ (u);
from this definition, we have*
Z (w) = E (a) - uE/K, 0 (u) = 0 (0) exp j T Z (t) dti .
22-732. The addition-formulae for E (u) and Z (u).
Consider the expression
--— f — ^ ; - — ^ ,-r + k- snu snv sn(u + v)
(»){u + v) 0(u) 0(v) V -r y
* The integral in the expression for 0 («) is not one-valued as Z (t) has residue 1 at its poles;
but the difference of the integrals taken along any two paths with the same end points is 2«7ri
where n is the number of poles enclosed, and the exponential of the integral is therefore one-
valued, as it should be, since 9 (u) is one-valued.
512 THE TRANSCENDENTAL FUNCTIONS [CHAP. XXII
qua function oi u. It is doubly-periodic* (periods 2K and liK') with simple
poles congruent to iK' and to iK' — v ; the residue of the first two terms at
iK' is — 1, and the residue of sn w sn ?; sn {u + ?;) is k~^ sn v sn {iK' -\- v) = k~\
Hence the function is doubly-periodic and has no poles at points
congi'uent to iK' or (similarly) at points congruent to iK' — v. By
Liouville's theorem, it is therefore a constant, and, putting u = 0, we see
that the constant is zero.
Hence we have the addition-formulae
Z (u) + Z (v) — Z (u + v) = k^ sn u sn v sn (u + v),
E (u) + E (v) — E (u + v) = k^ snu sn V sn (u + v).
[Note. Since Z (u) and £ («) are not doubly-periodic, it is possible to prove that no
algebraic relation can exist connecting them with sn w, en m and dn «, so these are not
addition-theorems in the strict senset.]
22"733. Jacobi's imaginary transformationl of Z (u).
From § 21 '51 it is fairly evident that there must be a transformation of
Jacobi's type for the function Z (u). To obtain it, we translate the formula
^2 (i^ I t) = (— ^"''")^ exp (— irx^jir) . ^4 {ixr \ t')
into Jacobi's earlier notation, when it becomes
H (iu + K, k) = (- It)^ exp ( t-^^t?-, ) B (u, k'),
TTU
.iKK'
and hence
cn i^u, k) = (- ^rf exp (^^, j ^^J^ ^ ^^-^ .
Taking the logarithmic differential of each side, we get, on making use of
§ 22-4,
Z (iu, k) = i dn (u, k') sc {u, k') — ^Z {u, k') — ■Triul{2KK').
22"734. Jacobi's imaginary transformation of E (u).
It is convenient to obtain the transformation of ^(w) directly from the
integral definition ; we have
E{iu, k)= jdn^{t, k)dt= \ dn^{it', k)idt'
Jo J a
= i[ dc^ (f, k') dt',
J 0
on writing t = it' and making use of § 22*4.
* 2/A" is a period since the additive constauts for the first two terms cancel.
+ A theorem due to Weierstrass states that an analytic function,/ (2;), possessing an addition-
theorem iu the strict i^ense must be either
(i) an algebraic function of z,
or (ii) an algebraic function of exp (irizju),
or (iii) an algebraic function of ^ (2 i co] , ^2);
where w, wj, wq are suitably chosen constants. See Forsyth, Tlieori/ of Functions, Chap. xiii.
X Fnndainenta Nova, p. IGl.
22-733-22-735] the jacobian elliptic functions 513
Hence, from § 2272 example 3, we have
E (lu, k) = i \u + dn {a, k') sc {u, k') - I "dn^ (tf, k') dt\ ,
and so E{iu, k) = iu + i dn (w, k') sc (a, k') — iE (u, k').
This is the transformation stated.
It is convenient to write E' to denote the same function of k' as E is of k,
i.e.E'=E(K',k'),sothat
E(2iK',k) = 2i(K'-E').
22'735. Legendre's relation*.
From the transformations of E{u) and Z{u) just obtained, it is possible
to derive a remarkable relation connecting the two kinds of complete elliptic
integrals, namely
EK' + E'K-KK'^lir.
For we have, by the transformations of §§ 22*733, 22*734,
E (iu, k)-Z (iu, k) = iu - i [E {u, k') - Z {u, k')] + triuJi^KK'),
and on making use of the connexion between the functions E{u, k) and
Z (w, k), this gives
iuEIK = iu - i [uE'jK'] + 7riu/{2KK').
Since we may take u=^0, the result stated follows at once from this
equation; it is the analogue of the relation ^Wg — 772<«^i = 0'""* which arose in
the Weierstrassian theory (§ 20*41 1).
Example 1. Shew that
E{u + K) — E{u) = E—k^snucdu.
Example 2. Shew that •
E (2m + 2iK') = E (2«) + 2t {K' - E').
Example 3. Deduce from example 2 that
E{u + iK') = \E{'lu-Y2iK')->r\k'^iin'{u + iK')sn (2u + 2iK')
= E(u) + cnudsu + i{K'-E').
Example 4. Shew that
E{u + K+iK')=E{u)-smidcu + E+i{K'-E').
Example 5. Obtain the expansions, valid when \I {x)\ <\itI{t),
ikAy.n^-u = K-^-KE-2.'^ I n,»cos|^.^ A'Z(.) = 2. I r^^ ,
(Jacobi.)
* Exercices de Calcul Integral, i. (1811).
W. M. A. 33
514 THE TRANSCENDENTAL FUNCTIONS [CHAP. XXII
22*736. Properties of the complete elliptic integrals, regarded as functions
of the modulus.
If, in the formulae E= I {l—k^siri^<p)^d<^, we differentiate under the
sign of integration (§ 4'2), we have
^1" = - p^'^-sin^ (/, (1 - yt^sin^ cfi)-^d(f>=^ ^1^ '
Treating the formula for K in the same manner, we have
dK fh^ 3 C^
= 1 k sin^ (f)(l-k' sm'' <l>)-id(f> = k sd^ udu
1 ( rK
dk JO jo
,,,,,, dn^wrfw
kk^ (Jo
lo
by § 2272 example 3 ; so that
dk ~ kk'"" k '
If we write k"^ = c, k''^ — c' , these results assume the forms
^dE ^E-K ^dK^E-Kc'
do c ' dc cc'
Example 1. Shew that
^dE^_ K'-E' ^dK^_ cK^- E'
dc c' ' dc cc'
Example 2. Shew, by diflferentiation with regard to c, that EK' + E'K- KK' is
constant.
Example 3. Shew that K and K' are sohitions of
dkX^" dk) "'''
and that E and E' - K' are solutions of
^"^•0'S)+^'^* = ^- (Legendre.)
22"737. The values of the complete elliptic integrals for small values of k.
From the integral definitions of E and K it is easy to see, by expanding
in powers of k, that
lim K=\imE=l'rr, lim (K - E)/k' = J tt.
k^O k^O k^O
In like manner, lim E' = cos (f)d(}> = 1.
A;-»0 Jo
It is not possible to determine lim K' in the same way because
(1 — F- sin"-^ </))"- is discontinuous at (f> = 0, k = 0; but it follows from
example 21 of Chapter xiv (p. 293) that, when |argA;j<7r,
lim {K' - log (i/k)] - 0.
k-fO
22-736-22*74] the jacobian elliptic functions 515
This result is also deducible from the formulae 2iK' = 7rTS3^ ,k=B2^/^3% by making
y -* 0 ; or it may be proved for real values of k by the following elementary method :
By §22-32, K'= j (t^-k^)'^ {l-fi)~^dt; now, when k<t<sjk, {\-t^) lies between
1 and \-k; and, when Jk<t<l, {t^-k^)/t^ lies between 1 and 1—^. Therefore K' lies
between
J k J ^k
and {l-k)-^U {t^-k^)-^dt+j t-^ (l-t^)-^dt\ ;
and therefore
^k
= (l-^/:)-i[21og{l + V(l--^)}-log^],
where O^^^l.
Now lim [2 (1 - <9yfc)~*log {I +v/(l -^)} -log 4] = 0,
lim{l-(l-^/(:)-*}log/5;=0,
and therefore lim {K' — log (4/^)} = 0,
which is the required result.
Example. Deduce Legendre's relation from § 22 '736 example 2, by making ^-*-0.
22'74. The elliptic integral of the third kind*.
To evaluate an integral of the type
f(l + Nt')-' {(A,t' + B,) {A,t^ + B^)] - ^ dt
/^
/^
in terms of known functions, we make the substitution made in the corre-
sponding integrals of the first and second kinds (§§ 22*72, 22"73). The
integral is thereby reduced to
-^ du = CM + (/3 - ai^) -— -— du,
where a, ^, v are constants ; if i/ = 0, — 1, oo or — A;- the integral can be
expressed in terms of integrals of the first and second kinds ; for other values
of 1/ we determine the parameter a by the equation v = — k^ sn^ a, and then
it is evidently permissible to take as the fundamental integral of the third
kind
„ , , P A;2 sn a en a dn a sn^ M ,
n (u, a) = — ^ yr, — :, du.
^ ' J Q 1 — «-^ sn^ a sn^ w
' To express this in terms of Theta-functions, we observe that the inte-
grand may be written in the form
2 k^ sn u sn a {sn (u + a) 4- sn {u — a)} =;^{Z{u — a) — Z {ii + a) + 2Z (a)\,
•* Legeudre, Exercices de Calcul Integral, i. p. 17; Fonctions Elliptiques, i. pp. 1-4-18, 74, 7o;
Jacobi, Fundamenta Nova, pp. 137-172 ; we employ Jacobi's notation, not Legendre's.
33—2
516 THE TRANSCENDENTAL FUNCTIONS [CHAP. XXII
by the addition-theorem for the Zeta-function ; making use of the formula
Z (u) = &' {u)/@ (u), we at once get
TT / X 1 1 ^ (U — a) ry / s
a result which shews that 11 (u, a) is a many-valued function of u with
logarithmic singularities at the zeros of 0 (w + a).
Example 1. Obtain the addition-formula*
n(w, a) + n(v, a)-n(u-\-v, a) = ilog_ ' ^,—~) — , ( ^ ; , — '
, , l-PsnasnusnvHn(u + v- a)
' • ss=ilosr — ^^ ..
^ ° l + k'^ai\asnu8nvsn{u + v + a)
(Legendre.)
(Take x -.1/ iz :iv = u :v : ±a -.u + v ±a in Jacobi's fundamental formula
[4] + [l] = [4]' + [l]'.)
Example 2. Shew that
n {u, a) — n (a, u) = uZ (a) —aZ (u).
(Legendre and Jacobi.)
[This is known as the formula for interchange of argument and parameter.]
Example 3. Shew that
n(u, a) + n(u,b)-n(u,a + b) = h log , — jr, t ) — -i i
^ ' ^ ^ ' ^ \ > -T^ / 2 ^ i + kiauaanbmusQ{a + b+u)
+ liJc^ sn a sn 6 sn (a + b).
(Jacobi.)
[This is known as the formula for addition of parameters.]
Example 4. Shew that
n {iu, ia + A", ^) = n («, a -H A'', k'). (Jacobi.)
Example 5. Shew that
n(M + i?, a + 6) + n(M-i;, a-6)-2n(M, a)-2n(^^, b)
70 If/ \ , -IS , N , 7^. 1 1 \ — k~8v?{u-a)fiV?{v-b)
= - «- sn a sn 6 . ■ (?« -fr?) sn (a-f- 6) - (« - v) sn (a - b)\ + i log , — ^s — 5-7 { — 5-7 r\ >
and obtain sijecial forms of this result by putting v orb equal to zero. (Jacobi.)
22'741. A dynamical application of the elliptic integral of the third kind.
It is evident from the expression for n (w, a) in terms of Theta-functions that if «, a, k
are real, the average rate of increase of 11 (li, a) as u increases is Z (a), since 6 (M±a) is
periodic with respect to the real period 2A'.
This result determines the mean precession about the invariable line in the motion of
a rigid body relative to its centre of gravity under forces whose resultant passes through
its centre of gravity. It is evident that, for purposes of computation, a resvilt of this nature
is preferable to the corresponding result in terms of Sigma-functions and Weierstrassian
Zeta-functions, for the reasons that the Theta-functions have a specially simple behaviour
with respect to their real jjeriod — the period which is of importance in Applied Mathe-
matics— and that the g'-series are much better adapted for computation than the product
by which the Sigma-function is most simply defined.
* No fewer than 96 forms have been obtained for the expression on the right. See Glaisher,
Messenger, x. p. 124.
22-741, 22-8] the jacobian elliptic functions 517
22*8. The lemniscate functions.
rx
The integi-al 1 (1 — t*)~^ c^^occurs in the problem of rectifying the arc of
J 0
the lemniscate*; if the integral be denoted by (f), we shall express the
relation between <f) and a; by writing^ w = sin lemn (f).
In like manner, if
J X " ^ 0
we write
a; = cos lemn (f>i,
and we have the relation
sin lemn (f> = cos lemn (2^ ~ 0 )•
These lemniscate functions, which were the first functions J defined by the
inversion of an integral, can easily be expressed in terms of elliptic functions
with modulus l/\/2 ; for, from the formula (§ 22"122 example)
fsAu
u= {( 1 - k'^if) ( 1 + ky^)] ~^dy,
■' 0
it is easy to see (on writing y=^t\JT) that
sin lemn <^ = 2 - ^ sd (<^ v/2, 1 / V2) ;
similarly, cos lemn <f) = cn(<f) \/2, 1 /\/2).
Further, 2^ ^^ ^^.e smallest positive value of <^ for which
cn(<^V2, l/\/2) = 0,
so that w = f^^Ko,
the suffix attached to the complete elliptic integral denoting that it is
formed with the particular modulus l/\/2.
This result renders it possible to express Ko in terms of Gamma-functions,
thus
^0=2* (\l-P)-^dt = 2-^ ru-^(l-u)-idu
Jo Jo
= 2-tr(i)rQ)/r(f) = i7r-i{r(i)h
a result first obtained by Legendre§.
Since k = k' when k = l/\/2, it follows that Kq = Kq, and so q^ = e'".
* The equation of the lemniscate being r'^ — a^co?,2d, it is easy to derive the equation
/d8\2 a4 , , ^ , /ds\2 , frddY
I ^- ) = — i 7 from the formula -;- = 1 + —,—
\drj a*-r* \dr/ \ dr J
t Gauss wrote si and cl for sin lemn and cos lemn {Werke, iii. p. 493).
X Gauss, Werke, iii. p. 404. The idea of investigating the functions occurred to Gauss on
January 8, 1797.
§ Exercices de Calcul Integral, i. p. 209. The value of A'q is 1-85407468..., while
'5r = 2-62205756....
518 THE TRANSCENDENTAL FUNCTIONS [CHAP. XXII
Example 1. Express Kq in terms of Gamma-functions by using Rummer's formula
(see Chapter xiv, example 12, p. 292).
Example 2. By writing ^ = (1 — m^)* in the formula
Eo=f\i-^t^)Hi-fiy^dt,
Jo
shew that 2^ Eo= j (l-it'^)-^du+ j u^{\-u*)~^du,
and deduce that 2^o - A'o = 27r^ {r (J)} - 2.
Example 3. Deduce Legendre's relation {% 22'735) from example 2 combined with
§ 22-736 example 2.
Example 4. Shew that
. 1 „ , 1 — coslemn'^d)
sm lemn^ m = , -. -^ri .
l+coslemn'^9
22*81. The vahies of K and K' for special values of k.
It has been seen that, when ^ = 1/^2, K can be evaluated in terms of Gamma-functions,
and K—K' ; this is a special case of a general theorem* that, whenever
K' _a + h ^n
K ~ c + dsjn^
where a, b, c, d, n are integers, k is a root of an algebraic equation with integral
coefficients.
This theorem is based on the theory of the transformation of elliptic functions and is
beyond the scope of this book ; but there are three distinct cases in which k, K, K' all
have fairly simple values, namely
(J) k=>j2-\, K' = K^2,
(II) k^am^iT, K' = K^Z,
(III) y5:=tan2l7r, K' = 2K.
Of these we shall give a brief investigation f.
(I) The quarter-periods with the modulus ^2 — \.
Landen's transformation gives a relation between elliptic functions with any modulus k
and those with modulus k-^ — {\ — k')l{\ -^-k') ; and the quarter-periods A, A' associated with
the modulus k^ satisfy the relation A'lA=^2K'/K.
If we choose k so that ki = k', then A = K' and k^ =k so that A' = K ; and the relation
A'/ A = 2K'/I{ gives A'2 = 2a2,
Therefore the quarter-periods A, A' associated with the modulus ki given by the
equation ki = {l-ki)l{l+k^) are such that A'=±Av^2; i.e. if ^i = v'2-l, then A' = Av/2
(since A, A' obviously are both positive).
(II) The quarter-periods associated with the modidus sin^oTr.
The case of k = s,\\i^^ir was discussed by LegendreJ ; he obtained the remarkable
result that, with this value of k,
K' = K^Z.
* Abel, Oeuvres, i. p. 377.
t For some similar formulae of a less simple nature, see Kronecker, Berlin. Sitzungsberichte,
1857, 1862.
J Exercices de Calcul Intigraly i. pp. 59, 210 ; Fonctions Elliptiques, i. pp. 59, 60.
22*81] THE JACOBIAN ELLIPTIC FUNCTIONS 619
This result follows from the relation between definite integrals
To obtain this relation, consider j(l-z^)~^dz taken round the contour formed by the
part of the real axis (indented at «=1 by an arc of radius R~'^) joining the points 0 and
R, the line joining Re^^^ to 0 and the arc of radius R joining the points R and /2e*"* ; as
R-*' 00 , the integral round the arc tends to zero, as does the integral round the indentation,
and so, by Cauchy's theorem,
r {l-.i^y^dx + if'^ {.7^-\)-^dx + e^'^j (l+x^)~^dx=0,
Jo J \ .' 00
on writing x and xe^^^ respectively for z on the two straight lines.
Writing
hi-x^)~^dx=ii, r {jfi-\)-^dx=i^, r{i+x^y^dx=r {l-x^y^dx=rs,
we have 7i + i/2=^(l+iV3)/3;
so, equating real and imaginary parts,
and therefore /x + Zs — /2v'3 = ^/3 + /3 — f /3 = 0,
which is the relation stated*.
Now, by § 22-72 example 6,
where the modulus is a{a'^+^y^ and
a2 = 2v/3-3, /32 = 2v'3 + 3,
so that F = i (2 - V3) = sin2 ^ «-.
