Cambruige Tracts in Mathematics
and Mathematical Physics
GlNIRAL EdITOM
G. H. HARDY, M.A., F.R.S.
J. G. LEATHEM, M.A.
No. 15
Complex Integration and Cauchy's
Theorem
CAMBRIDGE UNIVERSITY PRESS
C. F. CLAY, Manager
ILontlon: FETTER LANE, E.G.
Ebinfaurg!) : loo PRINCES STREET
»rrlin: A. ASHER AND CO.
ILeipjifl: F. A. BROCKHAUS
{fltin 8ork: G. P. PUTNAM'S SONS
BombaB anli ffalctttta: MACMILLAN AND CO., Ltd.
STornnto: J. M. DENT AND SONS, Ltd.
arokoa: THE MARUZEN-KABUSHIKI-KAISHA
//// rtif/ifs reserved
COMPLEX INTEGRATION
AND
CAUCHY'S THEOREM
by
G. N. WATSON, M.A.
FcUow of Trinity College, Cambridge
Cambridge :
at the University Press
1914
^Cambritigc :
PRINTED BY JOHN CLAY, M.A.
AT THE UNIVERSITY PRESS
Q1\
331
PREFACE
'^J'Y object in writing ihifi Tract wam to collect into a single
"*• volume those propositions which are employed in the
oourBe of a rigorous proof of Cauchy's theorem, together with
a brief account of some of the applications of the theorem to
the evaluation of definite integrals.
My endeavour has been to place the whole theory on a
definitely arithmetical basis without appealing to geometrical
intuitions. With that end in view, it seemed necessary to
include an account of various propositions of Analysis Situs,
on which depends the proof of the theorem in its most general
form. In proving thene propositions, I have followed the general
course of a memoir by Ames ; my indebtedness to it and to the
textbooks on Analysis by Ooursat and by de la Vall^ Poussin
will be obvious to those who are ac(fuainted with those works.
I must express my gratitude to Mr Hardy for his valuable
criticisms and advice ; my thanks are also due to Mr Littlewood
and to Mr H. Townshcnd, B.A., Scholar of Trinity College, for
their kindness in reading the proofs.
O. N. W.
Trinity Collbgk,
Frhmary 1914.
Digitized by tine Internet Arcinive
in 2007 witii funding from
IVIicrosoft Corporation
littp://www.arcliive.org/details/complexintegratiOOwatsuoft
CONTENTS
1 NTRODUCTION
CBAt.
I. Analysis 8iTt7S
II. Complex Introration .
III. CAUCHT'g ThKOREU
IV. MlHCKLLANBODS THEOREMS
V. The Calculus of Residues.
VI. The Evaluation op Definite Integrals
VII. Expansions in Series ....
VIII. Historical Summary ....
PAMI
1
3
17
30
41
46
64
73
77
INTRODUCTION
1. TiiRoi'oiiouT the tract, wherever it has seemed Advisable, for
the ttake uf cleamosK and brevity, to uae Uie la»Kuage of geometry,
I have not liesitated to do so ; but the reader should convince himself
that all the arguments employeil in Chapters I— IV are really arith-
metical arguments, and are not baneil on geometrical intuitions. Tliiia,
DO OM is made of the geometrical conception of an angle ; when it is
neoenary to define an angle in Chapter I, a purely analytical definition
is given. The fundamental theorems of the arithmetical theory of
liroitA are a^tumed.
A number of obvious theorems are implicitly left to the reader ;
eg. that a circle is a ' simple ' curve (the coordinates of any point on
ji* + y - 1 may be written x = cos ^ ^ = sin /, 0 ^ < ^ 2») ; tliat two
* simple ' curves with a common end-point, but with no other common
point, together fonu one ' simple ' curve ; and several others of a like
nature.
It in t4> \tc noted ttuit aluKiKt all the diffictilticH, whieli arixc in thoae
pn>)»lcuut of AHotjftit iSV/uj which arc liiHcuKHed in Chiipter I, disappear if tho
ctin'OH wliich arc em ployed in the following chapterH arc rcHtricted to he
Ktraight lines <ir cin-leH. This fiu-t in of houic pnu.-tiail in))>ortiiiice, Hiiiue, in
appliuatioiiM of CattchyH Theorem, it is iiMiially iMMwihle to employ only straight
liiKM aiMl circular hjkh uh cout4>iirH <»f integration.
2. Notation. If ; be a complex number, we shall invariably writ«
where x aiul tf are real ; with this definition of .r and y, we write'
x=y?(x), y = /(«).
If a complex number be denote<l by z with some suffix, it^ real and
imaginar>' partii will be denoted by x and y, rw»i»ec lively, with the same
suffix ; e.g.
s. = ^. + «>• ;
' The ajmboU R and / ar« read 'real part of and ' imat^inarj part oC
mpectivvly.
w. c. I. 1
2 INTRODUCTION
further, if ^ be a complex number, we write
Definitions. Paint. A 'point' is a value of the complex variable,
z'y it is therefore determined by a complex number, z, or by two
real numbers (iv, y). It is represented geometrically by means of the
Argand diagram.
Variation and Limited Variation^. If /{x) be a function of a
real variable x defined when a%x%b and if numbers ^i, a-j, ... a?„ be
chosen such that a^Xi^X2 •■• ^Xn'^b, then the sum
I /(^i) -/(«) I + 1 A^.) -/(^i) I + 1 A^z) -A^.) ! + ..• + 1 /(6) -/ W I
is called the variation off{x)for the set of values a, Xi, 0^2, ■■■ Xn, b.
If for every choice oi Xi, x^, ■•■ x„, the variation is always less than
some finite number A (independent of w), /(x) is said to have limited
variation in the interval a to b ; and the upper limit of the variation
is called the total variation in the. interval.
[The notion of the variation of f{.v) in an interval a to b is very much
more fundamental than that of the length of the curve y=/(.r) ; and through-
out the tract propositions will be proved by making use of the notion of
variation and not of the notion of length.]
2 Jordan, Cours d^ Analyse, §§ 105 et seq.
CHAPTER I
ANALYSIS SITUS
ft 3. Problenui uf AnalyM situs tu be diHciuwed. — § 4. Definitionn. — § fi. Pro
pertieB of oodUiiua. — § 6. TheurentH concenung the order, of a point.
— §. 7. Main ihoorem ; a r^^lar cl<M6d curve liati an interior and an
exterior. — § 8. MiHoellaneouM theorems ; definitionM of ouunterolockwiite
and orientati«>n.
3. The object of the present chapter is to give formal analytical
proofi) of various theorems of which simple cases seem more or less
obvious from geometrical considerations. It is convenient to summarise,
for purposes of reference, the general course of the theorems which
will be provetl:
A »impU eurr* m detenuined by the equations x^x(t),if ='y{t) (where t varieH
fntni to to 7^, the functioim x (t), y {t) lieing continuoiu ; and the cun^e has
no double |K>intM Have ({XMMibly) itM end ]>otntH ; if thetie coincide, the cun'e is
said t*> Iw doted. The order of a |xiint Q with respect to a cIiMod cun'e i«
defined to be n, where 2«rn is the amount by which the angle Ijetween QP aiul
Ox increaseH as P describes the cun-e onoe. It is then shewn that points in
the plane, not on the curve, can lie divided into two sets ; points (»f the firHt
Mt have order ± 1 with re.s])ect to the curve, points of the second Hct hiive
order zero ; the first set is ciilled the interior of the cun-e, and the second the
exterior. It is shewn that fvrry simple curve joining an interior point to an
exterior |toint must meet the given cun'e, but that simple cunea can l«
drawn, joining any two interior |)oints (or exterior jwtints), which have no
point in common with the given curve. It i.>s of course, not obvious that a
cloNod cune (defined as a cun-c with coincident end points) divides the plane
into two regions iMtssoMsing thcMC proiwrtios.
It is then ixMsiblc to distinguish the directit>n in which P describes the
cune (vix. ctiimterclockwiMO or cUxikwisc) ; the criterion which detenninc»«
the direction is the sign uf the order of an interior (loint.
The investigation just suuunarised is that due to Ames' ; the analysis
which will be given follows Iuh memoir closely. Other pnMjfiH that a closed cun'e
' Amw, AtHericoH Journal of Mathtmatitt, Vol. xxvn. (1905), pp. 84S-380.
1—2
4 ANALYSIS SITUS [CH. I
possesses an interior and an exterior have been given by Joi-dan^, Schoenflies^,
Bliss*, and de la Vall^ Poiissin'. It has l^een pointed out that Jordan's
proof is incomplete, as it assumes that the theorem is true for closed
polygons ; the other proofs mentioned are of less fundamental character than
that of Ames.
4. Definitions. A simple curve joining two points Zq and Z is
defined as follows:
Let" x = x{t\ y = y{t\
where x{f), y{t) are continuous one-valued functions of a real para-
meter t for all values of t such that' ^o ^ ^ ^ ^> the functions x (t), y (t)
are such that they do not assume the same pair of values for any two
different values of t in the range to<t < T; and
Zo^x(to) + iyito), Z = x{T)^iy{T).
Then we say that the set of points (x, y), determined by the set of
values of t for which ^o ^ ^ ^ ^> is a simple curve joining the points s©
and Z. If Zo = Z, the simple curve is said to be closed^.
To render the notation as simple as possible, if the parameter of
any particular point on the curve be called t with some suffix, the
complex coordinate of that point will always be called z with the same
suffix; thus, if
to^tr^'^HT,
we write c,<"> = x (^r<"0 + iy (^r<"0 = ^r*'*^ + iyr^'*^.
Regular curves. A simple curve is said to be regular^, if it can be
divided into a finite number of parts, say at the points whose para-
meters are ^i, ^2. ••• t„i where ^0 '^ ^1 ^ ^2 ^ ••• ^ ^m "^ ^j such that when
2 Jordan, Cours d'Analyse (1893), Vol. i. §§ 96-103.
3 Schoenflies, Gottingen Nachrichten, Math.-Phys. Kl. (1896), p. 79.
* Bliss, American Bulletin, Vol. x. (1904), p. 398.
» de la Vallde Poussin, Conm d' Analyse (1914), Vol. i. §§ 342-344.
^ The use of x, y in two senses, as coordinates and as functional symbols,
simplifies the notation.
' We can always choose such a parameter, t, that t^ <c T; ior \f this inequality
were not satisfied, we should put t = -t' and work with the parameter t'.
8 The word ' closed ' except in the phrase ' closed curve ' is used in a dififerent
sense ; tucloted set of points is a set which contains all the limiting points of the
set ; an open set is a set which is not a closed set.
0^ We do not follow Ames in assuming that x(t), y{t) possess derivatives with
regard to t.
8^] ANALYM18 HITUS 5
tr-t'it^ir. the r«U«ioii baiween * awl jr givw bjr the eqiMlioM
*-#(lX y-f (0 » eqniwilent to en equation jf-/(') w «fa»
'• 4(^X wher«>/ or 4 denotM a oontinuous one-valued function of ite
argoment, and r takes in turn the Taluen 1. 2, ... w + 1. while /«♦, » 7*.
It ia «MV to nee thai a chain of a finite nuin»*r of cunrea, giveo by the
6qtiati«nwt
|r-/i('). a,<*<a,^
*-/t(y). &i<y<ft.
.(A)
(where 6,-/i(<it)« «»-/t(Ai), .• »»»d /i. /t. ... are oonUnitouii one-valued
function* of their ar^piutentii), forma a simple cunre, if the chain haa no
doul>le fiointi* ; for we niay chiMMe a parameter f, auoh that
x-<, jf-/,(0. OiKt^fh;
If Hoine of the ineqiialitieN in eipuitiona (A) be reversed, it in poaable to ahew
in the Hame manner tliat tho cluiin forma a iiin)|>lc curve.
Elementary curves. Each of the two curves whose etiuatione are
(i) y = /(-r), {jr.^Jr^ *,) and (ii) * = * (jf ), (y. ^ y ^ yi), where / and ^
denote one-valued continuous functions of their respective arguments,
is calle<l an elementary curve.
Primitive ft^od. In the case of a closed simple curve let
u»= T-t»; we define the functions x(t),y(t) for all rwd values of /
by the relations
where n is any integer; m is calle<l the primitive period of the pair of
functions x(t), y{t).
Angle*. If £•, ;, be the complex coordinates of two distinct pointe
/^, /*!, we say that * /'«/'i makes an angle 9 with the axis of jr ' \(B
satisfies Itotk the eciuations**
cos tf - « (x, - X,), sin 9-^ K (y, - jr.),
where « is the positive number {(x, - x,)* + (yi - y.)*} ' *. This {lair of
equations has an infinite numlier of solutions such that if 6^ 9' be any
)* It ia suppoeed that the »ine and eo«tn« Mv defined hj the method indieatod
by Bromwlch. Thtotyof In/imiU .Sftin. f 60. (1) ; it ia eaay to deduce the abatemepU
made coooeminK tha solution* of the two equation* in queation.
6 ANALYSIS SITUS [CH. I
two different solutions, then {0'-6)l2ir is an integer, positive or
negative.
Order of a point. Let a regular closed curve be defined by the
equations x = x{t), y=y{t)y (to^t%T) and let w be the primitive
period of x (t), y (t). Let Q be a point not on the curve and let P
be the point on the curve whose parameter is t. Let B (f) be the
angle which QP makes with the axis of x\ since every branch of
arc cos {k (;ri - a-o)} and of arcsin {»c(yi-yj} is a continuous function
of ^, it is possible to choose 6{t) so that 6{t) is a continuous function
of t reducing to a definite number ^(^o) when t equals ^o- The
points represented by the parameters t and t + w are the same, and
hence B{t), d{t-^oi) are two of the values of the angle which QP
makes with the axis of x ; therefore
e{t + i^)-6{t)-- 2wx,
where n is an integer ; n is called the order of Q with respect to the
curve. To shew that n depends only on Q and not on the particular
point, P, taken on the curve, let t vary continuously; then d{t), $(t + o})
vary continuously; but since n is an integer n can only \&ry per saltus.