We therefore have
3-i.2^=3-*.2^' = 72=3*/i
= Z-^\-\\-tyUt=ln'^T{^)lT{l),
J 0
when the modulus k is sin-jij""-
(III) The quarter-periods \oith the modtdus tan^l^jr.
If, in Landen's transformation (§ 22-42), we take k=llJ2, we have A'lA = 2K'IK=2;
now this value of k gives
and the corresponding quarter-periods A, A' are ^(1 +2""*) ^o ^-nd (l + 2~*) A'q.
Example 1. Discuss the quarter- periods when k has the values (2^^2-2)', sin^j^jr,
and 2^(^/2-1).
♦ Another method of obtaining the relation is to express Jj, I2, I3 in terms of Gamma-
functions by writing t^, t~^, (t~i-l)* respectively for x in the integrals by which Ii, I2, I3 are
defined.
520 THE TRANSCENDENTAL FUNCTIONS [CHAP. XXII
Example 2. Shew that
n=0 re=l
(Glaisher, Messenger, v.)
Example 3. Express the coordinates of any point on the curve y'^=x^ -1 in the form
2 . 3* su w dn M
, 3*(l-cnM) 2.3*s
l + cn% ' -^ (1 +
(l + cntt)2 '
where the modulus of the eUiptic functions is sin^Tr, and shew that ^ = 3~*y.
By considering I 3('~io?^=3 *l (iw, evaluate iS" in terms of Gamma-functions when
i{; = sinJ2'r.
Example 4. Shew that, when y2 = ^3 _ j^
and thence, by using example 3 and expressing the last integral in terms of Gamma-
functions by the substitution x—t~^, obtain the formula of Legendre {Calcul Integral,
p. 60) connecting the first and second complete elliptic integrals with modulus sin yV it :
4-53=^(^-^4-
Example 5. By expressing the coordinates of any point on the curve F^ = 1 - X^ in
the form
3-(l-cnv) 2.3^snvdnv
"^~^ \+^n^' (H-cn?;)"2~'
in which the modulus of the elliptic functions is sin -f^ tt, and evaluating
+ ^ Y-^{l-XydX
in terms of Gamma-functions, obtain Legendre's result that*, when ^=sin j^tt,
"■ V3 „, ( J-,, v/3 — 1
'=^'{^'-^73'^'}
22*82. A geometrical illustration of the functions snu, cmt, dnu.
A geometrical representation of Jacobian elliptic functions with k = l/^j2 is aflforded by
the arc of the lemniscate, as has been seen in § 22"8 ; to represent the Jacobian functions
with any modulus k (0<^<1), we may make use of a cwve described on a sphere, known
as Seiffert's spherical spiral-^.
Take a sphere of radius unity with centre at the origin, and let the cylindrical polar
coordinates of any point on it be (p, <^, z), so that the arc of a curve traced on the sphere
is given by the formula J
{dsf = p^{dcl>Y + {l-p^)-'{dpf.
* It is interesting to observe that, when Legendre had proved by differentiation that
EK' + E'K - KK' is constant, he used the results of examples 4 and 5 to determine the constant,
before using the methods of § 22-8 example 3 and of § 22-737,
+ Seiffert, Ueber eine neue geometrische Einfiihnmg in die Theorie der elUptisclien Funktionen
(Charlottenburg, 1896).
X This is an obvious transformation of the formula [ds)"^ = {dp)"^ + p"^ {dcf))'^ + [dz)"^ when
p and z are connected by the relation p'^ + z'^ = l.
22-82] THE JACOBIAN ELLIPTIC FUNCTIONS 521
Seiffert's spiral is defined by the equation
(p = kg,
where s is the arc measured from the pole of the sphere (i.e. the point where the axis of z
meets the sphere) and ^ is a positive constant, less than unity*.
For this curve we have
and so, since s and p vanish together,
p = sn (s, k).
The cylindrical polar coordinates of any point on the ciu-ve expressed in terms of the
arc measured from the pole are therefore
(p, (f), z) — (sn s, ks, cu s) ;
and dn s is easily seen to be the cosine of the angle at which the curve cuts the meridian.
Hence it may be seen that, if K be the arc of the curve from the pole to the equator, then
sn s and en s have period 4K, while dn s has period 2K.
REFERENCES.
A. M. Legendre, Fonctions Elliptiques.
C. G. J. Jacobi, Fundamenta Nova Theoriae Functionum Ellipticarum (Konigsberg,
1829).
J. Tannery et J. Molk, Fonctions Elliptiques.
A. Catley, Elliptic Functions.
P. F. Verhulst, Traite elementaire des fonctions elliptiques (Brussels, 1841).
A. Enneper, Elliptische Funktionen, Zweite Auflage von F. Miiller (Halle, 1890).
Miscellaneous Examples.
1. Shew that one of the values of
J/dnM + cnM\* /dnM-cnM\*"l J/ 1— sn?i \^ / l + snw \*"l
|\ l+cuM / \ 1 — cn?i ) ) \\dnu — k' %uu) ydiwu + k' »nu) j
is 2(1 +/&'). (Math. Trip. 1904.)
2. If .r + iy = sn2(M+zv) and x — iy = iirfi {u — iv), shew that
{(^ - l)2+.^2|i= (^'2 + /)^ dn 22t + en 2m.
(M^th. Trip. 1911.)
3. Shew that
4. Shew that
,, , ,- ,, , ,, (en ti + cn vY
{1 ± en lu + v)} {1 ± en {u -v)]= ~~f^ — 5 -»- .
cn^ ^l + cn2 v
1 + en {u + v) en (;< - r) = — jr, — , ^ .
^ ' ^ \- k' sn- u sn'' v
(Jacobi.)
* If /c > 1, the curve is imaginai-y.
(Jacobi.)
622 THE TRANSCENDENTAL FUNCTIONS [CHAP. XXII
t- T. l + cn (u+v) cn (u — v) « .. ,.0 o
5. Express - — 3 — } .—, — ) ; as a function of sn^^s + sm-y.
^ l+dn{u + v)du{ii-v)
(Math. Trip. 1909.)
6. Shew that
, , , , , sn tt dn M en V — sn V dn v en w
sn (u-v) dn {u + v) = 1 — jr, — 5 5 •
(Jacobi.)
7. Shew that
{l-(l + /fc')sn«sn(w + A'')} {l-{l-k')snusn{u + K)} = {sn{u + K)-anuf.
(Math. Trip. 1914.)
8. Shew that
1 T^^s /, ,,, _i ^'snM + cnwdnM
sn {u+^iK') = k-^ {l+k).nu + icr^dnu^
^ ^ l+«sn^i6
9. Shew .that
. . , , , , , 2 sn ?t en n dn v
sin {am (w + v) -i- am (% - y)} = :, — j^. — 5 5— ,
^ ^ ' ^ ' \ — k'- sir M sn'^ y
,, en^ y — sn^ -i; dn^ It
cos {am (« + ^) -am {n-v)\ = -^-^^--^---^^- .
10. Shew that •
, V , , ^ ds^ wds^ v+F^'2
^"(^ + '^)^"(^^-^^= ns-^^ns^^-/:-^ '
and hence express
rg>(w + ^)-e2 g>(w--?;)-e2'j*
L(p (16 + w) - e, ■ g> (?t - y) - egj
as a rational function of ^ («) and f> (y). (Trinity, 1903.)
11. From the formulae for on (2/f-w) and dn {2K-u) combined with the formulae
for l+cn2i( and l+dn2?(, shew that
(l-cnfA')(l + cln§/ir) = l. (Trinity, 1906.)
12. With notation similar to that of § 22-2, shew that
Ci o?2 -c^di _ en {til + M2) — dn (mj + M2) _
Sj - ^2 sn («i + M2) '
and deduce that, if ?fi + tt2 + % + ^<4 = 2A', then
(Ci ^2 - C2C?l) (C3<^4 - C40?3) = ^'^ («1 " S2) (^S " S4)-
(Trinity, 1906.)
13. Shew that, if u + v + w=0, then
1 - dn^ u - dn^ v — dn^ iv + 2 dn « dn «; dn w = k^ sn^ « sn^ y sn^ w.
(Math. Trip. 1907.)
14. By Liouville's theorem or otherwise, shew that
dn ?( dn (?< + w) — dn v dn (y + to) = B (sn v en m sn (y + ?<■-) en (m + w)
- sn M en V sn (w + w) en (« + ?»)}.
(Math. Trip. 1910.)
15. Shew that
2 en W2 en % sn (m2 - %) dn ?<.i + sn (?<2 - ^3) sn (M3 - Ui) sn (?<i — u^) dn tti dn ?<2 dn ii^ = 0,
the summation applying to the suffices 1, 2, 3. (Math. Trip. 1894.)
18. If P(u)=i , ^ ,
P (u) + P (m + 2iX') an 2u en m
THE JACOBIAN ELLIPTIC FUNCTIONS 523
16. Obtain the formulae
Hndu=A/D, onSu=BID, dn3M=C/i),
where ^ = 3# - 4 (l + P)»3 + 6i{rV ->?:*«»,
B = c{\-4s^ + 6k'^.<t*-4k*ifi+k*s»},
C = d{l- 4/fc2a2 + Qk^s* _ 4;t2«6 + /^«8},
and 8 = 8nu, c=cnu, d=dnu.
17. Shew that
1 -dn 3m _ /I— dnwX /1 + ai dn M+a2dn2tt + a3dn3M + a4dn*«Y
l + dnSw \l+dnM/ \\. — aidnu + a2dn^u-a3dn^u + aidn*uJ^
where ai, a2, as, at are constants to be determined. (Trinity, 1912.)
a+dn3M\*
shew that „, , ,, , ., ..„^ -
P (?t) — /•(?« + 2iiA ) en iu sn m
Determine the poles and zeros of P{u) and the first term in the expansion of the
function about each pole and zero.
(Math. Trip. 1908.)
19. Shew that
sn(Mi + W2 + «3) = -4/i>, cn(Mi + ?<2 4-M3) = i?/-0, dn(?fj + «2 + %) = ^/A
where
^ =«ia283 { - 1 - F + 2F2si2 - (F + yH) 2S22S32 + 2yHSi2«22532}
+ 2 {Si 02C30?20?3 (1 + 2FS22S32 - F2«22«3^)},
5=CiC2C3 {1 -F2S22S32 + 2/^4^12522532}
+ 2 {Ci52S3^2t^3 ( - 1 +2/{;2s22V + 2P8i2-F2S22«3'')},
C =C?lO?2C?3 {1 - F2«22S32 4-2FS12S22S32}
+ F2 {C^1S2«3C2C3 ( - 1 +2Fs22«32 + 2Si2- ^252253^)},
i) = l - 2F2S22«32 + 4 (F +X-4) Si2522gg2_2;(45j2g22s322sj2 + ^455^4534^
and the summations refer to the suffices 1, 2, 3. (Glaisher, Messenger, xi.)
20. Shew that
sn {ui + u<i-\-U3) — A'jD', en {ui + U2 + us) = B'ID', ■dn{ui+tC2 + U3) = G'/D',
where J.' = 2siC2C3(^2''^3 - «i*2«3 {l + k'^ — k'^^s{^+k*Si^S2^s.J^),
B' = Ci C2C3 (1 — k^Si^s^^s^) — did2ds'2s2SzCidi ,
C" = 0?i (^2 (^3 ( 1 — ^^ 81^ ^2" *3") ~ ^^ ^1 ''2 C3 2*2 «3 Cl tl?i ,
Z)' = 1 - F 26-22532 + (F + A;*) 5^2^22 S32_ ^2 5j52S32Si C3C30?2£^3-
(Cay ley, Crelle, XLI.)
21. By applying Abel's method (§ 20-312) to the intersections of the twisted curve
^2^y2_i^ 22^^2^2 — 1 ^itj^ ^]jg variable plane lx-\-my-^nz=\, shew that, if
i(l + M2 + «3 + 2<4 = 0>
then
:0.
Si Ci dy 1
52 C., d^ 1
53 ^3 f/3 1
54 C4 di 1
Obtain this result also from the equation
(52 - Sl ) (C3 C^4 - C4 C?3) + (S4 - S3) (Cl C?2 - ^2 0?1 ) = 0,
which may be proved by the method of example 12.
(Cayley, Messenger, xiv.
524
22. Shew that
THE TRANSCENDENTAL FUNCTIONS
[chap. XXII
by expressing each side in terms of s^, S2, S3, S4 ; and deduce from example 21 that, if
Ul.+ U2 + U3 + Ui = 0,
then S4 Ci c/2 + *3 <^2 di + «2 ^3 di + .Sj C4 (^3 = 0,
«4 ^2 <^l + «3 Cl 0^2 + «2 <^4 <^3 + Si C3 0^4 = 0.
(Forsyth, Messenger, xiv.)
23. Deduce from Jacobi's fundamental Theta-function formulae that, if
Wl + «2 + % + i«4 = 0,
then ](f^ — k'^ F^ Si s^ »3 S4 + P Cj c^ C3 C4 — c^^ d^ d^ 0^4 = 0.
(Gudermann, Crelle, xviii.)
24. Deduce from Jacobi's fundamental Theta-function formulae that, if
Ui + U2 + U3 + Ui = 0,
then 1-2 (^g^ g^ (,g (.^ _ gj g^ 53 «4) — o?i o?2 + o?3 o?4 = 0,
^'2 (Sj«2 — 5384) + rfjO?2 0304 — 01^20^3 <3?4 = 0,
Sis^d^di — dy d-yS^Si -f C3C4 - CiC2= 0.
(H. J. S. Smith, Froc. London Math. Soc. (1), x.)
25. If ^ii + u^ + iis + Ui — O, shew that the cross-ratio of sn^i, sn«2j sn^s, snM4 is equal
to the cross-ratio of su{ni + K\ sn(M2-i-A'), sn(ji3-f-A'), sn(?<4-f A").
(Math. Trip. 1905.)
26. Shew that
sn2 (u + v) sn {u + v) sn (u — v) sn^ {u - v)
cn^ {u + v) en {u-\-v)Gn (u — v) cn^ (u-v)
dn^{u-{-v) dn (?(4-v) dn (w — i') dn^{u-v)
8k'^SiS2^CiC2did2
{i-k^sys2^f •
(Math. Trip. 1913.)
27. Find all systems of values of u and v for which sn^{u + {v) is real when m and v
are real and 0<F<1. (Math. Trip. 1901.)
28. If ^•' = |(a-i-a)2, where 0<a<l, shew that
4a3
sn^i-A:
(l-t-a''')(i + 2rt-a^)'
and that sn^^K is obtained by writing —a~^ for a in this expression.
(Math. Trip. 1902.)
29. If the values of en 2, which are such that cn3z=a, are Ci, C2, ... Cg, shew that
3^* n Cr + k'* 2 -0^ = 0.
r=l r=l
(Math. Trip. 1899.)
30. If
a+sn{u + v) b + cn (ri + v) c + dn (u + v)
a + Hn{u—v) b + cii {71-v) c + dn (?( — ?;)'
and if none of snv, cnw, dn «, I — k^ sn^ ii sn^ v vanish, shew that u is given by the
equation
F (/L-'2 a2 + &2 _ c2) sn2 u = k'-^ + Pb^-c^.
(King's, 1900.
THE JACOBIAN ELLIPTIC FUNCTIONS 526
31. Shew that
(Math. Trip. 1912.)
32. Shew that
l-sn(2^a;/7r)
_ " n -2g2n-Isip^^g4n-2^
{dn(2A'a7/jr)-/{:'sn(2A'j7/7r)}* »=i U 4-2'72»-i 8in^ + 5?*
(Math. Trip. 1904.)
33. Shew that if k be so small that ^ may be neglected, then
snu= sin u — ^^^ cos u . {u — sin m cos m),
for small values of u. (Trinity, 1904.)
34. Shew that, if \I{x)\< ^nl (t), then
logcn(2^^/7r) = logcos.r— 2
„=i n{l+ (-?)"}•
(Math. Trip. 1907.)
[Integrate the Fourier series for sn {'iKxjn) dc {2Kxlir).']
35. Shew that
_/ 0 cn-'wdn^w '^ ^ "
(Math. Trip. 1906.)
[Expi-ess the integrand in terms of functions of 2m.]
36. Shew that
/* en vdu ^ Si (^x+y-jv) Si (^x+y-W-^TTT) _ ^ ^i' (y+j^rr)
j snv-snM ^ ■9i(i^-iy)5i(i.«''-iy-iTr) ^i(y+iTr)'
where 2Kx=iru, 2% = »rv. (Math. Trip. 1912.)
37. Shew that
^^ + ^)^ jo (l+cn^Odn^?.-^-
(Math. Trip. 1903.)
38. Shew that
, f<^+P , , H-^snasn/3
a: I sn^^a2i = Iog, — , ^.
J a-/3 1 - AsnasujS
(St John's, 1914.)
39. By integrating Je^^dnw cs?/(f2 round a rectangle whose corners are ±^v,
±iir + cci (where 2Kz = Tru) and then integrating by parts, shew that, if 0<P<1, then
I cos {iriijK) log sn u du = ^K tanli (i ttzV).
Jo - .
(Math. Trip. 1902.)
40. Shew that K and K' satisfy the equation
where c = k'^; and deduce that they satisfy Legendre's equation for functions of degree
— J- with argument 1 — 'ik'^.
526 THE TRANSCENDENTAL FUNCTIONS [CHAP. XXII
41. Express the coordinates of any point on the curve a;^+^^ = l in the form
2.3*snwdnM-(l— cnw)2 _2t cos^tt (1— en w) {l+tan^n- cnw}
2.S^ snudnu + (l — cmi)^ 2 .S* Hmt6nu + {l-cnu)^
the moduhis of the elliptic functions being sin^Tr ; and shew that
J X Jo'
Shew further that the sum of the parameters of three collinear points on the cubic is a
period.
[See Richelot, Crelle, ix. and Cayley, Proc. Camh. Phil. Soc. iv. A uniformising variable
for the general cubic in the canonical form X^-\-Y^ + Z^ + QmXYZ=0 has been obtained
by Bobek, Einleitung in die Theorie der elliptischen Funhtionen (Leipzig, 1884), p. 251,
Dixon {Quarterly Journal, xxiv.) has developed the theory of elliptic functions by taking
the equivalent curve sfi+y^ - ^axy=\ as fundamental, instead of the curve
y2=(l-^-2)(l_^2^2).]
/2
42. Express / {{2x — x^) {Ax^-{-%))~^ dx in terms of a complete elliptic integi-al of the
first kind with a real modulus. (Math. Trip. 1911.)