Hence n is constant".
5. CoNTiNUA. A two-dimensional continuum is a set of points
such that (i) if Zo be the complex coordinate of any point of the set,
a positive number 8 can be found such that all points whose complex
coordinates satisfy the condition \z- Zo\<h belong to the set ; S is a
number depending on Zq, (ii) any two points of the set can be joined
by a simple curve such that all points of it belong to the set.
Example. The points such that | « | < 1 form a continuum.
" This argument really assumes what is known as Goursat's lemma (see § 12)
for functions of a real variable. It is proved by Bromwich, Theory of Infinite
Series, p. 394, example 18, that if an interval has the property that round every
point P of the interval we can mark oflf a sub-interval such that a certain inequality
denoted by {Q, P\ is satisfied for every point Q of the sub-intei-val, then we can
divide the whole interval into a finite number of closed parts such that each part
contains at least one point Pi such that the inequality {<^, P\} is satisfied for all
points Q of the part in which Pj lies.
In the case under consideration, we have a function, (f> (t) = 6 {t -^ u) - 6(t),
of t, which is given continuous ; the inequality is therefore \ <f>{t) - (f> (t') | < e ,
where e is an arbitrary positive number ; by the lemma, taking « < 2t, we can
divide the range of values of t into a finite number of parts in each of which
\4>(t) - <t> (^i) I < 2ir and is therefore zero ; ^ (t) is therefore constant throughout
each part and is therefore constant throughout the sub-interval.
4-5] amalths srruH 7
XmgAitotirhuud^ Xmr. If • point { be coniiectod with • Mi of
pointt in Mudi a way that * Mqaenoe (<•), nnmirting of points of the
•01, OAO be choeen mieh that { i« a limiting point of the leqaenoe, then
the point C i* iiaid to have points of the let in its nmgkbomrkood.
The itatenient ' all pointM imjficitntlif nnir a (mint C have a certain
property ' means that a positive number h exists such that all points t
iatisfjriuf; the inequality z-i<h have the property.
Boundarim^ Imtfrior amd Ertrrittr PmnU. Any point of a con-
tinuum it) railed an intrrior point. A point is said to be a lutuHdaiy
poimt if it lit not a {mint of the continuum, but has pointu of the
oontinuum in itM neighbourhou<l.
A potDt <,, mich that iibl*l« >* a boiind*ry |ioint of the onntiiiutun de-
fined by |«|<1.
A point which is not an interior point or a boundary point is called
an tJtierior point
If (O ^* * aeqiiciioe of pointJi belonging to • oi>ntiniiiun, then, if this
^HHpMtwe hA« a liuiitiiig point (, the pr>int ( im either on interior {mint or a
boundaiy }Miint ; for, even if ( \n not ah interior {mint, it hafl |>ointa of
the otNitiniuini in iu ii«ighbi>urh<Mid, viz. ]M>intM of the Heqiicncc, aim! w there-
fare • boundar}' )>oint.
AU point* ntficienUjf n«ar an exterior point are 4JUnor points ; for let <,
be «n exterior |>oint ; then, if ho {Kwitivc niinitier h ex\»\» Hiich that all {Miint«
Mtiafying the inequality ' t — t^\<h arc exterior (MtintM, it iM (xjiutible t4> find a
Mei{iience ((.) such that (. iH an interior |ioint or a Imundary point and
if,— <,I<i-»; and, whether f, in an interior |»oint or a ttouodary point, it ia
p'^H* to find an interior point (,' Kiich tliat ,C,'-C«i<2"'; m> thai
\(^'-to < 2' " ", and z, i« the limiting point tif the )*oquence f,' ; therefore «« ia
an interior point or a iKNuidary |»oint; thin iH eontnuy U* hyiMitheata; there-
fore, correN|tonding to any |iarticnilar |»oint :«, a ntinilicr h cxiNta. The theorem
IN therefore im)\'ed.
A oontinuum ix calletl lui" optrn rnjiim, a continuum with its
boundary in a r/iMk^i region.
ffmnmtpl^ lift S be a set of point* t ( ■• x -f ijf) denned bjf the rdations
jr-,<*<jr„ y-/(jr)+r (1),
wktrefis ome-evitud and eontinmotu, r lakea a// mine* mcA tAat 0<r<i, and
k i$ constant. TAen tke set o/ points S forms <i fontinnnm.
>* 8m note S on p. 4.
ANALYSIS SITUS
Let z' be any point of •S', so that
x^<oii < .ri , / =/(aO + r', where 0 < / < k.
[CH. I
Choose, e > 0, so that
2€ <r' <X'-2f (2).
Since /is continuous we may choose 8 > 0, so that
l/(-^0-/(-^')l<* (2a),
when \x — oi^\<h. It is convenient to take 8 so small that
j?o + ^ <.r'< .r,-8 (3).
Then otq < .t < x^ since | x—x' \ < 8.
Also, when \x — x'\<.hy ^■
/(:r)-e</(.r')</(.r) + e (3a),
so that if y be any number such that
y'-€<y<y' + f (4),
then f{^x')-\-r'-(<y<f{x') + r'-\-f (4a).
Adding (2), (3a) and (4a), we see that
f{x)<y<f{x)^-k.
Therefore the pt)int z=x + iy, chosen in this manner, is a point of the set iS*.
Hence, if S' be the smallei- of 8 and e, and if
\z-z\<b;
the conditions (2a) and (4) are both siitisfied, and hence z is a member of the
set. The first condition for a continimm is, consequently, Siitisfied.
Further, the points sf, z" (for which r' ^ r"), Ijelonging to S, can be joined
by the simple curve made up of the two curv&s defined by the relations
(i) .r=y, iy' ^y^y'+r"-t^),
(") y =f{^) + '■"> {^' < •^' < •^" or x" < X ^ A**).
Hence <S is a continuum.
b^] AXALYKIB RirVB 9
0. LntMA. Amif Umithg pmmt ^ »/ a ml of poimU om m timfU emrtt
lim ON tA4 rirrM.
Take *ity Mequmw ct ihm mH which Ium V •* '** unique HmtUiig point;
hH th0 panuueteni uf the pointe of the eaqueiioe be U, r,, ..^ Then the
•equenee (I.) bee at leiuit*' one limit r, and 1^ <r < T. Since x(0. jr (0 «i«
oontinuoua ftinctione. lint jr(0*'(vX li<ny(0"|f(W: <uid (jr(r), jf(r)) ieon
the cune nince r«<r<r: LA^ieno the ourve.
CotoUarf. If V^ be e fi&ed point not on the curve, the disUnoe of V* from
pointM mi the ourre haa e poeittTe lower limit i. Fur if i did i>ot exint we
ooaM find a aequenoe (P.) of |MiinU imi the cune such that (^Z*. • i < | ^^at
ao that ^ would U « limiting |M>int of the aequeiMW and would therafore Ue
on theeunre.
Thborkm I. I/a point if o/ordir n with rmptct to a eiomd m'mpU
mure, aU pnutt fufficinttljf ntar it art qfordeir n.
Let Q, be a point not on the curve and Qi any other point.
Then the distance of points on the cur>'e from <^, lias a pOHJtive lower
limit, 5; so tliat, if Q,Qi ^ A8, the line (?,<^i cannot meet the curve.
Let t be the parameter of any point, /*, on the given curve, and r
the {Mtrameter of a jwint^ Q, on <?.Q,, ami 6 (t, t) the angle QP makes
with the a.xi.s of x ; then 6 (/, t) is a continuous function of t, when / i*
fixed; therefore
<>(/ + «, T)-tf(/.T)
is a continuouft function'* of r; but the onier of a point (being an
integer) can only var}* per Mltim ; therefore B{t + •», t) — B(t, t) ia a
oonstaot, ao far tk» variations of r are concerned ; therefore the orders
of Q», Q, arc the tsame.
Tlie above argument has obviously proved the following more
general Uieorem:
TllBORKM II. //■ tiro fniintu Q,, Qt ran b* joinrd by a nimfile cmrrr
karing mt pifint in nnnmun irith a t/ivrn cUmtd $implr currr, tkf arder$
'tf' V«» U\ •'*»'^ regard to the cliteed curve are the tawte.
The following theorem ia now evident :
TilEORKM III. // tint poiiita Q„ Q^ A/ifv different ardent tritk
regard to a given rltt$fd simple riirre, every *imftie rurre Joining fAem
kaf at ieast one point in ctmimon tcitk the giren cto»ed rurre.
TlIRoRKM IV. Witkinan arhitrariijf itmall distance o/ any pt>int, P,,
of a regular chueti rurre, tkerr are tirtt points wkoite (trder* diffrr hy unity.
The cur>'e consiMts of u finite number of {tartu, each of which («u be
^ Yoang, SeU oi Poimu. pp. IH. 19. >« 8«e not* II on p. 6.
10
ANALYSIS SITUS
[CH. I
represented either by an equation of the form y =f(a;) or else by one
of the form x ^J (i/), where ./' is single-valued and continuous. First,
let Pq be not an end point of one of these parts.
Let the part on which Po lies be represented by an equation of the
form y = /(.r); if the equation be x=f{y), the proof is similar.
The lower limit of the distance of Po from any other part of the
curve ^"^ is, say, rj, where i\ > 0.
Hence if 0 < r < rj , a circle of radius r, centre Po, contains no
point of the complete curve except points on the curve y =/(dr) ; and
the curve y =/(^) meets the ordinate of P,, in no point except Po.
Let B be the point {x«, y^ + r), Pi the point {x^, y^ - r).
If P be any point of the curve whose parameter is t and if 6 (t), Oi (t)
be the angles which BP, Pi P make with the x axis, it is easily verified
that if BP = p, B^P = p, and <^ = ^ {t) - 6, (t),
{x-x^y + iy-y^J-i^
. , 2r(x-Xo)
sin ^ = - — ^^ ,
PPi
cos <^-
ppl
If w be the period of the pair of functions x (t), y (t) and if 8 be so
small that the distances from Po of the points whose parameters are
to ±8 are less than r, then", if x (to + 8)>x {t„),
4> (to) = (2wi +1)^, i> (t, 4- 8) > (2», + l)7r,
<f>(to + oi-8)< (2n.i + ])ir, <f>(to + <o) = (2th + 1)t.
" If a poritire number rj did not exist, by the corollary of the Lemma, P,
would coincide with a point on the remainder of the curve ; i.e. the complete curve
would have a double point, and would not be a simple cun-e.
'• If X (to -: i) < X (to), the inequalities involving 0 have to be reversed.
6] ANALTHia nrvs 11
Bat when l^<f<44>M^ ain^ vaiii»h«i oiily wheu ^-j^-iOi, mmI
tlMB eM4 M poAitive uum (x-x«)^"f(jf-jfj*> r».
HflnM 4'*'(Sii ^ l)v when l,<l«l. •*-••; therefore nnoe 4(0 i* *
rtintinuous fuiirtioii of f, n, - ii| « 0 or t !• But m,^$t,', for if Ni *■ «■
lien 4(l.4-<)>(iN, -t-l)*, 4(l,-^M-S)<(Sii,i- l)v and 4(0 would
tM|tt«l (S»i -•> 1 ) « for tome value of I between /« -^ i and t^-^m-i.
Therefore «« - *• ^ ± 1. and oomequeiitly
{#(«.♦-)-«(/,)}-{*, (I. + •)-«, (i,)}-4(i,*«)-4«;)-±i».
that is to iay the orders of tt, Hx differ by unity.
The theorem is therefore proved, except fur end pointa of the corve.
If /*! be an end point, a point i\ of the curve (not an end point)
tmMk be found such that /',/'. in arhitrarily xmall; then P,B < P*Pi
mnee P,B < r, < /'./',, and therefore /',// ; 2/^./',, so that FtB, and
•iiniUrly Ptttt, are arbitrarily small ; since the orders of B, B\ differ
by unity the theorem is proved.
Thbokbm V. (i) I/tirt) (rmtimma C,, C\ kaw a point Q !u eommom,
tkt ttiqf poinU^ S^ /«tmmi hy the two contimmi i» onf nmtinuum ; and
(ti) iftkt two comtinna C\, C\ kan> no point in rttmmon, Imt (f ail point*
mar any point, tkf ^td points rjxfffted, of thf eUmfntary
jf-yX'X ('•^•r ^'i). MoHtj to C, or to C\, or to tkf curw^ the
points amf^ttUty nnir and above ^^ the curef Mimgimj fu C, and tkom
nfficimtfy near and Mow it to V„ then the a*/ qf p(Hnt* S nnuti»ting
ff^Ci, Ct and tAr rwrcv (tAr end points ejvrpted) is one mtntiniinm.
(i) Let P be any point of S; if P belong to, say, (\ all points
xufiiciently near P belong U> C, and therefore to S. Hence »S' satisfies
the fintt condition for a continuum. Again if P, P' be uny two points
<jf 8tif P, /*' beloDK both to C\ or both to C„ they can be joined by
a aimple curve lying wholly in C, or T,, i.e. wholly in S. If P belong
to C| and P' to (\, each can be joined t<) Q by a simple curve lying
wholly in X If the curve** P(^, P'Q liave no point in (*ommon, save
(^, PQP' iH a Nimpio curve lying in *'. If l*Q, P'Q have a jwint in
common other tlian Q, let PQ, be an arc of PQ such tlutt (^i lie^ on
P'<^ but no other |>oint of PQ, lies on P'(^.
{The |H»iiit ^1 c\i>ttM ; fur A wt «>f pttiiiU csdUiiiion to iHith curvet* csistM :
let T \m tlio l«»wer biMtiHiAr}' '* «»f the |MinuueterN *»( the «ct, rt^gitnleil ait pttiiiU
•m PQ ; hj the lemma givcii ahnro, the {Mttiit Q^ with |ianuneter r it mi both
cQnr«R, aiid (MtiwtW!* titr iwcemAry ci»itditi<tii.]
" The temt* 'abov*' mmI 'below' are convention*]: (x, y| u above {m, y')
If f > t'.