43. If u=i {{t+\)lf + t + \)]-^dt,
J X
express x in terms of Jacobian elliptic functions of u with a real modulus.
(Math. Trip. 1899.)
44. If u=r {\-\-fi- 2t^) - ^ dt.
express x in terms of u by means of either Jacobian or Weierstrassian elliptic functions.
(Math. Trip. 1914.)
45. Shew that
,-+,-9.+,-25.^._(2i::i)r(i)_
2^^TT^
(Trinity, 1881.)
46. When a>.r>/3>y, reduce the integrals
j\{a-t){t-ff){t-y)]-^dt, j''^{(a-t){t-^){t-y)}-idt
by the substitutions
x — y=(a — y) dn^ u, x — y = {j3—y) nd^ v
respectively, where F^=(a — /3)/(a — y).
Deduce that, if u + v = K, then
1 - su- u — sn"-' V + k'^ an- u sn^ v = 0.
By the substitution y — [a — t) {t — (i)l{t-y) applied to the above integral taken between
the limits /3 and a, obtain the Gaussian form of Landen's transformation,
r (ai2cos2^ + 6i2sin2^)-*c^5= f {a^ coii^ 6 + h"^ am- 6)'^ d6,
where a^, hy are the arithmetic and geometric means between a and h.
(Gauss, Werke, ill. p. 352; Math. Trip. 1895.)
THE JACOBIAN ELLIPTIC FUNCTIONS 527
47. Shew that
Qcu=-kf-^{C{u-K)-i{u-K-^liK')-C{2iK')},
where the Zeta-functions are formed with periods 2<ai, 2a)2 = 2A', 4t,ff '.
(Math. Trip. 1903.)
48. Shew that E- k"^K satisfies the equation
4cc -Y-^=u,
where c = /^2, and obtain the primitive of this equation. (Math. Trip. 1911.)
49. Shew that n j k^'K'dk^in-l) j k^-^E'dk,
in+2) l^k^E'dk = {n + l) \^ t'K'dk. (Trinity, 1906.)
y 0 y 0
50. If v,= \l''{t{\-t){\-ct))-^dt,
Shew that c(c-l)^+(2c-l)^ + 5^. = -|^^f-^3|.
(Trinity, 1896.)
51. Shew that the primitive of
du %<?■ k _
_ A{E-K)+A'E'
^ ^~'AE+A'{E'-K'y
where A, A' are constants. (Math. Trip. 1906.)
52. Deduce from the addition-formula for E{u) that, if
Ml + ^2 + W34-tt4 = 0,
then (sn Ui sn -^2 — sn ?<3 sn ?«4) sn (Mi -|- u^)
is unaltered by any permutation of sufl&ces. (Math. Trip. 1910.)
53. Shew that
E{3ic)-SE{u) =
1 .- 6Fs* + 4 (/t-^ + k*) s^ - 3k*s» '
(Math. Trip. 1913.)
54. Shew that
Sk* I tt cd* udu = 2 K {(2 + /?•'') A' - 2 ( 1 + ^2) ^| ^
[Write ?<=A+v.] ' (Math. Trip. 1904.)
55. By considering the curves y^=x{\—x){\-k'^x), i/ = l + mx + nx% shew that, if
7ii + Ui + U3 + Ui = 0, then
E{Ui)+E{U2) + E{U3) + E(u^) = k\ 2 Sr^+2CiC2C3Ci-2Sj^S2S3Si-2i .
(Math. Trip. 1908.)
56. By the method of example 21, obtain the following seven expressions for
E{ii^-\-E{u^ + E{u3) + E{Ui) when u^ + u.^-\-U3-\-iti = 0:
Y'^Bs^HHS.rLx '' k'-' + d,d.d3d,rtl ''' k^C^C^C^C^- k'^ ^^^S,a,ICr,
k'-s^SiS^s^did^d^dj * ^ ^^ -F-c^c^c^c^d^d^d^d^ | s /(c (^ )
CjC2C3C4 + ^r'2s^S2S3«4 »•=!
- F{(5iS2S3«4)~' + (CjCoCaCi)-! -|-X-'(0fiC^2C?3^4)~^r^ 2 l/(s^C,,(/,.).
r=l
(Forsyth, Messenger, xv.)
528 THE TRANSCENDENTAL FUNCTIONS [CHAP. XXII
57. Shew that
™=i -I- ~ 9
when \ I(a7)\< irI{T) ; and, by differentiation, deduce that
6{2KJ7r)*ns*{2Kxlw) = 6cosec*a; + 4 {(1 +F) (2^/«-)2- 1} cosec2^
+ 64 ( 1 + P) A'3 ( A' - ^) TT-" - 32 ;{;2 j:4 ^-4
-32 i r^{4(l+F)A:2.-2-n2}2'"^^.
n=l *■ If
Shew also that, when \ I{x)\<^itI(t),
3/o/r / N " fl+^' {2n + lY / TT VI 27ry^+Hin(2n + l)^
(Jacobi.)
58. Shew that, if a be the semi-major axis of an eUipse whose eccentricity is sin ^v,
the perimeter of the eUipse is
(Ramanujan, Quarterly/ Journal, XLV.)
59. Deduce from example 19 of Chapter xxi that
T> Qf, -/r'-' + dn^ wdn.3?t ^^^.pcn^w cnSw
A;2cn3 2tt=- — rx^ — s :i — , an^2u=~ — j- — n ;r— .
l+fc^ sn^ u sn 3m 1 + «■' sn-* % sn 3%
(Trinity, 1882.)
60. Shew that the primitive of
||={|Psn2.-i(l+F)}^.
is u = {m^{C-z)cu^{C-z)dni{C-z)}-^{A+Bsn''^{C-z)},
where A and B are arbitrary constants and C=2K+iK'.
(Jamet, Comptes Rendus, cxi.)
[See also Brioschi, Comptes Rendus, lxxxvi. If v be written for z-2K—iK' the
equation becomes
of which the corresponding solution, v,^{i^' {\v)) *{J^i^(|i;) + 5i}, was given in 1880
by Halphen, Memoires par divers savants, xxvrii. (i), p. 105.
The equation is a special case of Lame's equation
(which assumes the form given on p. 199, when ^{v) is taken as a new independent
variable); this equation was solved in 1877, for integral values of n, by Hermite (who
used Jacobian elliptic functions), Stir quelques apflications des fonctions elliptiques
{Comptes Rendus, Lxxxv., ijublished separately, 1885). The corresponding solutions of
the equation with Weierstrassian fiuictions are given by Halphen, Fonctions elliptiques,
II., and also by Forsyth, Differential Equations, iv.]
APPENDIX
THE ELEMENTARY TRANSCENDENTAL FUNCTIONS
A'l. On certain results assumed in Chapters I-IV.
It was convenient, in the first four chapters of this work, to assume some of the
properties of the elementary transcendental functions, namely the exponential, logarithmic
and circular functions ; it was also convenient to make use of a number of results which
the reader would be prepared to accept intuitively by reason of his familiarity with the
geometrical representation of complex numbers by means of points in a plane.
To take two instances, (i) it was assumed (§ 2'7) that liin (exp2) = exp(lim0), and
(ii) the geometrical concept of an angle in the Argand diagram made it appear plausible
that the argument of a complex number was a many-valued function, possessing the
property that any two of its values differed by an integer mviltiple of 'Hn.
The assumption of results of the first type was clearly illogical ; it was also illogical to
base arithmetical results on geometrical reasoning. For, in order to put the foundations
of geometry on a satisfactory basis, it is not only desirable to employ the axioms of
arithmetic, but it is also necessary to utilise a further set of axioms of a more definitely
geometrical character, concerning properties of points, straight lines and planes*. And,
further, the arithmetical theory of the logarithm of a complex number appears to be
a necessary preliminary to the development of a logical theory of angles.
Apart from this, it seems unsatisfactory to the aesthetic taste of the mathematician to
employ one branch of mathematics as an essential constituent in the structure of another ;
particularly when the former has, to some extent, a material basis whereas the latter
is of a purely abstract nature t.
The reasons for pursuing the somewhat illogical and unaesthetic procedure, adopted in
the earlier part of this work, were, firstly, that the properties of the elementary transcen-
dental functions were required gradually in the course of Chapter ii, and it seemed
* It is not our object to give any account of the foundations of geometry in this work. They
are investigated by various writers, such as Whitehead, Axioms of Projective Geovietry (Cambridge
Tracts) and Mathews, Projective Geometry. A perusal of Chapters i, xx, xxii and xxv of the
latter work will convince the reader that it is even more laborious to develop geometry in a
logical manner, from the minimum number of axioms, than it is to evolve the theory of the
circular functions by purely analytical methods. A complete account of the elements both of
arithmetic and of geometry has been given by Whitehead and Kussell, Principia Mathematica.
+ Cf. Merz, History of European Thought in the Nineteenth Century, Vol. ii. pp. 631 (note 2)
and 707 (note 1), where a letter from Weierstrass to Schwarz is quoted. See also Sylvester,
Matli. Papers, m. p. 50.
W. M. A. 34
530 APPENDIX
undesirable that the courae of a general development of the various infinite processes
should be frequently interrupted in order to prove theorems (with which the reader was,
in all probability, already familiar) concerning a single particular function ; and, secondly,
that (in connexion with the assumption of results based on geometrical considerations)
a purely arithmetical mode of development of Chapters l-iv, deriving no help or illus-
trations from geometrical processes, would have very greatly increased the difficulties of
the reader unacquainted with the methods and the spirit of the analyst.
A'll. Summary of the Appendix.
The general course of the Appendix is as follows :
In §§ A'2-A*22, the exponential function is defined by a power series. From this
definition, combined with results contained in Chapter ii, are derived the elementary
properties (apart from the periodic properties) of this function. It is then easy to deduce
corresponding properties of logarithms of positive numbers (§§ A"3-A'33).
Next, the sine and cosine are defined by power series from which follows the connexion
of these functions with the exponential function. A brief sketch of the manner in which
the formulae of elementary trigonometry may be derived is then given (§§ A*4-A"42).
The results thus obtained render it possible to discuss the periodicity of the exponential
and circular functions hj purely arithmetical methods {^^ A"5, A'51).
In §§ A'52-A"522, we consider, substantially, the continuity of the inverse circular
functions. When these functions have been investigated, the theory of logarithms of
complex numbers (§ A '6) presents no further difficulty.
Finally, in § A'7, it is shewn that an angle, defined in a purely analytical manner,
possesses properties which are consistent with the ordinary concept of an angle, based on
our experience of the material world.
It will be obvious to the reader that we do not profess to give a complete account of
the elementary transcendental functions, but we have confined ourselves to a brief sketch
of the logical foundations of the theory*. The developments have been given by writers
of various treatises, such as Hobson, Plane Trigonometry ; Hardy, A course of Pure
Mathematics ; and Bromwich, Theory of Infinite Series.
A*12. A logical order of development of the elements of Analysis.
The reader will find it instructive to read Chapters i-iv and the Appendix a second
time in the following order :
Chapter i (omitting f all of § 1*5 except the first two paragraphs).
Chapter ii to the end of § 2-61 (omitting the examples in §§ 2-31-2-61).
Chapter in to the end of § 3-34 and §§ 3-5-3-73.
The Appendix, §§ A-2-A-6 (omitting §§ A-32, A-33).
Chapter ii, the examples of §§ 2'31-2'61.
Chapter iii, 5$5< 3-341-3-4.
Chapter iv, inserting §§ A-32, A-33, A-7 after § 4-13.
Chapter ii, ^§ 2-7-2-82.
^ In writing the Appendix, frequent reference has been made to the article on Algebraic
Analysis in the Encyldopildie der Math. Wissenschaften by Pringsheim and Faber, to the same
article translated and revised by Molk for the Encyclopedic des Sciences Math., and to Tannery,
Fonctions d'une Variable reelle.
t The properties of the argument of a complex number are not required in the text before
Chapter v.
A-ll-A'21] THE ELEMENTARY TRANSCENDENTAL FUNCTIONS 531
He should try thus to convince himself that (in that order) it is possible to elaborate
a purely arithmetical development of the subject, in which the graphic and familiar
language of geometry* is to be regarded as merely conventional.
A •2. The exponential function ex^ z.
The exponential function, of a complex variable z, is defined by the series t
z z^ :^ cc 2"
exp. = l+- + _ + _+...=l+^2^-.
This series converges absolutely for all values of z (real and complex) by D'Alembert's
ratio test (§ 2*36) since lim |(2/n)| = 0<l ; so the definition is valid for all values of z.
Further, the series converges uniformly throughout any bounded domain of values of z ;
for, if the domain be such that | « | ^ ^ when z is»in the domain, then
|(«»/%!)|$^/to!,
and the unifoi-mity of the convergence is a consequence of the test of Weierstrass (§ 3-34),
by reason of the convergence of the series 1+2 {R^jn !), in which the terms are indepen-
n=l
dent of z.
Moreover, since, for any fixed value of n, z'^jn ! is a continuous function of z, it follows
from § 3*32 that the exponential fvmction is continuous for all values of z ; and hence
(cf. § 3'2), if 2 be a variable which tends to the limit f, we have
lim exp z = exp f . .
A*21. The addition-theorem for the exponential function, and its consequences.
From Cauchy's theorem on multiplication of absolutely convergent series (§ 2-53), it
follows that J
(exp.0(exp^,) = (l + |5^ + |-; + ...)(l+|?+|-' + ...)
= 1 + ^^+ 2l + •••
= exp(2i+22), •
so that exp(zj+02) can be expressed in terms of exponential functions of z^ and of ^2 by
the formula
exp (%+32) = (exp zi) (exp Zj)-
* E.g. 'a point,' for 'an ordered number-pair,' 'the circle of unit radius with centre at the
origin' for 'the set of ordered number-pairs {x, y) which satisfy the condition x^-t-y^ — i^' <tijg
points of a straight line ' for ' the set of ordered number-pairs [x, y) which satisfy a relation of
the type Ax + Bij + C=Q,' and so on.
t It was formerly customary to define expz as lim ( l-f-- ) , cf. Gauchy, Cours cC Analyse, i.
p. 167. Cauchy {ibid. pp. 168, 309) also derived the properties of the function from the series,
but his investigation when z is not rational is incomplete. See also Schloinilcli, Handhuch der
dig. Analysis (1889), pp. 29, 178, 246. Hardy has pointed out (Math. Gazette, iii. p. 284) that
the limit definition has many disadvantages.
X The reader will at once verify tliat the general term in the product series is
(■2l" + nCl ^l"^' Z2 + n^\ ^l"~^ 22'' + . . . + Z2")/" ! = (^1 + ^i)"/" ! •
34—2
532 APPENDIX
This result is known as the addition-theorem for the exponential function. From it,
we see by induction that
(exp 2i) (exp 22) . . • (exp 2„) = exp (21 + ^2 + . . . + 2„),
and, in particular,
{exp z) {exp ( - 2)} = exp 0 = 1.
From the last equation, it is apparent that there is no value of 2 for which exp 2 = 0;
for, if there were such a value of 2, since exp ( - 2) would exist for this value of 2, we
should have 0=1.
It also follows that, when x is real, exp * > 0 ; for, from the series definition, exp ^ ^ 1
when x'^O; and, when x ^ 0, exp x= 1/exp ( - ^)>0.
Further, exp x is an increasing function of the real variable x ; for, if ^ > 0,
exp {x + k)- exp x = exp x . {exp ^ — 1} > 0,
because exp x>0 and exp ^ > 1.
Also, since {expA- l}/A=H-(A/2 !) + (A2/3 !) + ... ,
and the series on the right is seen (by the methods of § A'2) to be continuous for all
values of A, we have
lim {expA— 1}/A = 1,
, c?exp2 ,. exp (2 + A) — exp 2
and so -j = bm — ^ , — ^— =exp2.
dz h^o "-
A '22. Various properties of the exponential function.
Returning to the formula (exp2i) (exp22) ... (expz„) = exp(2i+22 + -.-+'2^n)> we see that,
when n is a positive integer,
(exp 2)"- = exp {nz),
and (exp 2) ~" = l/(exp 2)"= 1/exp (712) = exp (- ^2).
In particular, taking 2=1 and writing e in place of exp 1 = 271828..., we see that,
when m is an integer, positive or negative,
e™ = exp ?«-=!+ {mjl !) + (?/i2/2 !) + ....
Also, if p. be any rational number ( —plq, where p and q are integers, q being positive)
(exp /x)« = exp \iq = exp^ = e'",
so that the g'th power of exp/i is e^; that is to say, exp /a is a vahie of eP^i = e'^, and it is
obviously (§ A'21) the real positive value.
If X be an irrational-real number (defined by a section in which aj and ag ^^6 typical
members of the Z-class and the j^-class respectively), the irrational power e* is mostly
simply defined as exp x ; we thus have, for all real values of .r, rational and irrational,
an equation first given by Newton*.
It is, therefore, legitimate to write e^ for exp;p when x is real, and it is customary to
write e^ for exp 2 when 2 is com])lex. The function e' (which, of course, must not be
regarded as being a power of e), thus defined, is subject to the ordinary laws of indices, viz.
* De Analysi per aeqiiat. num. term. inf. (written before 1669, but not published till 1711) ;
it was also given both by Newton and by Leibniz in letters to Oldenburg in 1676 ; it was first
published by Wallis in 1685 in his Treatise on Algebra, p. 343. The equation when x is irrational
was explicitly stated by Schlomilch, Algehraische Analysis (1889), p. 182.
A-22-A-31] THE ELEMENTARY TRANSCENDENTAL FUNCTIONS 633
[Note. Tannery, Legons d'Algibre et d' Analyse (1906), i. p. 45, practically defines e*,
when X is irrational, as the only number X such that e"* ^X ^e^^, for every aj and Og.
From the definition we have given it is easily seen that such a unique number exists.
For exp x { = X) satisfies the inequality, and if X' {^X) also did so, then
exp flg — exp ai = e"2 - e"i ^ I X' — X i ,
so that, since the exponential function is continuous, a^ — a^ cannot be chosen arbitrarily
small, and so {a\, a^) does not define a section.]
A*3. Logarithms of positive numbers*.
It has been seen (§§ A'2, A'21) that, when x is real, exp a; is a positive continuous
increasing function of x, and obviously exp x-*- + oo as x-*'+ <x> , while
exp:p=l/exp(-ar)-»-0 as x-^ — cc.
If, then, a be any positive number, it follows from § 3"63 that the equation in x,
ex]ix=a,
has one real root and only one. This root (which is, of course, a function of a) will be
written t Logg a or simply Log a ; it is called the Logarithm of the positive number a.