>* Tb« lower boondar; etitta. Hobaon, Fumctiomt o/tt Heal VariabU, p. AM.
12 ANALYSIS SITUS [CH. 1
Then PQi, QiP' are simple curves with no point in common save Q,.
Hence PQiP' is a simple curve lying wholly in S. In either case, S
satisfies the second condition for a continuum. Hence S is a continuum.
(ii) Let the curve he AB ; draw CED parallel to Ch/ through any
point E oi AB (the end points excepted). If C and D be sufficiently
near to AB, C belongs to C, and D to C...
Then all points sufficiently near any point of C\ or of Cj belong
to >S^; and all points sufficiently near any point oi AB (the end points
excepted) belong to ^S. Hence S satisfies the first condition for a con-
tinuum.
Let P, P' belong to S. Then either ^« (a) P, P' both belong to C,
or to Ca; (6) P belongs to Ci, P' to ^o ; (c) P belongs to C,, P' to
AB; (d) P,P' both belong to AB.
In cases (a) and {d), PP' can obviously be joined by a simple
curve lying wholly in S. In case (6), simple curves PC, CD, DP' can
be drawn lying in 8, and a simple curve can be drawn joining PP' .
In case (c), simple curves PC, CE, EP' (the last being an arc of AB)
can be drawn lying in >S^, and a simple curve can be drawn joining
PP'. Hence S always .satisfies the second condition for a continuum.
Therefore 8 is a continuum.
Theorem VI. Given a continuum R and an elementary curve AB,
then : (a) If R contain all points of the ctwve except possibly its end
points, which may lie on the boundary of R, the set, R~, of points of R
which do not lie on A B form, at most, two continua.
(b) If one or both end points lie in R, R~ is one continuum.
(a) Let the equation of AB he y=f(ar). Through any point of
AB (not an end point) draw a line CD, parallel to Oy, bisected at the
•• There are several other cases which are obviously equivalent to one of these ;
e.g. P belongs to Co, P" to AD.
«]
ANALVIM srruM
18
point on AH, and lyin^; wkolljr in R\ chooM (\ I) wo that the orrlinate
of C M grMt«r thaii the ordinate of />.
Then //~ mdtfiea the fini cunditiun for a continuum (for if 2» be n
{loint of /f~ we can chooM S io that all point« aatiafying \s-t^\ <l
belong to H, and »ince ;« ii not on i4 if, we oan ohoote ff nnnller still if
r, to that uu iMiiut, z, of ^1 J9 mttafiea | £ - s« | < <'). Alao R-
the second cimditiou umttm a poimi P tff R' trUi* wkiek
' btjoitmi tu /) btf a timpU enrw fyinff tekuUy im R~. For if there
ia no such point, then if /', P' be any two points of /f~, they can each
be joined tu J) by a simple curve ; if tliese two curves do not interaect
eioept at I), PDP' in a simple curve ; if the two curves do intersect,
let Q be the first point o( intentectiun arrive<l at by a point which
daeeribes the curve PIJ. Then PQ, QP' are twu simple curves with
BO point in common except Q, no that PQP' \» a simple curve lying
wholly in R' ; hence R~ satisfies the second condition for a con-
tinuum.
Otherwise, join P to Dhy h simple curve lying wholly in R ; then
this curve has at lea«t one point not in R~ ; i.e. it haM at least one
point in common with AB.
Let ja be the first \mnt on AR which is reached by a point
deaeribing the curve P/) ; so tliat PK has no point on J if except E.
Ghooae an arc A'R' of AR, which contains E but not A or R.
CoDatmct two continua \* and X~ above and below A'R' respectively
as in the example of $ 5, each continuum ly'uxg wholly in R. Then
JV*, A'" and the curve A'R' with the end }K)ints omitted obviously
form one continuum, so that if a point F bo taken on EI* sufticiently
near E^ it will lie on N* or N~ ; for F cannot lie on A'R'. Suppose
that F lies in A'" ; choose a point G on CD lyinj? in A'' ; then FCi
can be joined by a simple curve lying in A'". Now PF, F(i, (il> are
14 ANALYSIS SITUS [CH. I
three simple curves lying in iV^ and N~ ; and hence a simple curve
PFGD can be drawn lying in iV+ or N~ ; i.e. PD has been joined by
a simple curve lying in R~ ; but this is impossible. Hence F must lie
in iV* : and then it can be shewn by similar reasoning that P can
be joined to C by a simple curve lying wholly in R~.
Hence the points of R~ can be divided into two sets :
(i) The points which cannot be joined to Z) by a simple curve
Ijdng wholly in R~ ; these points can be joined to C by a simple curve
lying wholly in R~.
(ii) The points of R~ which can be joined to i) by a simple
curve lying wholly in R~,
Each of these sets is easily seen to satisfy both the conditions for a
continuum. Hence the points of R~ form at most two continua.
{b) If B lies in R, a line BBi may be drawn parallel to Ox lying
wholly in R. Then by {a) the points of R not on A BBx form at most
two continua ; if they form only one continuum, the theorem is
granted ; for this continuum with the points on BB^ {B excepted)
forms one continuum ; if they form two continua^", these two continua
with the boundary points BB^^ {B excepted) form one continuum by
Theorem V.
7. The main Theorem. The points of the plane not on a given
regular closed curve foi'm two continua of which the entire curve is the
complete houndai-y.
Within an arbitrarily small distance of any point of the curve
there are two points of different orders with regard to the curve, by
Theorem IV of § 6. Hence by Theorem III of § 6, the points of the
plane not on the curve form at least two continua. Divide the curve
into a finite number of elementary curves and take these in the order
in which they occur on the curve as t increases from #« to 7" ; then by
the second part of Theorem VI of § 6 each of these elementary curves,
except the last, does not divide the region consisting of the plane less
the points of the elementary curves already taken ; the last divides
the plane into at most two continua, by the first part of Theorem VI
of § 6. Hence there are exactly two continua ; and the points of these
two continua are of different orders with regard to the curve.
^ It is easily seen that if there are two continua the points of one of them,
which are sufl&ciently near JSfij, are above BBx, while the points of the other, which
are sufficiently near BB\, are below BB^ .
fi Kl ANALV8I8 SITUS 16
Any i>"iut of the curve \» a boundary point of either oontinaum,
by TheoreuiM III and IV of ^6 ; and any point not on the curve is a
point of one continuum by Theorem I of H 6, and \» therefore not
a boundary |M.iint.
8. TiiKoREM I. AU itufficifntly dukmt points am qf order zero
with riyard to a ijivrn rf(fu/ar cloted rurte.
Let P {j;i/) bo any jwint on the curve, and Pi (j*,, y,) be any other
point ; the angle which PPt makes with the axis of x is given by
ooatf = «(jr-d',), sintt = K(y-yi),
where «»{(a:-a',)' + (y-y,)*}"*.
If x,» + Vi* be sufficiently large, either \Xi\ or | y, I must be so large
that either cos 6 or sin 6 never vanishes ; hence the change in ^ as P
goes round the curve cannot be numerically so great as ir ; but this
change is 2«ir where n is an integer and is the order of /*, ; hence n - 0.