Since a one-one correspondence has been established between x and a, and since a is
an increasing function of x, x must be an increasing function of a ; that is to say, the
Logarithm is an increasing function.
Example. Deduce from § A '21 that Log a + Log 6= Log a6.
A'31. The continuity of the Logarithm.
It will now be shewn that, when a is positive. Log a is a continuous function of a.
Let Loga=a;, Log(a + A) = .r+^,
so that e^ = a, e== + *^ = a + A, H-(/i/a) = e*.
First suppose that A > 0, so that ^ > 0, and then
]+(/i/a) = l + /& + |P+...>l4-;{;,
and so 0 < /{; < A/a,
that is to say 0 < Log (a + A) - Log a < hja.
Hence, h being positive. Log (a + Zi) — Loga can be made arbitrarily small by taking h
suflSciently small.
Next, suppose that A<0, so that ^•<0, and then al{a + h) = e~^.
Hence (taking 0 < — A < |a, as is obviously permissible) we get
al{a + h) = \ + {-k) + \B-\- ... >\-k,
and so -k< -l + a/(a + /i)= -h/(a + h)< -2h/a.
Therefore, whether h be positive or negative, if e be an arbitrary positive number and
if.| A I be taken less than both ia and iae, we have
I Log (a + A) - Log a I < f ,
and so the condition for continuity (§ 3*2) is satisfied.
* Many mathematicians define the Logarithm by the integral formula given in § A-3'2. The
reader should consult a memoir by Hurwitz (Math. Ann. lxx.) on the foundations of the theory
of the logarithm.
t This is in agreement with the notation of most text-books, iu which Log denotes the
principal value (see § A-6) of the logarithm of a complex nnmber.
534 APPENDIX
A*32. Differentiation of the Logarithm.
Retaining the notation of § A'31, we see, from results there proved, that, if A^O
(a being fixed), then also ^-*-0. Therefore, when a>0,
c^Loga_ V ^ _ 1 _ 1
da ~ jc-».Q^'''^^-^~ «^~ ct'
Since Log 1 =0, we have, by § 4'13 example 3,
(a
Loga= I t~^dt.
A '33. The expansion of Log(l + a) in powers of a.
From § A*32 we have
Log(H-a)= j"'{l + t)--^dt
= r{i-t+t^- ... +(-)»-ir-i+(-)"p(i+o-»}f^i;
= a-ia^ + W- ...+(-)"-^ ^a" + ^„,
where Rn^i-^ j"'t''{l + t)-^dt.
Now, if - 1< a < 1, we have
rial
l^ni^l t"{l-\a\)-^dt
= |ai" + i{(?i + l)(l-|a|)}-i
-••0 as ri -» 00 .
Hence, when — 1 < a < 1, Log (l+a) can be expanded into the convergent series*
Log(l+a) = a-|aHJa^- ... = 2 (-)»-ia'V'«-
If a= + l,
\Rn\=l f^il + t)-^dt< I t'^dt = {n-\- 1) - ^ -^ 0 a,s n-* CO,
Jo Jo
so the expansion is valid when a= + 1 ; it is not valid when a= -1.
Example. Shew that lim ( 1 + - ) = e.
[We have Jin.^ ,.log (l + i) = \im^ (i - i. + 2_ _ ...)
= 1,
and the result required follows from the result of § A-2 that lim e^ = e^]
A*4. The definition of the sine and cosine.
The functions + sinz and cosz are defined analytically by means of power series, thus
^3 2'') 00 ( — )" s2" + 1
sinz=z-- +--...=^^2^ (2n+l)! '
^2 2* ^ ( — )" S^"
cos2 = l- — + -—- ...=1+ 2 ™vr ;
2! 4! „=i (270!
these series converge absolutely for all values of z (real and complex) by § 2"36, and so the
definitions are valid for all values of z.
* This method of obtaining the Logarithmic expansion is, in effect, due to Wallis, Phil.
Trans, ii. (1668), p. 754.
+ These series were given by Newton, Be Analysi... (1711), see § A-22 footnote. The other
trigonometrical functions are defined in the manner with which the reader is famiUar, as
quotients and reciprocals of sines and cosines.
A-32-A*5] THE ELEMENTARY TRANSCENDENTAL FUNCTIONS 535
On comparing these series with the exponential series, it is apparent that the sine and
cosine are not essentially new functions, but they can be expressed in terms of exponential
functions by the equations*
2i sin 2 = exp (iz) — exp ( - iz), 2 cos z = exp {iz) + exp ( - iz).
It is obvious that sin z and cos z are odd and even functions of z respectively ; that is
to say
sin { — z)=— sin z, cos ( - 2) = cos z.
A '41. The fundamental properties of sin 2 and cos 2.
It may be proved, just as in the case of the exponential function (§ A*2), that the series
for sin 2 and cos 2 converge uniformly in any bounded domain of values of 2, and con-
sequently that sin 2 and cos 2 are continuous functions of 2 for all values of 2.
Further, it may be proved in a similar manner that the series
3!'''5! •••
defines a continuous function of z for all values of 2, and, in particular, this function
is continuous at 2=0, and so it follows that
lim (2~isin2) = l.
A*42. The addition-theorems for sin 2 and cos 2.
By using Euler's equations (§ A'4), it is easy to prove from properties of the exponential
function that
sm{zi + Z2) = smzi cos 22 + cos 21 sin 22
and cos {zi + z>>) — cos 21 cos 22 — sin 2i sin 22 ;
these results are known as the addition-theorems for sin 2 and cos 2.
It may also be proved, by using Euler's equations, that
sin^z-\-coii^z=l.
By means of this result, sin (21 + 22) can be expressed as an algebraic function of sin^i
and sin 22, while 008(21 + 22) can similarly be expressed as an algebraic function of cos2i
and cos 22 ; so the addition-formulae may be regarded as addition-theorems in the strict
sense (cf. §§ 20-3, 22-732 note).
By differentiating Euler's equations, it is obvious that
ds'niz dcosz
J = cos 2, -. = — SHI z.
dz dz
Example. Shew that
sin 22 = 2 sin 2 cos 2, cos 22 = 2 cos^ 2 - 1 ;
these results are known as the duplication-formulae.
A'5. The periodicity of the exponential function.
If 2i and 22 are such that exp 21 = exp 22, then, multiplying both sides of the equation by
exp (-22), we get exp (21- 22) =1 ; and writing y for 21-20, we see that, for all values of 2
and all integral values of n,
exp (2 + ny) = exp 2 . (exp y)" = exp z.
* These equations were derived by Euler [they were given in a letter to Johann Bernoulli in
1740 and published in the Hist. Acad. Berlin, v. (1749), p. 279] from the geometrical definitions
of the sine and cosine, upon which the theory of the circular functions was then universally
based.
536 APPENDIX
The exponential function is then said to have period y, since the effect of increasing
z by y, or by an integral multiple thereof, does not affect the value of the function.
It will now be shewn that such numbers y (other than zero) actually exist, and that all
the numbers y, possessing the property just described, are comprised in the expression
2;i»ri, {n=±l, ±2, ±3, ...)
where tt is a certain positive number* which happens to be greater than 2 v/2 and less
than 4.
A'51. The solution of the equation exp y= 1.
Let 7=0 + 1^, where a and /3 are real ; then the problem of solving the equation
expy=l is identical with that of solving the equation
expo. expi/3=l.
Comparing the real and imaginary parts of each side of this equation we have
expo, cos /3=1, expo, sin /3=0.
Squaring and adding these equations, and using the identity cos'^/3 + sin2;3= 1, we get
exp2a=l.
Now if a were positive, exp 2a would be greater than 1, and if a were negative, exp 2a
would be less than 1 ; and so the only possible value for a is zero.
It follows that cos/3=l, sin/3=0.
Now the equation sin/3 = 0 is a necessary consequence of the equation cos^=l, on
account of the identity cos^/S + sin^/S^l. It is therefore sufficient to consider solutions
(if such solutions exist) of the equation cosi3 = l.
Instead, however, of considering the equation cos/3=l, it is more convenient to
consider the equation t cos;r=0.
It will now be shewn that the equation cos^' = 0 has one root, and only one, lying
between 0 and 2, and that this root exceeds v'2 ; to prove these statements, we make use
of the following considerations :
(I) The fvmction cos*' is certainly continuous in the range 0 ^^^2.
(II) When 0 ^ .r ^ ^2, we have +
and so, when Q^x^ J2, cos x > 0.
(Ill) The value of cos 2 is
'-^+5-|i(i-.'r8)-,^,(i-n^)--=-i--«'-
(IV) When0<.r^2,
sin X
, x'^\ X* ( ^ x"^ \ T •^^^ 1
and so, when 0 ^ a* ^ 2, sin x ^ \x.
* The fact that tt is an irrational number, whose value is 3-14159..., is irrelevant to the
present investigation. For an account of attempts at determining the value of tt, concluding
with a proof of the theorem that tt satisfies no algebraic equation with rational coefficients, see
Hobson's monograph Squaring the Circle.
t If cosx=:0, it is an immediate consequence of tlie duplication-formulae that cos 2a; = -1
and thence that cos4.r = l, so, if a; is a solution of cos a; = 0, 4a; is a solution of cos/3 = l.
X The symbol 2j may be replaced by > except when x = j2 in the first place where it occurs,
and except when .x- = 0 in the other places.
A'51, A-52] THE ELEMENTARY TRANSCENDENTAL FUNCTIONS 537
It follows from (II) and (III) combined with the results of (I) and of § 3 '63 that the
equation cos.r = 0 has at least one root in the range v'2<^<2, and it has no root in the
range 0^^^^2.
Further, there is not more than one root in the range x/2<a'<2; for, suppose that
there were two, .»i and X2 (^2 > a;i) ; then 0 < ^^2 - ajj < 2 - v'2< 1, and
sin (x2 - ^i) = sin 1C2 cos jti — sin Xi cos Xz = 0,
and this is incompatible with (IV) which shews that sin (a^g -x^'^\ {x^ — x^).
The equation cos ^=0 therefore has one and only one root lying between 0 and 2. This
root lies between «y2 and 2, and it is called \ n ; and, as stated in the footnote to § A"5, its
actual value happens to be 1*57079
From the addition-formulae, it may be proved at once by induction that
cos w TT = ( — 1 )", sin nir = 0,
where n is any integer.
In particular, cos 2nir = 1, where n is any integer.
Moreover, there is no value of 3 (other than those values of the form inn) for which
008/3 = 1; for if there were such a value, it must be real* and so we can choose the
integer m so that
— w ^2TO7r —^<n.
We then have
sin I WITT - iiS I = ± sin (tott - J^) = ± sin ^/3= ± 2 ~ ^ (1 - cos /3)*=0,
and this is inconsistent t with sin | mTr — ^/3 | ^ ^ | win- — ^^ | unless ^=2m7r.
Consequently the numbers 2??jr, (w=0, ±1, ±2,...), and no others, have their cosines
equal to unity.
It follows that a positive number n- exists such that exp^ has period 2iTi and that
exp z has no period fundamentally distinct from 27ri.
The formulae of elementary trigonometry concerning the periodicity of the circular
functions, with which the reader is already acquainted, can now be proved by analytical
methods without any difficulty.
Example 1. Shew that sin |n- is equal to 1, not to — 1.
Example 2. Shew that tan x>x when 0<.x< ^tt.
[For cos ^ > 0 and
00 ^n-l
„=i(4»-l)
and every term in the series is positive.]
x'^ x^ x^ . . 25 X'^ x^
Example 3. Shew that 1 — tt + :t7 — ^^t^k ^^ positive when x = --i, and that l—\+7r-
z 24 7aU lb 2 24
vanishes when .r = (6 — 2^/3)^ = 1" 5924... ; and deduce that J
3-125 <7r< 3-185.
A'52. The solution of a pair of tngonometneal equations.
Let X, ju. be a pair of real numbers such that X'^ + /li^= 1.
* The equation cos/3=l implies that exp t/3=l, and we have seen that this equation has no
complex roots.
t The inequality is true by (IV) since 0 ^ | thtt- J/3 i ■$ |7r<2.
X See De Morgan, A Budget of Paradoxes, pp. 316 et seq., for reasons for proving that
7r>3i.
sin:r — ;?;cos:r= 2 tt- tt-; ^ 4?i - 2 - ■, ~[ ,
4n + lj '
538 APPENDIX
Then, if X4= - 1, the equations
cos^=X, Bin x=fi
have an infinity of solutions of which one and only one lies between* — n and tt.
First, let k and /x be not negative ; then (§ 3"63) the equation cos^=X has at least one
solution Xi such that O^J7^^|7r, since cos 0=1, co8^ = 0. The equation has not two
solutions in this range, for if Xi and x.^ were distinct solutions we could prove (cf. ^ A"51)
that sin (^i — ^2) = 0> and this would contradict § A"51 (IV), since
Further, sin a^i= 4-/^/(1— cos2^i)= +v'(l -\^) = fi, so ^j is a solution of 6of/i equations.
The equations have no solutions in the ranges ( — tt, 0) and (^rr, tt) since, in these
ranges, either sin a,' or cos^ is negative. Thus the equations have one solution, and only
one, in the range ( — n, tt).
If X or )u. (or both) is negative, we may investigate the equations in a similar manner ;
the details are left to the reader.
It is obvious that, if x^ is a solution of the equations, so is Xi + 2mT, where n is any
integer, and so the equations have an infinity of real solutions.
A '521. The principal solution of the trigonometrical equations.
The unique solution of the equations cos.r=X, sin.r = ^ (where X2 + /i^ = l) which lies
between — tt and n is called the principal solution-^, and any other solution differs from it
by an integer multiple of 27r.
The principal value % of the argument of a complex number ^ ( 4= 0) can now be defined
analytically as the principal solution of the equations
\z\coii(li = R{z), |s I sin (^ = 7(2),
and then if z = \z\. (cos 6-\-i sin 6),
we must have d — cji + 2nn, and $ is called a value of the argument of 2, and is written
arg2 (cf. § 1*5).
A'522. The continuity of the argument of a complex variable.
It will now be shewn that it is possible to choose such a value of the argument 6 (z), of
a complex variable z, that it is a continuous function of z, provided that z does not pass
through the value zero.
Let zo be a given value of z and let ^0 t)e any value of its argument ; then, to prove that
6(z) is continuous at Zq, it is sufficient to shew that a number di exists such that ^j = arg2i
and that | ^1 - ^0 ( can be made less than an arbitrary positive number e by giving 1 21 - 20 I
any value less than some positive number t].
Let Zq = x^t + ii/o , ■^i = -^i + '!?/ 1 •
Also let 1 2i - Zq I be chosen to be so small that the following inequalities are satisfied § :
(I) I .i"i - .To I < -^ I ^0 I ) provided that Xo =t= 0,
(II) I y 1 - yo ! < i 1 .yo I ) pro vided that 3/0 + 0,
(III) l^-i-^ol <ie|2u|, \->/i-2/o\<h\^o\-
* If X = - 1, ±7r are solutions and there are no others in the range ( - tt, tt).
t If X=: - 1, we take +7r as the principal solution ; cf. p. 9.
X The term principal value was introduced in 1845 by Bjorling ; see the Archiv der Math,
unci Phys. (1) ix. (1847), p. 408.
§ (I) or (II) respectively is simply to be suppressed in the case when (i) X(, = 0, or when
(ii) 2/0-0.
A-521-A-7] THE ELEMENTARY TRANSCENDENTAL FUNCTIONS 539
From (I) and (II) it follows that .roO^i and y^y^ are not negative, and
80 that -^o^i + yo^i ^ i I «o I ^.
Now let that value of di be taken which differs from 6q by less than n ; then since
^0 and Xi have not opposite signs and y^ and yi have not opposite signs*, it follows from
the solution of the equations of § A-52 that di and 6^ differ by leas than ^rr.
Now . tan(^i-^o) = '^"^'~'^''^",
^o^i+yoyi
and 80 (§ A'51 example 2),
I /} z, I ^ Uo.yi--yi.yo I
''1 ~ t'o ^ ;
^o^i+yo^i
^ l-^o(.yi-yo)-yo(^i-^o)l
^•o^i +yoyi
=^ 2 I zo TMko I • I yi -^0 i + 1 3/0 I . I ^1 -J^o |}-
But I ^0 I ^ I ■^o I and also I ^o I < i ■^o I > therefore
|(9i-(9ol=$2|2o|~Mlyi-yo| + |.«'i-.«?ol} < «•
Further, if we take \zi-Zq\ less than J | a:o | , (if ^o =*= 0) and ^|yo l» (if yo=t=0) and ie | Zq |>
the inequalities (I), (II), (III) above are satisfied ; so that if rj be the smallest of the
three numberst -11 ■3:*o I, i i ^o h jf I ■^^o I? "by taking | 2i -^o I < '7) we have | ^i- ^o I < « ! and
this is the condition that 6 {£) should be a continuous function of the complex variable z.
A*6. Logarithms of complex numbers.
The number ( is said to be a logarithm of z if z = e^.
To solve this equation in f, write t= ! + *'?> where ^ and rj are real ; and then we have
z = e^ (cos T) ■\- i sin rf).
Taking the modulus of each side, we see that | a | = e% so that (§ A '3), | = Log \z\; and
then
2= 1^1 (cos 7/ +1 sin rj),
so that T] must be a value of argz.
The logarithm of a complex number is consequently a many-valued function, and it
can be expressed in terms of more elementary functions by the equation
log z = Log z + i arg z.
The continuity of logz (when z^O) follows from § A"31 and § A'522, since \z\ is a
continuous function of z.
The differential coefficient of any particular branch of logz (§ 5*7) may be determined
as in § A'32 ; and the expansion of § A'33 may be established for log (1 +a) when | a | < 1.
Corollary. If a' be defined to mean e'^''^'^, a^ is a continuous function of z and of a
when a =1=0.
A'7. The analytical definition of an angle.
Let 2i, ^2) h ^^ three complex numbers represented by the points Pi, P.^, P3 in the
Argand diagram. Then the angle between the lines (§ A"12, footnote) PiF^ and P1P3 is
defined to be any value of arg (23 — ^j) — arg (Zi-Zi).
* The geometrical interpretation of these conditions is merely that zq and zi are not in
different quadrants of the plane.
t If any of these numbers is zero, it is to be omitted.
540 APPENDIX
It will now be shewn* that the area (defined as an integral), which is bounded by two
radii of a given circle and the arc of the circle terminated by the radii, is proportional to
one of the values of the angle between the radii, so that an angle (in the analytical sense)
possesses the property which is given at the beginning of all text-books on Trigonometry t.