That continuum which cont^iins these sufficiently distant ))oiut6
is called the e^rteriar of the curve ; the other continuum is called the
interior.
Since the order of any point of the interior of a regular closed
curve differs from the order of any point of the exterior by unity, the
order of any ytoxnt of the interior is + 1. If the order of any {wint of
the interior is + 1, we say that the point (-r (t), i/ {t)) 'describes the
curve in the counterclockwi^ direction as t increases from f, to TJ
If the order be - 1, we say that the point describes the curve in the
clocktcige direction.
\jQtt' =-t ; and let 6' {i') be the angle that AP makes with the
axis of X, A being a point of the interior and P being the }x>int whose
parameter is t or t'.
Then B' (f ) -9{t) - 2»i», and, if we take e'{t') to varj- continuously
as t varies continuously, m is constant, since m can only vary per
taltm. Consequently
B'{-T^m)-d'{-T) = -\6{t,^»)-e{t,)\.
Tlierefore the order of the interior point when t' is the parameter is
minus the order of the \yo\ui when t is the i>arameter.
Dbpinition. Oriented curcett. Orientation. Ijet P, S hfi the end
points of a simple curve. Let one of them, say P, be called ihefintt
pt/int. If Q, It be two other points on the curve (^ is said to be
16 ANALYSIS SITUS [CH. I
b*>fore R\itp<tq< t^ or if tp>t^i> tR. The points of the curve have
thus been ordered^\ Such an ordered set of points PS is called an
oriented curve ; it diifers from the oriented curve SP in which S is the
first point.
Two oriented curves Ci, Cj with a common arc <r liave the same
orientation if the points of <r are in the same order whether <r is
regarded as belonging to Ci or to C.,. If the points are not in the
same order, the curves have opposite orientations.
It is easy to see that if P, Q, R be three points on a regular closed
curve, the curves PQRP, PRQP have opposite orientations.
We agree to choose the parameter of an oriented curve so that the
first point has the smallest parameter. This can be done by taking a
new parameter t' --t, if necessary.
It is convenient always to choose that orientation of a closed curve
which makes the order of interior points + 1 ; that is to say that an
oriented closed curve is such that a point describes it counterclockwise
as t increases from ^o to ^o + ^^
Theorem II^. Let two continua Ri, R^ be the interiors of two
regular closed curves Ci, Ci respectively. Let a segment o-i of Cj
coincide with a segment o^ of C^; then (i) if Ri, R.^ have no point in
common the orientations of a-^, a-^ are opposite ; and (ii) if R^ be wholly
interior^ to R^, the orientations of o-i, a.^ are the same.
(i) If the orienta,tions of o-j and o-o are the same, by Theorem IV
of § 6 it follows that arbitrarily near any point P^ of o-i and o-j (not an
end point) there are two points J5, B' such that the order of B with
regard to either Cj or C^, exceeds that of J5' by unity ; so that B is an
interior point of both curves which is impossible. Hence the orienta-
tion of o-i is opposite to that of o-j.
(ii) If the orientations of o-j, o-^ are different, we can find
points B, B' arbitrarily near any point P^ of o-j and o-j such that
(a) the order of B' with regard to Cj exceeds that of ^ by + 1, {b) the
order of B with regard to C-i exceeds that of 5' by + 1. Consequently
^ is a point of Ri but not of R^ ; this is impossible. Hence the
orientations of cr,, o-, are the same.
'' Hobson, Functioim of a Real Variable, § 122.
" Ames points out that Goursat tacitly assumes this theorem.
'^ I.e. if every point of li^ is a point of R^.
CHAPTER II
OOMPLKX INTEGRATION
I 9l The intogml at m fiuiotion of • real varUKic ; cxtotMitiii to oomplex
TMiahlon ; rB«tricti<Hi of the |Mth of iiit«Kratioii.— 1( la Definition of m
ooQiplex integmL— H II. Kxiittenoe theorcuut.— § li. Qoiumt'* leniiiM.
— g IS. VariouM Miuplo tbeoramik
9. The integral' of a continuouR fiinctiou, /(x), of a real
%>anable x, U defined by means of the limit of a tturo in the following
manner :
Divide &ii intenr*! a to 6 (a < 6) into 2* eqiuU porta and let y, be the rth
|iart. Let i/,, Jk^ be the upper and lower limitti o(/{x) in y,.; let
« _ 5;/ *-" *^- iiL *""
r-I " r-l »"
Then (8J ia a noD-iuoreaaing aequenoe and (O itt a uon-deereaaing aequence,
and £•>«■: oonaeqtiently <$«, «. have finite liniitn aj* n -*«o ; and if /(x) is
oontintwMM it oan )« proved that thonc two liniitM arc the Hawe ; the common
vahw of thenc two UmiOt in oallod the integral of /(x) taken between the
end'traluea or limit* a and 6, and in written
f.
Further, it oan W iihewn that if < ia arl>itrar>', a numher i can be found mioh
that if the inter\-al a to 6 lie divided int^i anjf Hul>-iiiti>r\-nlit i;,, 7,, ... ly, each
IcM than A, and if x, be any \)oiut in the rth intcnal, then
If.
/(x)cir- X ih./(x,) <#,
When we ntuily the theory of funetion>i of complex variahleH, we
naturally enquire whether it is not |»u.H»iible tu ^'eneraliae thiti definition ;
for the interval a to /' may be regardetl a8 a segment of a particular
curve in tlie Argand diagram, namely the real axis.
* Biomwieh't Tluor^ of ti^miU Strit$ (190H), f| 157 163. abould be consulted ;
the analjaU given above b qocMHl from | IAS.
w. c. I. 2
18 COMPLEX INTEGRATION [CH. II
This suggests that we should define the integral of a continuous ^
function, /(«), of the complex variable z, taken along a curvilinear
path AB'va. the Argand diagram by the natural extension of the above
definition, namely that the integral oi f{z), taken between the limits
«o and Z, is the number S (if that number exist) such that it is possible
to make
IV I
/S*- 2 {Zr+i-Zr)f{Zr)
r=0 I
less than an arbitrary positive number e by taking v points c, , Cg, . . • s,-
in order on the cmwQ A B{zy+^ being interpreted as meaning Z) in any
way such that
I Zr+i ■-Zr\ < 8 for r = 0, 1, 2, ... v,
8 being a number depending on c (so that v also depends on e), and
the point Zr being any point on the curve between Zr and c,.+i .
[Note that we do not say
V
S=\\m 2 (c,.+l-^r)/(2-r').
v~^<*i r=0
because the summation on the right is a function of 2»' + 1 independent
variables Zi, z.^, ••• Zy, Zo, z-^, ... zj, and so S is not an ordinary limit of
a function of one variable.]
It is, however, necessary to define exactly what is meant by the
phrase 'points in order on the curve AB.'
To ensure that the limit, by which we shall define an integral, may
exist, we shall restrict the curve on which the points Zi, z^, ... lie, to
be an ' oriented simple curve.' And a further restriction is convenient,
namely that the curve should have limited variations* ; that is to say
that the functions x{t), y{t) should have limited variations in the
interval ^o to T.