Let (^1, _yi) be any point (both of whose coordinates are positive) of the cii'cle
ifi+y'^ = a^{a>0). Let 6 be the principal value of arg(^i + iyi), so that 0<^<-^7r.
Then the area bounded by OX and the line joining (0, 0) to (.ri, yx) and the arc of the
circle joining {xi,y-^ to {a, 0) is I f{x) dx, wherej
Jo
/ (x) = X tan 6 {O^x^a cos 6),
f{x) = {ofi-x^)^ (a cos ^<a;^«),
if an area be defined as meaning a suitably chosen integral (cf. p. 61).
It remains to be proved that I f{x) dx is proportional to 6.
J 0
/"a racosB Ca ,
Now I f{x)dx=\ xi&uddx+l {a?-x^y dx
Jo' Jo y a cose
= ^a=^sin^cos<9-f I I \a^{a^~x^)~^ + -j-x{a^ - x'^)^\ dx
= ^a^r {a^-x^)-^dx
J a cos 6
= ^a^U\l-fiy^ dt- r"^\l-fi)-^ dt\
on writing x^at and using the example worked out on p. 64.
That is to say, the area of the sector is proportional to the angle of the sector. To
this extent, we have shewn that the popular conception of an angle is consistent with
the analytical definition.
* The proof here given applies only to acute angles ; the reader should have no difficulty in
extending the result to angles greater than ^w, and to the case when OX is not one of the
bounding radii.
t Euclid's definition of an angle does not, in itself, afford a measure of an angle ; it is shewn
in treatises on Trigonometry (of. Hobson, Plane Trigonometry, Chapter i) that an angle is
measured by twice the area of the sector which the angle cuts off from a unit circle whose centre
is at the vertex of the angle.
J The reader will easily see the geometrical interpretation of the integral by drawing a
figure.
LIST OF AUTHORS QUOTED
[The numbers refer to the pages. Initials which are rarely used are
given in italics]
Abel, N. H., 16, 17, 50, 57, 58, 205, 223, 224,
347, 412, 422, 435, 489, 498, 505, 518
Adamoff, A., 345, 348
Adams, J. C, 126, 229, 325, 399
Airey, J. E., 371
Aldis, W. S., 371
Alexeiewsky, W. P., 258
Amigues, E. P. M., 122
Anding, E., 371
Appell, P. E., 274, 292, 294, 330, 393, 411
Argand, J. R., 9
Barnes, E. W., 128, 159, 230, 258, 273, 280,
283, 290, 293, 320, 324, 332, 340, 341, 345,
362
Basset, A. B., 366, 377
Bateman, H., 224, 346, 392, 395
Bauer, G., 327, 394
Beau, 0., 187
Berger, A., 185
Bernoulli, Daniel (1700-1782), 160, 350
Bernoulli, Jakob (1654-1705), 126, 422, 455
Bernoulli, Joliaun (1667-1748), 535
Bertrand, J. L. F., 71
Bessel, F. W., 198, 335, 336, 350, 351, 355, 360
Besso, D., 346
Biermann, W. G. A., 447
Binet, J. P. M., 242, 243, 244, 245, 247, 255,
256, 257, 308
Bjorling, E. G., 538
Blades, E., 396
Bobek, K., 526
Bocher, M , 81, 197, 206, 215, 223, 224, 225,
354
Bolzano, B.,' 13
Bonnet, P. Ossian, 66, 161
Borchardt, C. W., 105, 448, 456
Borel, E., 53, 108, 140, 141, 144, 154, 155, 159
Bouquet, J. C, 448, 455, 480
Bourget, J., 372, 389
Bourguet, L., 240, 254, 255, 258
Bridgeman, P. W., 374
Brioschi, F., 528
Briot, J. A. A., 448, 455, 480
Bromwioh, T. J. I'a., 10, 25, 26, 33, 38, 45,
50, 59, 75, 80, 144, 156, 159, 236, 284,
530
Brouncker, William (Viscount), 16
Burgess, J., 336
Burkhardt, H. F. K. L., 317, 392
Burmann, Heinrich, 129
Burnside, William, 450
Burnside, W. S., 206, 355, 446
Cajori, F., 59, 60
Callandreau, O., 358
Cantor, Georg F. L. P., 4, 161, 177, 180, 181
Carda, K., 126
Catalan, E. C, 326, 327
Cauchy, (Baron) A. L., 13, 16, 21, 27, 29, 40,
42, 59, 71, 77, 83, 85, 86, 91, 93, 96, 99,
105, 119, 122, 123, 161, 257, 455, 531
Cayley, A., 192, 292, 427, 432, 448, 480, 489,
491, 501, 521, 523, 526
Cesaro, E., 38, 39, 58, 155, 156
Chapman, S., viii, 159
Charpit, P., 384
Chartier, J., 72, 77
Chree, C, 374
Christoffel, E. B., 327
Chrystal, G., 23
Clausen, T., 292
Clebsch, R. F. A., 448
Corey, S. A., 108
Craig, T., 202
Cunningham, E., 204, 347
Curzon, H. E. J., 204, 345
D'Alembert, J. le Bond, 20, 22, 160
Daniels, A. L., 448
Dantscher, V. von, 4, 10
Darboux, J. G., 43, 61, 96, 125
De Brun, F. D., 450
Debye, P., 362
Dedekind, J. W. Richard, 4, 10
De la ValMe Poussin, Ch. J., 39, 59, 72, 73,
80, 81, 108, 184
De Morgan, A., 23, 537
542
LIST OF AUTHORS QUOTED
Descartes, E. du P., 4
Dini, U., 374
Dinnik, A., 371
Dirichlet, P. G. Lejeune, 17, 60, 71, 76, 77, 80,
81, 161, 167, 177, 241, 242, 252, 273, 308,
309
Dixon, A. C, 295, 526
Dolbnia, J. P. (Dolbnja, Iwan), 454
Dougall, J., 295
Du Bois Eeymond, P. D. G., 66, 81
Eisenstein, F. G. M., 52, 427
Emde, F., 336, 371
Encke, J. F,, 336
Enneper, A., 448, 521
Euclid, 540
Euler, L., 16, 69, 119, 127, 128, 151, 155, 159,
160, 229, 230, 231, 235, 247, 254, 255, 256,
259, 260, 265, 266, 385, 350, 455, 480, 488,
505, 535
Faber, G., 530
Fagnano, (II Marchese) Giulio Carlo de Toschi
di, 422
Feaux, B., 243
F^jer, L., 161, 164
Ferrers, N. M., 317, 318, 324
Filon, L. N. G., 392
Floquet, A. M. G., 405, 406, 419
Ford, L. E., 448
Forsyth, A. R,, 188, 195, 197, 198, 202, 279,
381, 383, 384, 393, 429, 448, 512, 524, 527,
528
Fourier, J. B. Joseph (Baron), 160, 182, 205,
374, 392, 456
Fr^chet, M., 224
Fredholm, E. I., 206, 207, 209, 224
Freeman, A., 392, 456
Frend, W., 4
Fricke, K. E. E., 474, 480, 501
Frobenius, F. G., 191, 195, 202, 313, 317, 414,
439, 451
Fuchs, L L., 191, 202, 405
Fiiratenau, E., viii
Fuss, P. H., 231
Gambioli, D., 148
Gauss, Carl F. (Johann Friedrich Karl), 7, 9,
16, 234, 240, 241, 242, 275, 277, 288, 290,
291, 302, 313, 422, 455, 505, 517, 526
Gegenbauer, L., 323, 329, 354, 378
Glaisher, James, 336
Glaisher, J. W. L., 484, 487, 489, 491, 500.
502, 509, 516, 520, 523
Gmeiner, J. A., 14
Goldbach, C, 231
Goursat, Edouard J. B., 53, 59, 61, 80, 85,
108, 121, 144
Grace, J. H., 123
Gray, A., 367, 371
Green, George, 387
Gregory, James, 16
Gudermann, C, 484, 487, 524
Guichard, C, 148
Gutzmer, C. F. A., 109
Hadamard, J., 108, 206
Halm, J. K. E., 204
Halphen, G. H., 480, 528
Hamburger, M., 405
Hamilton, Sir William Rowan, 8
Hancock, H., 474, 480, 485
Hankel, H., 10, 238, 261, 359, 362, 363, 364,
367, 371
Hansen, P. A., 370
Hardy, G. H., 10, 17, 38, 50, 59, 69, 81, 156,
159, 266, 291, 508, 530, 531
Hargreave, C. J., 375
Hargreaves, R., 303
Harkness, J., 480
Heffter, L. IF. J., 41
Heine, H. E., 53, 54, 302, 313, 315, 320, 323,
324, 361, 371, 455
Hermite, C, 198, 263, 264, 265, 294, 295, 327,
344, 410, 450, 451, 480, 528
Heun, K., 325
Heymann, K. W., 292
Hey wood, H. B., 224
Hicks, W. M., 394
Hilbert, D., 207, 221, 224
Hill, G. W., 36, 40, 399, 406, 407, 408, 409,
410, 417, 419
Hill, M. J. M., 97, 291
Hobson, E. W., 3, 10, 53, 56, 67, 75, 80, 160,
161, 184, 286, 317, 319, 320, 324, 328, 358,
374, 375, 530, 536, 540
Hocevar, F., 335
Hodgkinson, J., 306
Holder, 0., 66, 230
Hurwitz, Adolf, 161, 175, 259, 262, 263, 533
Hurwitz, Wallie Abraham, 76
Inoe, E. Lindsay, 417, 419, 420
Isherwood, J. G., 371
Jackson, F. H., 455
Jacobi, Karl (Carl) G. J., 39, 108, 109, 124,
308, 362, 422, 455, 456, 457, 460, 461, 462,
403, 467, 468, 471, 472, 473, 480, 481, 482,
484, 487, 489, 491, 498, 501, 502, 503, 505,
507, 509, 510, 512, 513, 515, 516, 521, 522,
528
Jacobsthal, W., 332, 345
Jahuke, P. R. E., 336, 371
Jamet, E. V., 528
Jeti'ery, G. B., 396
LIST OF AUTHORS QUOTED
643
Jensen, J. L. W. V., 264, 273
Je2ek, 0., 147
Jordan, If. E. C, 17, 115, 121, 480
Kalahne, A., 371
Kapteyn, W., 146, 367
Kelvin (Sir William Thomson), Lord, 371,
387, 392, 394
Kiepert, L., 453
Klein, C. Felix, 197, 198, 202, 203, 277, 474,
480, 501
Kluyver, J. C, 142
Kneser, J. C. G. A., 217
Koch, N. F. H. von, 36, 37, 412, 416
Kronecker, L., 123, 456, 461, 518
Kummer, E. E., 244, 256, 279, 290, 292, 332
Lagrange, J. L., 96, 132, 133
Laguerre, E. A'^., 335
Lalesco (Lalescu), T., 224
Lamb, H., 62, 394
Lame, G., 198, 394, 398, 410
Landau, E. G. H., 11, 270, 273, 335
Landen, J., 469, 500, 501
Landsberg, G., 124, 468
Laplace, P. S. (Le marquis de), 205, 306, 308,
379
Laurent, Paul Mathieu Hermann, 123, 315
Laurent, Pierre Alphonse, 99
Leaute, H., 330
Lebesgue, H., 63, 167, 184
Legendre, A. M., 122, 198, 229, 234, 235, 247,
254, 296, 297, 298, 299, 310, 320, 324, 329,
335, 422, 488, 492, 505, 508, 511, 513, 514,
515, 516, 517, 518, 520, 521
Leibniz (Leibnitz), G. W., 67, 532
Lerch, M., 81, 108, 110, 148, 265, 273,
274
Le Vavasseur, R., 292
Levi-Civit^, T., 144
Liapounofif, A., 175
Lie, M. Sophus, 41
Lindelof, Ernst L., 108, 121, 253, 273, 277
Lindemann, C. L. F., 204, 410, 417, 419
Lindstedt, A., 419
Liouville, J., 105, 205, 215, 424, 448, 455
Lipschitz, R. O. S., 371
Littlewood, J. E., 156, 159
Loramel, E. C. J., 351, 358, 370, 371, 373,
374, 376
London, F., 26
Love, A. E. H., 392
McClintock, E., 133
Macdonald, H. M., 121, 329, 375, 376, 378
Maclaurin, Colin, 71, 77, 94, 127, 128, 151,
422
Maclaurin, R. C, 399, 419
Malmst^n, C. J., 243
Mangeot, S., 147
Manning, H. P., 25
Mansion, P., 43
Mascheroni, L., 229
Maseres, Francis (Baron), 4
Mathews, G. B., 367, 371, 529
Mathieu, E. L., 198, 397, 398, 404, 419, 420
Maxwell, J. Clerk, 397
Mehler, F. G., 308, 309, 361, 376
Meissel, D. F. E., 370
Mellin, R. Hj., 280, 290
Merz, J. T., 629
Meyer, F. G., 80
Miidner, R., 148
Milne, A., 225, 345, 348
Minding, E. F. A., 119
Mittag-Leffler, M. G., 134
Molk, C. F. J., 448, 457, 480, 521, 530
Moore, E. H., 230
Morley, F., 295, 480
Miiller, H. F., 448, 521
Murphy, R., 218, 305, 306
Netto, E,, 64
Neumann, Franz Ernst, 314
Neumann, Karl (Carl) Gottfried, 215, 316, 323,
351, 365, 367, 368, 369, 371, 373, 377^
378
Newman, F. W., 230
Newton, Sir Isaac, 532, 534
Nicholson, J. W., 362, 371, 374, 376
Nielsen, N., 142, 253, 371, 374, 377
Niven, Sir William D., 394
Olbricht, R., 318, 324, 328, 371
Oldenburg, H., 532
0!^good, W. F., 45, 49, 59, 72, 87
Painleve, P., 452
Panton, A. W., 206, 355, 446
Papperitz, J. E., 200, 201, 290
Parseval, M. A., 177
Peano, G., vii
Pearson, Karl, 347
Peirce, B. 0., 371
Picard, C. E., 405
Pierpont, J., 75
Pincherle, S., 109, 142, 149, 329
Plana, G. A. A., 146
Pochhammer, L., 250, 286, 293
Pockels, F. C. A., 392
Poincare, J. Henri, 36, 151, 159
Poisson, S. D., 161, 362, 389, 468
Porter, M. B., 354
Pringsheini, A., 17, 26, 27, 28, 33, 38, 243, 253,
530
Prym, F. £., 335
544
LIST OF AUTHORS QUOTED
Eaabe, J. L., 126, 255
Bamanujan, S., 528
Ravut, L., 87
Rayleigh (J. W. Strutt), Lord, 184, 389, 392
Reiff, R. A., 16
Richelot, F. J., 526
Riemann, G. F. Bernhard, 26, 38, 63, 80, 84,
85, 161, 166, 177, 178, 179, 180, 181, 184,
200, 201, 202, 203, 259, 260, 263, 266, 267,
268, 273, 288, 290
Riesz, M., 156
Eodrigues, 0., 297
Routh, E. J,, 326
Russell, Hon. Bertrand A. W., 5, 10, 529
Saalschiitz, L., 237, 238, 295
Salmon, G., 448
Savidge, H. G., 371
Schafheitlin, P., 363
Scheibner, W., 108, 109
Schendel, L., 330
Sobering, E. C. J., 505
Schlafli, L., 207, 324, 327, 328, 351, 356, 357,
366
Scblesinger, L., 202
Schlomilch, O. A'., 142, 144, 159, 223, 236,
253, 258, 335, 346, 349, 369, 370, 531, 532
Schmidt, 0. J. E., 217, 221, 224
Schonholzer, J. J., 372
Schumacher, H. C, 505
Schwarz, K. H. A., 181, 427, 448, 529
Seidel, P. L., 44
Seiffert, L. G. A., 520
Silva, J. A. Martins da, 325
Simon, H., 38
Smith, B. A., 371
Smith, H. J. S., 461, 480, 524
Soldner, J. von, 335, 336
Sommerfeld, A. J. W., 371
Sanine (Sonin, Ssonin), N. J., 346, 357, 375,
376
Stickelberger, L., 439, 451
Stieltjes, T. J., 255, 335, 410, 413, 417, 419
Stirling, James, 94, 151, 245, 247
Stokes, Sir George G., 44, 73, 77, 81, 152, 198,
371
Stolz, O., 10, 14, 27
Stormer, F. CM., 122
Sylvester, J. J., 393, 412, 529
Tait, P. G., 387, 392
Tannery, J., 448, 457, 480, 521, 530, 533
Taylor, Brook, 93
Teixeira, F. G., 131, 132, 146
Thome, L. W., 191, 202, 316
Thomson, Sir William, see Kelvin
Todhunter, I., 198, 324
Transon, A. E, L., viii
Verhulst, P. F., 521
Volterra, V., 207, 212, 215, 224
Walhs, John, 11, 275, 532, 534
Watson, G. N., 43, 53, 59, 77, 108, 159, 292,
345, 347, 419, 455
Weber, H., 198, 225, 336, 341, 345, 375, 480
Weierstrass, Karl (Carl) T. W., 4, 13, 34, 41,
44, 49, 99, 108, 110, 137, 230, 427, 447,
448, 479, 480, 512, 529
Wessel, Caspar, 9
Whitehead, A. N., 10, 529
Whittaker, E. T., 225, 331, 333, 334, 341, 345,
347, 381, 392, 400, 404, 417, 419, 448,
495
Wilson, R. W., 371
Wolstenholme, J., 123
Wronski, J. Hoen6, viii, 147
Young, A. W., 399, 419
Young, W. H., 56
Zach, (Freiherr) F. X. von, 335
GENERAL INDEX
[The numbers refer to the pages. Re/erences to theorems contained in a fetv
of the more important examples are given hy numbers in italics^
Abel's discovery of elliptic functions, 422, 505 ; inequality, 16 ; integral equation, 223, 224: ;
method of establishing addition theorems, 435, 489, 490, 523, 527; special form, ^^(z),
of the confluent hypergeometric function, 347 ; test for convergence, 17 ; theorem on con-
tinuity of power series, 57 ; theorem on multiplication of convergent series, 58, 59
Abridged notation for products of Theta-functions, 461, 462; for quotients and reciprocals of
elliptic functions, 487
Absolute convergence, 18, 28; Cauchy's test for, 21; D'Alembert's ratio test for, 22; De
Morgan's test for, 23
Absolute value, see Modulus
Absolutely convergent double sei'ies, 28; infinite products, 32; series, 18, (fundamental
property of) 25, (multiplication of) 29
Addition formulae, distinguished from addition theorems, 512
Addition formula for Bessel functions, 351, 373; for Gegenbauer's function, 329; for Legendre
polynomials, 320, 387 ; for Legendre functions, 322 ; for the Sigma-function, 444 ; for
Theta-functions, 460; for the Jacobian Zeta-function and for E(ii), 511, 527; for the
third kind of elliptic integral, 516 ; for the Weierstrassian Zeta-function, 439
Addition theorem for circular functions, .535 ; for the exponential function, 531; for Jacobian
elliptic functions, 487, 490, 523; for the Weierstrassian elliptic function, 433, 450; proofs
of, by Abel's method, 435, 489, 490, 523, 527
Affix, 9
Air in a sphere, vibrations of, 391
Amplitude, 9
Analytic continuation, 96, (not always possible) 98 ; and Borel's integral, 141 ; of the hyper-
geometric function, 282. See alao Asymptotic expansions
Analytic functions, 82-110 (Chapter v); defined, 83; derivates of, 89, (inequality satisfied by) 91;
represented by integrals, 92; Riemann's equations connected with, 84; values of, at points
inside a contour, 88; uniformly convergent series of, 91
Angle, analytical definition of, 539 ; and popular conception of an angle, 540
Angle, modular, 485
Area represented by an integral, 61, 540
Argand diagram, 9
Argument, 9, 538 ; principal value of, 9, 538 ; continuity of, 538
Associated function of Borel, 141 ; of Riemann, 177 ; of Legendre [P, ™ (z) and Q^™ {z)], 317-319
Asymptotic expansions, 150-159 (Chapter viii); differentiation of, 153; integration of, 153;
multiplication of, 152; of Bessel functions, 361, 306, 367 ; of confluent hypergeometric
functions, 336, 337 ; of Gamma-functions, 245, 270 ; of parabolic cylinder functions, 342 ;
uniqueness of, 154
Asymptotic inequality for parabolic cylinder functions of large order, 348
Asymptotic solutions of Mathieu's equation, 418
Auto- functions, 220
Automorpbic functions, 448
Axioms of arithmetic and geometry, 529
W. M. A. 35
546 GENERAL INDEX
Barnes' contour integrals for the hypergeometric function, 280,283; for the confluent hyper-
geometric function, 337-339
Barnes' G-function, 258, 272
Barnes' Lemma, 283
Basic numbers, 455
Bemoullian numbers, 126 ; polynomials, 126, 127
Bertrand's test for convergence of infinite integrals, 71
Bessel coefficients [J„ (z)], 100, 349 ; addition formulae for, 351 ; Bessel's integral for, 355 ;
differential equation satisfied by, 351; expansion of, as power series, 350; expansion of
functions in series of (by Neumann), 367, 377, (by Schlomilch), viii, 369; expansion of
{t - 2)~i in series of, 368 ; expressible as a confluent form of Legendre functions, 361 ;
expressible as confluent hypergeometric functions, 352; inequality satisfied by, 372;
Neumann's function O,^ {z) connected with, see Neumann's function ; order of, 351 ; recur-
rence formulae for, 353 ; special case of confluent hypergeometric functions, 352. See
also Bessel functions
Bessel functions, 349-378 (Chapter xvii), J„(2) defined, 353; addition formulae for, 373;
asymptotic expansion of, 361, 366, 367; expansion of, as an ascending series, 353, 364;
expansion of functions in series of, 367, 369, 374 ; first kind of, 353 ; Hankel's integral
for, 35'J ; integral connecting Legendre functions with, 358, 394 ; integral properties of,
373, 378 ; integrals involving products of, 373, 376, 378 ; notations for, 350, 365 ; order
of, 351 ; products of, 372, 373, 376, 378, 421 ; recurrence formulae for, 353, 366, 367 ;
relations between, 354, 364; relation between Gegenbauer's function and, 378; Schlafli's
form of Bessel's integral for, 356, 366; second kind of, Y,,[z) (Hankel), 363, YC) (2)
(Neumann), 365; second kind of modified, K^{z), 367; solution of Laplace's equation by,
388 ; i-olution of the wave-motion equation by, 390 ; tabulation of, 370 ; whose order is
large, 362, 376 ; whose order is half an odd integer, 358 ; with imaginary argument,
I^{z),K^{z), 366, Z(S1,377; zeros of, 354, 360, 371, 373. See also Bessel coefficients and
Bessel's equation
Bessel's equation, 198, 353, 366 ; fundamental system of solutions of (when n is not an in-
teger), 353, 367; second solution when n is an integer, 363, 367. See also Bessel functions
Binet's integrals for log r (z), 242-245
Binomial theorem, 95
Bdcher's theorem on linear differential equations with five singularities, 197
Bolzano's theorem on limit points, 13
Bonnet's form of the second mean value theorem, 66
Borel's associated function, 141; integral, 140; integral and analytic continuation, 141; method
of 'summing' series, 154; theorem (the modified Heine-Borel theorem), 53
Boundary, 44
Boundary conditions, 380 ; and Laplace's equation, 386
Bounds of continuous functions, 55 •
Branch of a function, 106
Branch-point, 106
Burmann's theorem, 129 ; extended by Teixeira, 131
Cantor's Lemma, 177
Cauchy's condition for the existence of a limit, 13; discontinuous factor, 123; formula for
the remainder in Taylor's series, 96; inequality for derivates of an analytic function, 91;
integral, 119; numbei's, 372; tests for convergence of series and integrals, 21, 71
Cauchy's theorem, 85 ; extension to curves on a cone, 87 »
CeU, 423
Ces^ro's method of 'summing' series, 155; generalised, 156
Change of order of terms in a series, 25 ; in an infinite determinant, 37
Change of parameter (method of solution of Mathieu's equation), 417
Characteristic functions, 220 ; numbers, 213 ; numbers associated with symmetric nuclei are
real, 220
GENERAL INDEX 547
Ohsurtier'B test for convergence of infinite integrals, 72
Circle, area of sector of, 540 ; limiting, 98 ; of convergence, 30
Circular functions, 428, 534 ; addition theorems for, 535 ; continuity of, 535 ; differentiation
of, 535 ; duplication formulae, 535 ; periodicity of, 537 ; relation with Gamma-functions,
233
Circular membrane, vibrations of, 389
Class, left (L), 4 ; right (B), 4
Closed, 44
Coefficients, equating, 59 ; in Fourier series, nature of, 173 ; in trigonometrical series, values
of, 163
Coefficients of Bessel, see Bessel coefficients
Comparison theorem for convergence of integrals, 71; for convergence of series, 20
Complementary moduli, 472, 486 ; elliptic integrals with, 472, 494, 513
Complete elliptic integrals [£, K, E', A''] (first and second kinds), 491, 492, 511 ; Legendre's re-
lation between, 513 ; properties of (qua functions of the modulus), 477, 491, 492, 494, 514 ;
series for, ;295 ; tables of, 511 ; the- Gaussian transformation, 5i;;6; values for small values
of I A; I, 514; values (as Gamma functions) for special values of k, 517-520; with comple-
mentary moduli, 472, 494, 513
Complex integrals, 77; upper limit to value of, 78
Complex integration, fundamental theorem of, 78
Complex numbers, 3-10 (Chapter i), defined, 6 ; amplitude of, 9 ; argument of, 9, 538 ; de-
pendence of one on another, 41; imaginary part of (I), 9; logarithm of, 539; modulus of, 8;
real part of (R), 9 ; representative point of, 9
Complex variable, continuous function of a, 44
Computation of elliptic functions, 478
Conditional convergence of series, 18 ; of infinite determinants, 408. See also Convergence
and Absolute convergence
Condition of integrability (Eiemann's), 63
Conditions, Dirichlet's, 167
Conduction of Heat, equation of, 380
Confluence, 196, 331
Confluent form, 197, 331
Confluent hypergeometric function ['^V, «»(-)]> 331-348 (Chapter xv) ; equation for, 331;
general asymptotic expansion of, 336, 339 ; integral defining, 333 ; integrals of Barnes'
type for, 337-339 ; Kummer's formulae for, 332 ; recurrence formulae for, 346 ; relations
with Bessel functions, 352; the functions ^Vj^^j^(z) and Mjc^^{z), 332, 333; the relations
between functions of these types, 340; various functions expressed in terms of Wic,m{^)y
334, 346, 347, 352. See also Bessel functions and Parabolic cylinder functions
Confocal coordinates, 398
Congruence of points in the Argand diagram, 423
Constant, Euler's or Mascheroni's, [7], 229, 240, 242
Constants e^, c-i, 63, 436; E, E', 511, 513 ; of Fourier, 175 ; iji, 772- ^139, (relation between t)^ and
7?2) 439; G, 465; K, 477, 491, 492; K' , 477, 494, 496
Construction of elliptic functions, 426, 471, 485; of Mathieu functions, 402, (second method) 413
Contiguous hypergeometric functions, 288
Continua, 43
Continuation, analytic, 96, (not always possible) 98 ; and Borel's integral, 141 ; of the hyper-
geometric function, 282. See also Asymptotic expansions
Continuity, 42 ; of power series, 57, (Abel's theorem) 57 ; of the argument of a complex variable,
538; of the circular functions, 535; of the exponential function, 531; of the logarithmic
function, 533, 539 ; uniformity of, 54
Continuous functions, 41-60 (Chapter in), defined, 42 ; bounds of, 55 ; integrability of, 63 ; of a
complex variable, 44; of two variables, 67
Contour, 85; roots of an equation iu the interior of a, 119, 123
35—2
548 GENERAL INDEX
Contour integrals, 85; evaluation of definite integrals by, 112-124; the'Mellin-Barnes type of,
280, 337; see also tinder the special function represented by the integral
Convergence, 11-40 (Chapter ii), defined, 14; circle of, 30; conditional, 18; of a double series,
27 ; of an infinite determinant, 86; of an infinite product, 32 ; of an infinite integral, 70,
(tests for) 71, 72; of a series 15, (Abel's test for) 17, (Dirichlet's test for) 17; of the geo-
metric series, 19 ; of the hypergeometric series, 24 ; of the series S?i~*, 19 ; of the series
occurring in Mathieu functions, 415 ; of trigonometrical series, 161 ; principle of, 14 ; radius
of, 80; theorem on (Hardy's), 156. See also Absolute convergence, Non-tmiform convergence
ajid Uniformity of convergence
Coordinates, coufocal, 398 ; orthogonal, 394
Cosecant, series for, 185
Cosine, see Circular functions
Cosine-integral [Ci (z)], 346 ; -series (Fourier series), 171
Cotangents, expansion of a function in series of, 139
Cubic function, integration problem connected with, 445, 505
Cunningham's function [w„,^(^)], 347
Curve, simple, 43 ; on a cone, extension of Cauchy's theorem to, 87 ; on a sphere (Seiffert's
spiral), 520
Cut, 275
D'Alembert's ratio test for convergence of series, 22
Darboux' formula, 125
Decreasing sequence, 12 ^
Dedekind's theory of irrational numbers, 4
Deficiency, 448
Definite integrals, evaluation of, 111-124 (Chapter vi)
Degree of Legendre functions, 296, 301, 318
De la Valine Poussin's test for uniformity of convergence of an infinite integral, 72
De Morgan's test for convergence of series, 23
Dependence of one complex number on another, 41
Derangement of convergent series, 25; of double series, 28; of infinite determinants, 37; of
infinite products, 33, 34
Derivates of an analytic function, 89; Cauchy's inequality for, 91; integrals for, 89
Derivates of elliptic functions, 41^3
Determinant, Hadamard's, 206
Determinants, infinite, 36 ; convergence of, 36, (conditional) 408; discussed by Hill, 36, 408;
evaluated by Hill in a particular case, 408; rearraugement of, 37
Difference equation satisfied by the Gamma-function, 231
Differential equations satisfied by elliptic functions and quotients of Theta-functions, 429, 463,
485; (partial) satisfied by Theta-functions, 463; Weierstrass' theorem on Gamma-functions
and, 230. See also Linear differential equations and Partial diflerential equations
Differentiation of an asymptotic expansion, 153 ; of a Fourier series, 174 ; of an infinite
integral, 74 ; of an integral, 67 ; of a series, 79, 91 ; of elliptic functions, 423, 486 ; of
the circular functions, 535 ; of the exponential function, 532; of the logarithmic function,
534, 539
Dirichlet's conditions, 167; form of Fourier's theorem, 167; formula connecting repeated
integrals, 75 ; integral, 252 ; integral for r// (z), 241 ; integral for Legendre functions,
308 ; test for convergence, 17
Discontinuities, 42 ; and non-uniform convergence, 47 ; of Fourier seiies, 175 ; ordinary, 42 ;
regular distribution of, 20G ; removable, 42
Discontinuous factor, Cauchy's, 123
Discriminant associated with Weierstrassian elliptic functions, 437
Divergence of a series, 15 ; of infinite products, 33
Domain, 44
Double circuit integrals, 250, 287
GENERAL INDEX 549
Double Integrals, 68, 249
Double series, 26 ; absolute convergence of, 28; convergence of (Stolz' condition), 27; methods
of summin<,', 27; a particular form of, 51; rearrangement of, 28
Doubly periodic functions, 422 et seq. See Jacoblan elliptic functions, Theta-functlons and
Weierstrassian elliptic functions
Duplication formula for the circular functions, 535; for the Gamma- function, 234; for the
Jacobian elliptic functions, 491 ; for the Sigma-function, 452, 453 ; for the Theta-functions,
481 ; for the Weierstrassian elliptic function, 434 ; for the Weierstrassian Zeta-function, 452
Electromagnetic waves, equations for, 397
Elementary functions, 82
Elementary transcendental functions, 529-540 (Appendix). See also Circular functions,
Exponential function and Logarithm
Elliptic cylinder functions, see Mathieu functions
Elliptic functions, 422-528 (Chapters xx-xxii) ; computation of, 478 ; construction of, 426, 471 ;
derivate of, 423 ; discovery of, by Abel, Gauss and Jacobi, 422, 505, 517 ; expressed by
means of Theta-functions, 466 ; expressed by means of Weierstrassian functions, 441-444 ;
general addition formula, 450; number of zeros (or poles) in a cell, 424, 425; order of,
425; periodicity of, 422, 472, 493, 495, 496; period parallelogram of, 423; relation be-
tween zeros and poles of, 426 ; residues of, 424, 497 ; transformations of, 501 ; with no
poles (are constant), 424; with one double pole, 425, 427; with the same periods
(relations between), 445 ; with two simple poles, 425, 484. See also Jacobian elliptic
functions, Theta-functions and Weierstrassian elliptic functions
Elliptic integrals, 422, 505 ; first kind of, 508 ; function E (u) and, 510 ; function Z (u) and,
611 ; inversion of, 422, 445, 447, 473, 477, 505, 517 ; second kind of, 510, (addition formulae
for) 511, 512, 527, (imaginary transformation of) 512 ; third kind of, 515, 516, (dynamical
application of) 516, (parameter of) 515 ; three kinds of, 507. See also Complete elliptic
integnrals
Elliptic membrane, vibrations of, 397
Equating coefficients, 59
Equation of degree m has m roots, 120
Equations, iudicial, 192; number of roots inside a contour, 119, 123; of Mathematical Phy-
sics, 197, 379-396 ; with periodic coefficients, 405. See also Difference equation, Integftal
equations, Linear differential equations, and under the names of special equations
Equivalence of curvilinear integrals, 83
Error-function [Erf(x) and Erfc(a;)], 335
Essential singiUarity, 102 ; at infinity, 104
Eta-function [H (u)], 472
Eulerian integrals, first kind of [B (m, n)], 247 ; expressed by Gamma-functions, 248 ;
extended by Pochhammer, 250
Eulerian integrals, second kind of, 235 ; see Gamma-function
Euler's constant [7], 229, 240, 242 ; expansion (Maclaurin's), 127 ; method of ' summing '
series, 155 ; product for the Gamraa-function, 231 ; product for the Zeta-function of
Riemann, 265
Evaluation of definite integrals and of infinite integrals, 111-124 (Chapter vi)
Evaluation of Hill's infinite determinant, 408
Even functions, 115, 171; of Mathieu [ce„(z, q)], 400 ■
Existence of derivatives of analytic function, 89 ; -theorems, 381
Expansions of functions, 125-149 (Chapter vii) ; by Burmann, 129, 131; by Darboux, 125; by
Euler and Maclaurin, 127 ; by Fourier and Bessel, 374 ; by Lagrange, 132, 149 ; by Laurent,
99; by Maclaurin, 94 ; by Vlnna, 145; by Taylor, 93 ; by Wronski, viii, 147 ; in infinite
products, 136 ; in series of Bessel coefficients or Bessel functions, 368, 369, 374, 377 ; in
series of cotangents, 139 ; in series of inverse factorials, 142 ; in series of Legendre
polynomials or Legendre functions, 304,316,5^4,5^5, 329; in series of Neumann func-
tions, 369, 377; in series of parabolic cylinder functions, 345; in series of rational
550 GENERAL INDEX
functions, 134. See also Asjrmptotic expansions, Fourier series. Series, and under the
names of special functions
Exponential function, 531 ; addition theorem for, 531 ; continuity of, 531 ; differentiation of,
532 ; periodicity of, 535
Exponential-integral [Ei(2)], 346
Exponents at a regular point of a linear differential equation, 192
Exterior, 44
External spheroidal harmonic, 396
Factor, Caucliy's discontinuous, 123 ; periodicity-, 456
Factorials, expansion in a series of inverse, 142
Factor-theorem of Weierstrass, 137
F^jer's theorem on the summability of Fourier series, 164
Ferrers' associated Legendre functions {Pn^{z) and Q,^(z)'\, 317
First kind, Bessel functions of, 353 ; elliptic integrals of, 508, (complete) 511, (integration of)
508 ; Eulerian integral of, 247, (expressed by Gamma-functions) 248 ; integral equation
of, 215 ; Legendre functions of, 301
First mean-value theorem, 65, 96
Floquet's solution of differential equations with periodic. coefficients, 405
Fluctuation, 56 ; total, 57
Foundations of arithmetic and geometry, 529
Fourier-Bessel expansion, 374; integral, 377
Fourier constants, 175
Fourier series, 160-187 (Chapter ix) ; coefficients in, 173 ; differentiation of, 174 ; discon-
tinuities of, 175 ; expansions of a function in, 167-169 ; expansions of Jacobian elliptic
functions in, 503, 504; expansion of Mathieu functions in, 402, 404, 407, 413; F^jer's
theorem on, 164; Hurvvitz-Liapounoff theorem on, 175; series of sines and series of
cosines, 171 ; summability of, 164 ; uniformity of convergence of, 169. See also Trigono-
metrical series
Fourier's theorem, Dirichlet's statement of, 167
Fourier's theorem on integrals, 182, 205
Fredholm's integral equation, 207-211, 222
Functionality, concept of, 41
Functions, branches of, 106 ; identity of two, 98 ; limits of, 42 ; principal parts of, 102 ;
without essential singularities, 105; which cannot be continued, 98. See also under the
names of special functions or special types of functions, e.g. Legendre functions. Analytic
functions
Fundamental formulae of Jacobi connecting Theta-functions, 460, 481
Fundamental period parallelogram, 423 ; polygon (of automorphic functions), 448
Fundamental system of solutions of a linear differential equation, 191, 194. See also under
the names of special equatiojis
Gamma- function [r(2;)], 229-258 (Chapter xii) ; asymptotic expansion of, 245, 270; circular
functions and, 233 ; complete elliptic integrals and, 517-520, 528; contour integral (Hankel's)
for, 238 ; difference equation satisfied by, 231 ; differential equations and, 230 ; duplication
formula, 234 ; Euler's integral of the first kind and, 248 ; Eulei-'s integral of the
second kind, 235, (modified by Hankel) 238, (modified by Saalschiitz) 237; Euler's
product, 231 ; incomplete form of, 335 ; integrals for, (Binet's) 242-245, (Euler's) 235 ;
minimum value of, 247; multiplication formula, 234; sei'ies, (Rummer's) 244, (Stirling's)
245 ; tabulation of, 247 ; trigonometrical integrals and, 250 ; Weierstrassian product, 229.