[It can be proved ' that a necessary and suflBcient condition that a simple
curve should have a finite length is that it should have limited variations, but
this proposition will not be required ; the lemma below will be sufficient for
the purposes of this work.]
A function f{z) of a complex variable z is said to be ' continuous
on a simple curve' if /(*) is a continuous function of t.
2 Young, Sets of Points, §§ 140-141. Jordan, Cours dWnalyse, t. i. p. 90. It
will be obvious that the definition may be extended to cover the case when the path
of integration consists of a finite number of simple curves with limited variations.
=' Young, Sets of Points, § 167.
9-10] COMPLEX INTBOIAVIOV 19
We oui now prove the foUoviiig iaportMit laouM :
liKMMA. !m t,, tu *%t ••' '•♦I ^ <Mf «7iMiM» q^ poimit in mdtr
m
tma mmpU> ntrtt, T%m 1 \(t,^x'' B^\ it km thorn or 9qwil Ut tht mtm
r-«
^l4tf toto/ variatiimt </ j* (/) him/ jf (1) m I Ponm/inm t^ to I.,,-
Since the luoiltiltu of a Rum doM not exceed the mini of the
Bodali, it follows that
1 |(V|-«r)|- 5 l|(«r*,-»r)^»(jfrO-^)ll
< 5 [iK..-*r)l + |U(^*.-fr)}|)
Bot /r«.i><ri since the points £,, ;,, s,, ... are in order; and
conMiiuentljr the fintt of the«e Mummations is lesM than or equal to
the total rariation of x{t\ and the second summation is less than or
equal to the total variation of jf(t) ; that is to say, 1\ (Zr^i-z,)\ is
leee than or etiual to the sum of the total variations of r {t) and jr (/).
10. We are now in a poeition to give a formal definition of a
complex integral and to discuss its properties. The notation which
hai been introduced in ^ 3, 4 and 5 will be employed throughout
DbfURTIOH. Lft AB b« a fimplt' curve trt/A limited varialiomt
4kmnt in the Argand diatjram. l^ei /{z) be a function if the complex
tariable z which it eomtinucmt on the curve AB. Let z^ bt tht
eoordinatt qf A, and Z tke cimiplex niordinate of B. Let a
<^ ptintt am A B be clkttim, and urkett n tf tkemt point* have
been taken^ let the points taken in order be catted Zx*^\ z^'\ ... s»<*> {to
that ifm^n, z,**' it ome q/' the points z,<-', c,'"», ... i"*—..!); thetequence
qf' points may be cJ^oten according to antf definite late wkaterer*, pro-
vided ontjf tttat tite points are att different and tluU, gitrn anif positive
mmnher h, we ran find an integer n, sucA tkat wken n > ii«,
where r » 0, I. 2. ... n aim/ /.«■> = r„ ^..••' - T.
* If (•■0. Tml, the ftimpkHt Uw is given by Uking l|(*i. IJ*K ...l.<^ to b*
lbs fliM n of Um nombera i : |. | : )• !• !• I; -.- «bcn Umm n number* •!«
inmyipil la ooUr of magnitod*.
20 COMPLEX INTEGRATION [CH. II
Then the complex integral I /{z)dz is denned as meaning the
Jollowing limit:
jy{z)dz = lim [(5rx'"> -2o'"V (W"') + (W*> -«i<"Oy (^i'"0
+ (2;,W-W"')/(^."*>) + •- +(^-^»<">)/(«n<">)]
= lim 2 [(:^r.,<")-2:.<">)y(«r'"')].
[It is pennissible to speak of the limit of
2 [(2r + ,<")-^r<'")/(^rW)]
r = 0
because these expressions form a sequence (depending on n), each member of
the sequence being determinate when the form of / and the law, by which
the points «,.<") are chosen, are given.]
The integral is said to be taken along the path AB, and the path
AB is usually called the contour of integration ; and if the path AB
be called C, we sometimes write / J (z)dz in the form / J (z) dz or
Ja jub)
/{z)dz.
/<
11. It is next necessary to prove (Theorem I) that the limit, by
which an integral is defined, exists.
When we have proved Theorem I we shall prove (Theorem II) that
if a positive number t be taken arbitrarily, it is possible to find a
number Sj such that, when any v numbers ti, t^, ■■• t^ are taken so that
to^ti^ti^ ... ^tv'^ ty+i = T and #p+i - ^p $ 8j (j» = 0, 1, ... v), and when
Tp is such that tp%Tp% tp^,, then
\l^/(z)dz- 2 (Zp,r-Zp)/{Zp) <€.
I J A p = 0
Theorem I. Let /S; (z) = 2 [(;2,^i<") - ^r'"0/(^r<'")] ; then lim S^ (z)
eansts.
To prove the existence of the limit, we shall prove that, given
an arbitrary positive number e, we can choose an integer n such that»
when m> n,
\S„{z)-S^(z)\<€.
This establishes' the existence of lim S^ (z).
tt-»OC
« Bromwich, Theory of Infinite Seriet, §§ 3, 75, 151.
10-11] OOMPLBX limORATIOK SI
L«t L tw the turn of the total raruitiorui of » (#) mm! jr (I) for the
iatenrd <• to T of r.
In virtue uf tJie routiituity of /(£) tfta foootioo of I, eocreqwodiiig
to an arbitrmry po«itive number «. we can find a poeittve nanbar h gaeh
that, if s be any particular point on Ali, and if £' be on AH, Umd*
l/(«V/(«)l t;|./i (6)
whaoerer i r' - f ' < S ; it t« obviooji that, in general, S it a function of #.
Let Of ammmt for the prenent' that, when « is taken arbitrarily, a
number K (independent of f, but depending on <) exista, such tlwt. for
<ii/ valuee of t under consideration,
that is to say, we assume that/(s) is a umiformiif amtinmrns* function
of I.
Now choose m so Urge that
0 <<,.,'•» -/,<*»<«w.
forr-0^1, 2, ...n; this is possible by reason of the h>'pothesis made
eoDoarning the law by which the numbers /r'"' were chosen.
Let m be any iut4*Ker hucIi that m> n ; and let those of the points
s,**' which lie between «,'•♦ and «,♦"• l»e called C|. «, *^,, ... x«,*i.«, where
x^,^ = z^*'\ s«,»i.»*=-i'"* ; •"*^» generally, let those of the points r,'"'
which lie between Cr'"* »nd Zr*/"' be called r,. ,, :% r, ••• -■,..i.r, where
•1. r "^ *r » ••,..1. r — ♦r*! •
Tlieu *-. - 1 [(c,.."' - s,"»)./(5r'"»)]
since the |)oint< :^ , are the ftauie sji the points £,*"'>
AIm. S, - 2 f *2 {(s..,. . - c^ ,)./(.% r)sl.
r-oLa-l J
mitlmt (N.-.S.)| = | 1 r 2 (-...., -.Vr)|/(s,"')-/(--.r)}l|
< J "s |(r..,r-r..,){/W»)-/(-^r)} .