See also Eulerian integrals and Logarithmic derivate of the Gamma-function
Gauss' discovery of elliptic functions, 422, 505, 517; integral for V (z)lT (z), 240; lemniscate
functions, see Lemniscate functions ; transformation of elliptic integrals, 626
Gegenbauer's function [C'^" (z)], 323 ; addition formula, 329 ; differential equation for, 323 ;
recurrence formulae, 324 ; relation with Legendre functions, 323 ; relation involving
GENERAL INDEX 551
Bessel functions and, 378 ; Rodrigues' formula (analogue), 323 ; Schlafli's integral
(analogue), 323
Ctonus, 448
Geometric series, 19
Olalsher's notation for quotients and reciprocals of elliptic functions, 487
Greatest of tbe limits, 13
Green's functions, 387
Hadamard's lemma, 206
Half-periods of Weierstrassian elliptic functions, 437
Hankel's Bessel function of the second kind, Y^{z), 363; contour integral for V(z), 238;
integral for J„(i;), 359
Hardy's convergence theorem, 156 ; test for uniform convergence, 50
Harmonics, solid and surface, 385 ; spheroidal, 396 ; zonal, 296 ; Sylvester's theorem con-
cerning integrals of, 393
Heat, ecjuation of conduction of, 380
Heine-Borel theorem (modified), 53
Heine's expansion of {t-z)~'^ in series of Legendre functions, 315
Hermite's equation, 198, 203, 336, 341. See also Parabolic cylinder functions
Hermite's formula for the generalised Zeta-function f (s, «), 263
Hill's equation, 399, 406-410; Hill's method of solution, 406
Hill's infinite determinant, 36, 40, 408 ; evaluation of, 408
Hobson's associated Legendre functions, 319
Homogeneity of Weierstrassian elliptic functions, 432
Homogeneous integral equations, 211, 213
Hurwitz' definition of the generalised Zeta-function f(s, a), 259; formula for ^{s,a), 262;
theorem concerning Fourier constants, 175
Hypergeometric equation, see Hypergeometric functions
Hypergeometrlc functions, 275-295 (Chapter xiv) ; Barnes' integrals, 280, 283 ; contiguous,
288; continuation of, 282; contour integrals for, 285; differential equation for, 196, 201,
277; functions expressed in terms of, 275, 305; of two variables (Appell's), 294; relations
between twenty-four expressions involving, 278, 279, 284 ; Eiemann's P-equation and,
202, 277; series for (convergence of), 24, 275; squares and products of, 292; value of
F{a, b; c; 1), 275, 287; values of special forms of hypergeometric functions, 292, 295.
See also Bessel functions, Confluent hypergeometric functions and Legendre functions
Hypergeometric series, see Hypergeometric functions
Hypothesis of Riemann on zeros of f (s), 266
Identically vanishing power series, 58
Identity of two functions, 98
Imaginary argument, Bessel functions with [In{z) and A'„(2)], 366, 367, 377
Imaginary part (J) of a complex number, 9
Imaginary transformation (Jacobi's) of elliptic functions, 498, 499 ; of Theta-functions, 124,
467; of 7<;(!<) and Z(u), 512
Improper integrals, 75
Incomplete Gamma-functions [7 {n, x)], 335
Increasing sequence, 12
Indicial equation, 192
Inequality (Abel's), 16; (Hadamard's) 206; satisfied by Bessel coefficients, 372; satisfied by
Legendre polynomials, 297; satisfied by Parabolic cylinder functions, 348; satisfied by
f (s, a), 268, 269
Infinite determinants, see Determinants
Infinite integrals, 69; convergence of, 70, 71; differentiation of, 74; evaluation of, 111-124;
functions represented by, see under the navies of special functions ; representing analytic
552 GENERAL INDEX
functions, 92 ; theorems concerning, 73 ; uniform convergence of, 70, 72, 73. See also
Integrals and Integration
Infinite products, 32 ; absolute convergence of, 32 ; convergence of, 32 ; divergence to zero,
83; expansions of functions as, 136, 137 {see also 'under the names of special functions);
expressed by means of Theta-functions, 466, 481 ; uniform convergence of, 49
Infinite series, see Series
Infinity, 11, 103 ; essential singularity at, 104 ; point at, 103 ; pole at, 104 ; zero at, 104
Integers, positive, 3 ; signless, 3
Integrability of continuous functions, 63 ; Eiemann's condition of, 63
Integral, Borel's, 140 ; and analytic continuation, 141
Integral, Cauchy's, 119
Integral, Dirichlet's, 252
Integral equations, 205-225 (Chapter xi) ; Abel's, 223, 224; Fredholm's, 207-211, 222; homo-
geneous, 211, 213 ; kernel of, 207 ; Liouville-Neumann method of solution of, 215 ; nucleus
of, 207; numbers (characteristic) associated with, 213; of the first and second kinds, 207,
215 ; satisfied by Mathieu functions, 400 ; satisfied by parabolic cylinder functions, 225 ;
Schlomilch's, 223 ; solutions in series, 222 ; Volterra's, 215 ; with variable upper limit,
207, 215
Integral formulae for the Weierstrassian elliptic function, 430 ; for the Jacobian elliptic
functions, 485, 487
Integral functions, 106; and Mathieu's equation, 411
Integral properties of Bessel functions, 373, 378 ; of Legendre functions, 219, 299, 318 ; of
Mathieu functions, 404; of Neumann's function, 378; of parabolic cylinder functions,
344
Integrals, 61-81 (Chapter iv); along curves (equivalence of), 87; complex, 77, 78; differentiation
of, 67; double, 68, 249; double-ch-cuit, 250, 287; evaluation of, 111-124; for derivates of
an analytic function, 89 ; functions represented by, see under the names of the special
functions; improper, 75; lower, 61; of harmonics (Sylvester's theorem), 393; of irrational
functions, 445 ; of periodic functions, 112 ; principal values of, 75, 117 j regular, 195 ;
repeated, 68, 75 ; representing analytic functions, 92 ; representing areas, 61, 540 ; round
a contour, 85 ; upper, 61. See also Elliptic integrals, Infinite integrals, and Integration
Integral theorem, Fourier's, 182, 205 ; of Fourier-Bessel, 377
Integration, 61 ; complex, 77 ; contour-, 77 ; general theorem on, 63 ; general theorem on
complex, 78 ; of asymptotic expansions, 153 ; of integrals, 68, 74, 75 ; of series, 78 ; pro-
blem connected with cubics or quartics and elliptic functions, 445, 505. See also Infinite
integrals and Integrals
Interior, 44
Internal spheroidal harmonics, 396
Invariants of Weierstrassian elliptic functions, 430
Inverse factorials, expansions in series of, 142
Inversion of elliptic integrals, 422, 445, 447, 473, 477, 505, 517
Irrational functions, integration of, 445, 505
Irrational-real numbers, 5
Irreducible set of zeros or poles, 423
Irregular points (singularities) of differential equations, 191, 196
Iterated functions, 216
Jacobian elliptic functions [sn u, en «, dnu], 425, 471, 484-528 (Chapter xxii) ; addition theorems
for, 487, 490, 523, 528; connexion with Weierstrassian functions, 498; definitions of am m,
A(p, snu (sin amw), cnii, dnu, 471, 485, 487; differential equations satisfied by, 463, 485;
differentiation of, 486 ; duplication formulae for, 491 ; Fourier series for, 503, 504, 528 ;
geometrical illustration of, 517, 520 ; general description of, 497 ; Glaisher's notation for
quotients and reciprocals of, 4S7; infinite products for, 501, 525; integral formulae for,
4:8o, 487 ; Jacobi's imaginary transformation of, 498,499; Landen's transformation of,
500; modular angle of, 485 ; modulus of, 472, 485, (complementary) 472, 486 ; parametric
GENERAL INDEX 658
representation of points on curves by, 517, 520, 520, 526', periodicity of, 472, 493, 495, 496;
poles of, 425, 496, 497; quarter periods, K, iK', of, 472, 491, 492, 494; relations between,
485 ; residues of, 497 ; Seiffert's spherical spiral and, 520 ; triplication formulae, 523, 527,
528; values of, when u is ^K, ^iK' or ^(K + iK'), 493, 499, 500; values of, when the
modulus is small, 525. See also Elliptic functions, Elliptic integrals, Lenmiscate fonctions,
Theta-functions, and Welerstrassian elliptic functions
Jacobi's discovery of elliptic functions, 422, 505 ; earlier notation for Theta-functions, 472 ;
fundamental Theta-function formulae, 460, 481 ; imaginary transformations, 124, 467,
498, 499, 512 ; Zeta-function, see under Zeta-function of Jacobi
Jordan's lemma, 115
Kernel, 207
Klein's theorem on linear differential equations with five singularities, 197
Kummer's formulae for confluent hypergeometric functions, 332 ; series for log T (z), 244
LacuDiary function, 78
Lagrange's expansion, 132, 149; form for the remainder in Taylor's series, 96
Lamp's equation, 198 ; generalised, 198 ; solution of, 452, 528
Landen's transformation of Jacobian elliptic functions, 469, 500, 526
Laplace's equation, 379 ; its general solution, 381 ; solutions involving functions of Legendre
and Bessel, 384,388; solution with given boundary conditions, 386; symmetrical solution
of, 392; transformations of, 394
Laplace's integrals for Legendre polynomials and functions, 306, 307, 308, 313, 320, 328
Laurent's expansion, 99
Least of limits, 13
Lebesgue's lemma, 167
Left (L-) class, 4
Legendre's equation, 198, 298 ; for associated functions, 318 ; second solution of, 310. See
also Legendre functions and Legendre polynomials
Legendre functions, 296-330 (Chapter xv); P„(z), Q.„(z), P,i'»(^), Q™™ («) defined, 300, 310, 317,
319; addition formulae for, 322, 387; Bessel functions and, 358, 361, 394; degree
of, 301, 318 ; differential equation for, 198, 300, 318 ; distinguished from Legendre poly-
nomials, 300 ; expansions in ascending series, 305, 320 ; expansions in descending series,
296, 311, 320, 328; expansion of a function as a series of, 328; expressed by Murphy as
hypergeometric functions, 305, 306 ; expression of Q„ (z) in terms of Legendre polynomials,
313, 314, 327 ; Ferrers' functions associated with, 317, 318 ; first kind of, 301; Gegenbauer's
function, C^" (2), associated with, see Gegenbauer's function; Heine's expansion of (t-z)-''-
as a series of, 315 ; Hobson's functions associated with, 319 ; integral connecting Bessel
functions with, 358 ; integral properties of, 318 ; Laplace's integrals for, 306, 307, 313
320, 328 ; Mehler-Dirichlet integral for, 308 ; order of, 318 ; recurrence formulae for,
301, 312; Schlafli's integral for, 298, 300; second kind of, 310-314, 319, 320; summation
of 2;i»P„(2) and S/t''Q„(2), 296, 315; zeros of, 297, 310, 329. See also Legendre poly
nomials and Legendre's equation
Legendre polynomials [P„(2)], 95, 296 ; addition formula for, 320, 387; degree of, 296; differ
ential equation for, 198, 298 ; expansion in ascending series, 305 ; expansion in descending
series, 296, 328 ; expansion of a function as a series of, 304, 316, 324, 325, 326, 329
expressed by Murphy as a hypergeometric function, 305, 306 ; Heine's expansion of
(t-2)~i as a series of, 315; integral connecting Bessel functions with, 358; integral
properties of, 219, 299 ; Laplace's equation and, 384 ; Laplace's integrals for, 306, 308
Mehler-Dirichlet integral for, 308 ; Neumann's expansion in series of, 316 ; numerical
inequality satisfied by, 297; recurrence formulae for, 301, 303; Rodrigues' formula for, 219
297; Schlafli's integral for, 297,298; summation of ^/t»P„(£), 296; zeros of, 297, 310
See also Legendre functions
Legendre's relation between complete elliptic integrals, 513
Lenmiscate functions [sin lemn </) and cos lemu <p], 517
554 GENERAL INDEX
Liapounoff's theorem concerning Fourier constants, 175
Limit, condition for existence of, 13
Limit of a function, 42; of a sequence, viii, 11, 12; -point (the Bolzano-Weierstrass theorem),
13
Limiting circle, 98
Limits, greatest of and least of, 13
Limit to the value of a complex integral, 78
Lindemann's theory of Mathieu's equation, 410
Linear differential equations, 188-204 (Chapter x), 379-396 (Chapter xvni) ; exponents of, 192 ;
fundamental system of solutions of, 191, 194; irregular singularities of, 191, 196; ordinary
point of, 188; regular integral of, 195 ; regular point of, 191; singular points of, 188, 191,
(confluence of) 196 ; solution of, 188, 191, (uniqueness of) 190 ; special types of equations :
— Bessel's for circular cylinder functions, 198, 336, 351, 352, 366; Gauss' for hypergeo-
metric functions, 196, 201,277; Gegenbauer's, 323 ; Hermite's, 198, 836, 341; Hill's, 399,
406 ; Jacobi's for Theta-functions, 463 ; Lame's, 198 ; Laplace's, 379, 381 ; Legendre's for
zonal and surface harmonics, 198, 298, 318 ; Mathieu's for elliptic cylinder functions, 198,
399 ; Neumann's, 377 ; Riemann's for P-functions, 200, 277, 285, 288 ; Stokes', 198 ;
Weber's for parabolic cylinder functions, 198, 203, 336, 341 ; Whittaker's for confluent
hypergeometric functions, 331 ; equation for conduction of Heat, 380 ; equation of Tele-
graphy, 380 ; equation of wave motions, 380, 390, 395 ; equations with five singularities (the
Klein-Bocher theorem), 197 ; equations with three singularities, 200 ; equations with two
singularities, 202 ; equations with r singularities, 203 ; equation of the third order with
regular integrals, 204
Liouville's method of solving integral equations, 215
Liouville's theorem, 105, 424
Logarithm, 533 ; continuity of, 533, 539 ; differentiation of, 534, 539 ; expansion of, 534, 539 ;
of complex numbers, 539
Logarithmic derivate of the Gamma-function [^{z)], 234, 235; Binet's integrals for, 242-245;
circular functions and, 234 ; Dirichlet's integral for, 241 ; Gauss' integral for, 240
Logarithmic derivate of the Riemann Zeta-f unction, 273
Logarithmic-integral function [Liz], 335
Lower integral, 61
Lunar perigee and node, motions of, 399
Maclaurln's (and Euler's) expansion, 127 ; test for convergence of infinite integrals, 71 ;
theorem, 94
Many-valued functions, 106
Mascheroni's constant [7], 229, 240, 242
Mathematical Physics, equations of, 197, 379-396 (Chapter xviii). See aho under Linear dif-
ferential equations and the names of special equations
Mathieu functions [ce„(2, q), se^^{z, q), in„(z, 7)], 397-421 (Chapter xix) ; construction of, 402,
413 ; convergence of series in, 415 ; even and odd, 400 ; expansions as Fourier series, 402,
404, 418; integral equations satisfied by, 400, 402; integral formulae, 404; order of, 403;
second kind of, 420
Mathieu's equation, 198, 397-421 (Chapter xix) ; general form, solutions by Floquet, 405, by
Lindemann and Stieltjes, 410, by the method of change of parameter, 417 ; second solution
of, 406, 413, 420 ; solutions in asymptotic series, 418 ; solutions which are periodic, see
Mathieu functions ; the integral function associated with, 411. See also Hill's equation
Mean-value theorems, 65, 66, 96
Mehler's integral for Legendre functions, 308
Mellin's (and Barnes') type of contour integral, 280, 337
Membranes, vibrations of, 389, 397, 398
Mesh, 423
Methods of ' summing ' series, 154-156
Minding's formula, 119
GENERAL INDEX 555
Minimum value of T{x), 247
Modified Helne-Borel theorem, 53
Modular angle, 485 ; function, 474, (equation connected with) 475 ; -surface, 41
Modulus, 423 ; of a complex number, 8 ; of Jacobian elliptic functions, 472, 485, (complementary)
472, 486 ; periods of elliptic functions regarded as functions of the, 477, 491, 492, 494,
514
Monogenic, 83
Motions of lunar perigee and node, 399
M-test for uniformity of convergence, 49
Multiplication formula for T(z), 234; for the Sigma-f unction, 453
Multiplication of absolutely convergent series, 29 ; of asymptotic expansions, 152 ; of convergent
series (Abel's theorem), 58, 59
Multipliers of Theta-functions, 456
Murphy's formulae for Legendre functions and polynomials, 305, 306
Neumann's definition of Bessel functions of the second kind, 365 ; expansions in series of
Legendre and Bessel functions, 316, 367 ; integral for the Legendre function of the second
kind, 314 ; method of solving integral equations, 215
Neumann's function [0„(2)], 368; differential equation satisfied by, 377; expansion of, 368;
expansion of functions in series of, 369, 377 ; integral for, 368 ; integral properties of,
378 ; recurrence formulae for, 368
Non-uniform convergence, 44 ; and discontinuity, 47
Normal functions, 218
Notations, for Bessel functions, 350, 365 ; for Legendre functions, 319, 320 ; for quotients
and reciprocals of elliptic functions, 487; for Theta-functions, 457, 472, 480
Nucleus of an integral equation, 207 ; symmetric, 217, 222
Numbers, 3-10 (Chapter i) ; basic, 455; Bernoulli's, 126; Cauchy's, 372; characteristic, 213,
(reality of) 220; complex, 6; irrational, 6; irrational-real, 5; pairs of, 6; rational, 3, 4;
rational-real, 5 ; real, 5
Odd fanctions, 115, 171; of M&thien,- [se,^{z, q]], 400
Open, 44
Order (O and o), 11; of BemouUian polynomials, 126; of Bessel functions, 351; of elliptic
functions, 425 ; of Legendre functions, 318 ; of Mathieu functions, 403 ; of poles of a
function, 102 ; of terms in a series, 24 ; of zeros of a function, 94
Ordinary discontinuity, 42
Ordinary point of a linear differential equation, 188
Orthogonal coordinates, 394 ; functions, 218
Oscillation, 11
Parabolic cylinder functions [7)„ (z)], 341 • contour integral for, 343 ; differential equation for,
198, 203, 341 ; expansion in a power series, 341 ; expansion of a function as a series of, 345;
general asymptotic expansion of, 342 ; inequalities satisfied by, 348 ; integral equation
satisfied by, 225 ; integral properties, 344 ; integrals involving, 347 ; integrals repre-
senting, 347; properties when h is an integer, 344, 347, 348; recurrence formulae, 344;
relations between different kinds of [l>,^{z) and D_„_i (±J2)], 342; zeros of, 348. See also
Weber's equation
Parallelogram of periods, 423
Parameter, change of (method of solving Mathieu's equation), 417; connected with Theta-
functions, 450; of a point on a curve, 435, 489, 490, 520, 523, 526; of third kind of
elliptic integral, 515 ; thermometric, 398
Parseval's theorem, 177
Partial differential equations, property of, 383, 384 ; see also Linear differential equations
Partition function, 455
Parts, real and imaginary, 9
556 GENERAL INDEX
Pearson's function [a^, ,„ (z)], 347
P-equatlon, Riemann's, 200, 331 ; connexion with the hypergeometric equation, 202, 277 ; solu-
tions of, 277, 285, (relations between) 288; transformations of, 201
Periodic coefficients, equations with (Floquet's theory of), 405
Periodic functions, integrals involving, 112, 250. -See also Fourier series and Doubly periodic
functions
Periodicity factors, 456
Periodicity of circular and exponential functions, 535-537; of elliptic functions, 422, 427,
472, 493, 495, 496 ; of Theta-functions, 456
Periodic solutions of Mathieu's equation, 400
Period-parallelogram, 423 ; fundamental, 423
Periods of elliptic functions, 422 ; qua functions of the modulus, 477, 491, 492, 494, 514
Pincherle's functions (modified Legendre functions), 329
Plana' s expansion, 145
Pochhammer's extension of Eulerian integrals, 250
Point, at infinity, 103 ; limit-, 13 ; representative, 9 ; singular, 188, 196
Poles of a function, 102 ; at infinity, 104 ; irreducible set of, 423 ; number in a cell, 424 ;
relations between zeros of elliptic functions and, 426 ; residues at, 425, 497 ; simple,
102
Polygon, (fundamental) of automorphic functions, 448
Polynomials, expressed as series of Legendre polynomials, 304 ; of Abel, 347; of Bernoulli, 126 ;
of Legendre, sec Legendre polynomials ; of Sonine, 346
Popular conception of an angle, 539 ; of continuity, 42
Positive integers, 3
Power series, 29 ; circle of convergence of, 30 ; continuity of, 57, (Abel's theorem) 57 ; expan-
sions of functions in, see under the names of special functions; identically vanishing, 58;
Maclaurin's expansion in, 94; radius of convergence of, 30; series derived from, 31;
Taylor's expansion in, 93 ; uniformity of convergence of, 57
Principal part of a function, 102 ; solution of a certain equation, 475 ; value of an integral,
75, 117 ; value of the argument of a complex number, 9, 538
Principle of convergence, viii, 14
Pringsheim's theorem on summation of double series, 28
Products of Bessel functions, 372, 373, 376, 378, 421 ; of hypergeometric functions, 292. See
also Infinite products
Quarter periods K, iK', 472, 491, 492, 494. See also Elliptic integrals
Quartic, canonical form of, 506 ; integration problem connected with, 445, 505
Quasi-periodicity, 438, 440, 456
Quotients of elliptic functions (Glaisher's notation), 487, 504 ; of Theta-functions, 470
Radius of convergence of power series, 30
Rational functions, 105 ; expansions in series of, 134
Rational numbers, 3, 4 ; -real numbers, 5
Real functions of real variables, 56
Reality of characteristic numbers, 220
Real numbers, rational and irrational, 5
Real part (JR) of a complex number, 9
Rearrangement of convergent series, 25 ; of double series, 28 ; of infinite determinants, 37
Reciprocal functions, Volterra's, 212
Reciprocals of elliptic functions (Glaisher's notation), 487, 504
Recurrence formulae, for Bessel functions, 353, 366, 366, 367; for confluent hypergeometric
functions, 346; for Gegenbauer's function, 324; for Legendre functions, 301, 303, 312;
for Neumann's function, 368 ; for parabolic cylinder functions, 344. Sec also Contiguous
hypergeometric functions
Region, 44
GENERAL INDEX 557
Regular, 83 ; distribution of discontinuities, 206 ; integrals of linear differential equations,
195, (of the third order) 204 ; points (singularities) of linear differential equations, 191
Relations between Bessel functions, 354, 364; between confluent hypergeometric functions
ir±t,^(±2) and M;^., ±m(^)' ^■^O; between contiguous hypergeometric functions, 288; be-
tween parabolic cylinder functions D,j(±2) and D_„_j (±12), 342; between poles and
zeros of elliptic functions, 426; between Riemann Zeta-functions f(«) and f(l-«), 263.
See also Recurrence formulae
Remainder after n terms of a series, 15 ; in Taylor's series, 95
Removable discontinuity, 42
Repeated integrals, 68, 75
Representative point, 9
Eo5idues, 111-124 (Chapter vi), defined, 111 ; of elliptic functions, 425, 497
Riemann's associated function, 177, 178; condition of integrability, 63; equations satisfied by
analytic functions, 84; hypothesis concerning ^{$), 266; lemmas, 166, 178, 180; P-equa-
tion, 200, 277, 285, 288, (transformation of) 201, (and the hypergeometric equation) 202,
see also Hypergeometric functions ; theory of trigonometrical series, 177-182 ; Zeta- function,
see Zeta-function (of Riemann)
Riesz' method of ' summing ' series, 156
Right (R) class, 4
Rodrigues' formula for Legendre polynomials, 297 ; modified, for Gegenbauer's function, 323
Roots of an equation, number of, 120, (inside a contour) 119, 123 ; of Weierstrassian elliptic
functions (ei, e^, ^3), 436
Saalschiitz' integral for the Gamma-function, 237
Schlafli's integral for Bessel functions, 356, 366 ; for Legendre polynomials and functions, 297,
298, 300 ; modified, for Gegenbauer's function, 323
ScMomilch's expansion in series of Bessel coefficients, viii, 369; integral equation, 223
Schmidt's theorem, 217
Schwarz' lemma, 181
Second kind, Bessel function of, (Hankel's) 363, (Neumann's) 365, (modified) 367 ; elliptic
integral of [E (xC), Z (u)], 510, (complete) 511 ; Eulerian integral of, 235, (extended) 238 ;
integral equation of, 207, 215; Legendre function of, 310-314, 319, 320
Second mean-value theorem, 66
Second solution of Bessel's equation, 363, 365, (modified) 367 ; of Legendre's equation, 310 ; of
Mathieu's equation, 406, 420; of the hypergeometric equation, 280, (confluent form) 337;
of Weber's equation, 341
Section, 4
Seififert's spherical spiral, 520
Sequences, 11 ; decreasing, 12 ; increasing, 12
Series (infinite series), 15 ; absolutely convergent, 18 ; change of order of terms in, 25 ; con-
ditionally convergent, 18 ; convergence of, 15 ; differentiation of, 79, 92 ; divergence of, 15;
geometric, 19 ; integration of, 78 ; methods of summing, 154-156 ; multiplication of, 29,
58, 59; of analytic functions, 91; of cosines, 171; of cotangents, 139; of inverse factorials,
142; of powers, see Power series; of rational functions, 134; of sines, 172; of variable
terms, 44 (see also Uniformity of convergence) ; order of terms in, 24; remainder of, 15 ;
representing particular functions, sec under the name of the function; solutions of
differential and integral equations in, 188-196, 222; Taylor's, 93. See also Asymptotic
expansions. Convergence, Expansions, Fourier series, Trigonometrical series and Uniformity
of convergence
Set, irreducible (of zeros or poles), 423
Sigma-functions of _\Veierstrass [cr (z), 0-1(2), <ro{z), cr^iz)], 440, 441; addition formula for, 444,
451, 453; analogy with circular functions, 440; duplication formulae, 452, 453; four
types of, 441 ; expression of elliptic functions by, 443 ; quasi-periodic properties, 440 ;
singly infinite product for, 441 ; three-term equation involving, 445 ; Theta-functions con-
nected with, 441, 466, 480 ; triplication formula, 452
558 GENERAL INDEX
Signless integers, 3
Simple curve, 43 ; pole, 102 ; zero, 94
Simply- connected region, 448
Sine, product for, 137. See also Circular fimctions
Sine-integral [Si (z)], 346, -series (Fourier series), 172
Singly -periodic functions, 422. See also Circular functions
Singularities, 83, 84, 103, 188, 191, 196; at infinity, 104; confluence of, 197, 331; equations
with five, 197 ; equations with three, 200, 204 ; equations with two, 202 ; equations with
)•, 203; essential, 102, 104; irregular, 191, 196; regular, 191
Singular points (singularities) of linear differential equations, 188, 196
Solid harmonics, 385
Solution of Riemann's P-equation by hypergeometric functions, 277, 282
Solutions of differential equations, see Chapters x, xviii, and under the names of special
equations
Solutions of integral equations, see Chapter xi
Sonine's polynomial [T^n'' (z)], 346
Spherical harmonics, see Harmonics
Spherical spiral, Seiffert's, 520
Spheroidal harmonics, 396
Squares of hypergeometric functions, ^9^; of Jacobian elliptic functions (relations between), 485;
of Theta-functions (relations between), 459
Statement of Fourier's theorem, Dirichlet's, 167
Steadily tending to zero, 17
Stieltjes' theory of Mathieu's equation, 410
Stirling's series for the Gamma-f unction, 245
Stokes' equation, 198
Stolz' condition for convergence of double series, 27
Strings, vibrations of, 160
Successive substitutions, method of, 215
Sum-formula of Euler and Maclaurin, 128
Summahility, methods of, 154-156 ; of Fourier series, 164 ; uniform, 156
Surface harmonic, 385
Surface, modular, 41
Surfaces, nearly spherical, 326
Sylvester's theorem concerning integrals of harmonics, 393
Symmetric nucleus, 217, 222
Tabulation of Bessel functions, 370; of complete elliptic integrals, 511; of Gamma-functions,
247
Taylor's series, 93 ; remainder in, 95
Teixeira's extension of Burmann's theorem, 131
Telegraphy, equation of, 380
Tests for convergence, see Infinite integrals. Infinite products and Series
Thermometric parameter, 398
Theta-functions [^i [z), %2 {z), ^3 [z), ^4 (z) or ^ (z), 9 (;/)], 455-483 (Chapter xxi) ; abridged nota-
tion for products, 461, 462 ; addition formulae, 460 ; connexion with Sigma-functions, 441,
466, 480 ; duplication formulae, 481 ; expression of elliptic functions by, 466 ; four types
of, 456 ; fundamental formulae (Jacobi's), 460, 481 ; infinite products for, 462, 466, 481 ;
Jacobi's first notation, 9 (u) and H (u), 472 ; multipliers, 456 ; notations, 457, 472, 480 ;
parameters q, t, 456 ; partial differential equation satisfied by, 463 ; periodicity factors,
456 ; periods, 456 ; quotients of, 470 ; quotients yielding Jacobian elliptic functions, 471 ;
relation &i' = &2^3^4' 463; squares of (relations between), 459 ; transformation of, (Jacobi's
imaginary) 124, 467, (Landen's) 469 ; triplication formulae for, 483 ; with zero argument
(^2. ^3, ^4, ^I'j. 457; zeros of, 458
Third kind of elliptic integral, IT (u, a), 515 ; a dynamical application of, 516
GENERAL INDEX 559
Tliird order, linear differential equations of, 204, 292, 411, 421
Tliree kinds of elliptic integrals, 507
•mree-term equation involving Sigma-functions, 445
Total fluctuation, 57
Transcendental functions, see under the names of special functions
Transformations of elliptic functions and Theta-functions, 501 ; Jacobi's imaginary, 467, 498,
499, 512 ; Landen's, 469, 500 ; of Eiemaun's P-equation, 201
Trigonometrical equations, 537, 538
Trigonometrical integrals, 112, 257; and Gamma-functions, 250
Trigonometrical series, 160-187 (Chapter ix) ; convergence of, 161 ; values of coefficients in,
163 ; Riemanu's theory of, 177-182. See also Fourier series
Triplication formulae for Jacobian elliptic functions and E (ii), 523, 527 ; for Sigma-functions,
452; for Theta-functions, 483; for Zeta-functions, 452
Twenty-four solutions of the hypergeometric equation, 278 ; relations between, 279, 282, 284
Two-dimensional continuum, 43
Two variables, continuous functions of, 67 ; hypergeometric functions (Appell's) of, 294
Unicursal, 448
Uniformisation, 447 .
Uniformising variable, 448
Uniformity, concept of, 52
Uniformity of continuity, 54 ; of summability, 156
Uniformity of convergence, 41-60 (Chapter iii), defined, 44 ; of Fourier series, 169 ; of infinite
integrals, 70, 72, 73 ; of infinite products, 49 ; of power series, 57 ; of series, 44, (condi-
tion for) 45, (Hardy's test for) 50, (Weierstrass' iH-test for) 49
Uniformly convergent infinite integrals, properties of, 73; series of analytic functions, 91,
(differentiation of) 92
Uniqueness of an asymptotic expansion, 154 ; of solutions of linear differential equations, 190
Upper bound, 55 ; integral, 61
Upper limit, integral equation with variable, 207, 215 ; to the value of a complex integral,
78, 91
Value, absolute, see Modulus ; of the argument of a complex number, 9, 538 ; of the co-
efficients in trigonometrical series, 163 ; of particular hypergeometric functions, 275, 287,
292, 295; of Jacobian elliptic functions of i/iT, ^iA", ^{K+iK'), 493, 499, 500; of K, K'
for special values of k, 514, 517, 518 ; of f (s) for special values of s, 261, 263
Vanishing of power series, 58
Variable, uniformising, 448; terms (series of), see Uniformity of convergence; upper limit,
integral equation with, 207, 215
Vibrations of air in a sphere, 391 ; of circular membranes, 389 ; of elliptic membranes, 397,
398; of strings, 160
Volterra's integral equation, 215 ; reciprocal functions, 212
Wave motions, equation of, 380 ; general solution, 390, 395 ; solution involving Bessel
functions, 390
Weber's equation, 198, 203, 336, 341. See also Parabolic cylinder functions
Weierstrass' factor theorem, 137 ; 3/-test for uniform convergence, 49 ; product for the
Gamma-function, 229 ; theorem on limit points, 13
Weierstrassian elliptic function [^(2)], 422-454 (Chapter xx), defined and constructed, 4:25,
426 ; addition theorem for, 433, (Abel's method) 435 ; analogy with circular functions,
431; definition of {(^[z)- e^)^, 444; differential equation for, 429; discriminant of, 437;
duplication formula, 434 ; expression of elliptic functions by, 441 ; expression of ^(z) - ^t) (y)
by Sigma-functions, 444 ; half-periods, 437 ; homogeneity properties, 432 ; integral formula
for, 430 ; integration of irrational functions by, 445 ; invariants of, 430 ; inversion
560 GENERAL INDEX
problem for, 477; Jacobian elliptic functions and, 498; periodicity, 427; roots ej, 62, 63,
436. See also Sigma-functlons and Zeta-function (of Weierstrass)
WMttaker's function 'Fj., „j(z), see Confluent hypergeometric functions
Wronski's expansion, viii, 147
Zero argument, Thetafunctions with, 457; relation between, 463
Zero of a function, 94 ; at infinity, 104 ; simple, 94
Zeros of a function and poles (relation between), 426 ; connected with zeros of its derivate,
121, 123; irreducible set of, 423; number of, in a cell, 424; order of, 94
Zeros of functions, (Bessel's) 354, 360, 371, 373, (Legendre's) 297, 310, 329, (parabolic
cylinder) 34:8, (Eiemann's Zeta-), 262, 263, 266, (Theta-) 458
Zeta-function, Z (m), (of Jacobi), 611; addition formula for, 511; connexion with E (u), 511;
Fourier series for, 513 ; Jacobi's imaginary transformation of, 512. See also Jacobian
elliptic functions
Zeta-function, f (s), f(s, a), (of Riemann) 259-274 (Chapter xiii), (generalised by Hurwitz) 259;
Euler's product for, 265 ; Hermite's integral for, 263 ; Hurwitz' integral for, 262 ; in-
equalities satisfied by, 268, 269 ; logarithmic derivate of, 273 ; Riemann's hypothesis
concerning, 266 ; Riemann's integrals for, 260, 267 ; Riemann's relation connecting f (s)
and f{l-s), 263; values of, for special values of s, 261, 263; zeros of, 262, 263, 266
Zeta-function, f(z), (of Weierstrass), 438; addition formula, 439; analogy with circular
functions, 439; constants rji. 172 connected with, 439; duplication formulae for, 452; ex-
pression of elliptic functions by, 442 ; quasi-periodicity, 438 ; triplication formulae, 452.
See also Weierstrassian elliptic functions
Zonal harmonics, 296
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