* Tb« rwuon (or cbooaiitK ^^ iuulUpli«r | will be Mrn when we come to
TtMorvcn II.
' A (oniuU proof U given in f 13.
* Th* eontinuily U mdd to b* uniform bpc«uae. a« r' •» f , f(:') tend* to the limit
/(:) uattoniily with roepect to the v«ri«bU I.
22 COMPLEX INTEGRATION [CH. II
But ^r+i*"' ^t,^T> tr^*\ 80 that 0 ^ /,. r - tj^^^ ^ Sfl, and consequently
r = 0 »=1 r = 0
and consequently
r=0 »=1
in
r=0
since 2 \ {ZrJ"^^ - Zr^""^) \ ^ L,
by the Lemma of § 9. That is to say that, given an arbitrary positive
number c, we have found n such that when m>n, \Sn-Sm\<f; and
consequently we have proved that lim Sn exists ; the value of this
limit is written
/.
'/{z)dz.
We can now prove the following general theorem :
Theorem II. Given any positive number c, it is possible to find
a positive number 8^ such that, when any v numbers ti, t^, ... tv are
taken so that 0 ^ tp^.■^ -tp%^, (p = 0, 1, ••• v, and t^+i = T), while Tp is
such that tp $ Tp % tp+i, th^n
L
^f{z)dz- 2 {zp,^-Zp)f{Zp)
Zp, Zp being the points whose parameters are tp, Tp respectively.
Choose So and n to depend on € in the same way as in the proof of
Theorem I ; we shall prove that it is permissible to take 8j = So-
For, assuming that 0 ^ ^;,+, -tp^^ 8„, we can find an integer r
corresponding to each of the numbers tp, (p=¥v+l), such that
^r**' ^ ^p < ^r+i^"' ; let the numbers tp which satisfy this inequality for
any particular value of r be called in order t^^r, t^^r, ■•■ ttf^, r-
Then we may write
2 [(^,.,-Cp)/(Zp)]= 2 [(c,.,.-cW)/(Zo..) + (c^r-c,.)/(Z,.)
p=0 r=0
^iz,,r-Z^r)/{Z^r)+...+{ZrJ''^-Zy^,r)/{Z,.,r)].
11] COMPLEX INTEORATIUN S8
The fullowiitK ODUvetitioiM have to be adopted in interpivtiiig tlia
nnmation od the nKht-hand side :
(i) l^r < 7Vr ^ 'i.r ; whero U,r mean* that number of the set
U, ^if-^ which immediately preoadee li.r'
(ii) txr, r ^ Tx^. r < <jr^«i. r i whew <jr,«i.r meaD« that number of the
set lit'ii •• 'r»i which immediately follows rjr^, r.
(iii) If, for any value of r, there is no number t, such that
t/^ ^tf< fr*/"\ the term of the summation corresponding to that
value of r is (s,,,<"> - «,"») /{Z^ ,), where Ur^T^r^t,,r and <^ r, «i. r
are respectively the largest and smallest numbers of the set <«, l|, ... tp^i
which satisfy the inequalities
With thexe conventions, if i$w has the same meaning as in
Theorem I, we may write
= J^U-..r-5r'-»){/(^..r)-/(«r'-»)} + (*^r-:r,..){/(^,..)-/(z.<-')}
if for any value of r, there is no number tp such that O*^ %t,< #r*i**'»
the term of the summation corres|X)nding to that value of r is
(r...W-*.«-0{/(^^r)-/(^«-»)}.
Now if « « 0, 1, ... Nrt we have
and T^r>t^ur-i,>0'^-i.i
hence I 7^ r - <r<"M < 2fi..
Tberefoie, if <'-i(r^r-(-^<"0. we have
so that, since the modulus of a sum does not exceed the sum of the
moduli,
I /(Z. ,) Atr») 1 1; I /{Z^r)^Az') I + [ /(*)-/(«.«)
by equation (5) of Theorem I.
24 COMPLEX INTEGRATION [CH. II
It follows that
^ 2 [|(5..,-;?,W)l.icZ-'
r=0
+ |(^r-;tl.r)|.i^X-^+...+|(W"-^.V,.r)l.|«i^-']
r=0
Now, by the Lemma of § 9, the general term of this last summation
is less than or equal to the sum of the variations of .r (t) and y (t) in
the interval #,.<") to ^,+i*"', since the points
are in order; and, hence, since the numbers ^o^"*, ^i*"', ... #„+i*"> are in
order, the whole summation is less than or equal to the sum of the
variations of x {t) and y (t) in the interval ^o*"* to #„+i<"' ; that is to say
2 (Zp+i - Zp) / (Zp) - Sn
p = 0
But, by Theorem I, with the choice of w which has been made
I ^m - 'S^n I ^ if,
when m> n. Hence, since c is independent of m,
I( lim^„.)-^„i^|c.
1.6.
Jzt
dz-Sn
^K
Therefore
f
f{z)dz- 5 {Zp,r-Zp)f{Z,)
p=0
^1 /V(^)^^-'S^«
1 /^o
+ i.S;-2 {zp^,-z,)f{Z,)\
1> = 0
^h-
That is to say that, corresponding to an arbitrary positive number
€, we have been able to find a positive number Sj (namely, the number
denoted by So in Theorem I), such that if
0 ^ tp^, - tp % 8,, {p = 0, 1, 2, ... v, and ^,^, = T\
then
/ f{z)dz- 2 {z,.,,-z,)nZp)
Jzt p=i)
11-12] COMPLEX INTEGRATION 16
PrOQi tiiiH ironpntl tilivinMii, wn cau di'^uee the following |MiHinulAr
tfaeorent
Theorem III. Tkt' valu» qf I /(<) dz dom not depend on the
jfMtrticular Uttr by wkirk the point* Xr*"' ar$ ckottn, provided that the law
eeUi^iee ike muditione t^ % 10.
Let poinU chosen aooording to any other Uw tliau that alreaiiy con>
eidered be called ^w, (^.q. 1, ... k ; t^, ^, ^^,w « Z) ; then if t be
the parameter of the point C, ^o can find a number y, "uch that when
► >!'*, 0^T,./'*-''|.**^^^ ; hence we may take the numbers t^ of
Theorem II to be the numbers r,!"* respectively, uiui we will take
Z,=^(ff^; therefore, by the result of Tlieorem II,
aiui, corre8iK>ndinK to any positive number «, we can always find the
number k, such tliat this inequality in satisfied when !'>»*,.
Therefore lim 2 [(C.*/-»- ^-^/(CpW)]
!>-»« pmO
exists* and is equal to / / {z) dz, which lias been proved to be the
value of
lim 5 (r,..«-»-s;-')/(VO;
and this is the result which liad to be proved, namely to shew that the
rZ
value of I /(z) dz does nt»t tle|)end on the particular law by which we
choose the |)oints Zr^'K
12. It was assumed in the course of proving Theorem I of § 1 1
that if a function of a real variable was continuous at all {K)ints of a
finite closed interval, then the function was uniformly oontinnons in
the interval.
A formal proof of tlii»« a.'v>iiuii»tinn is now necessary"; l>ut u is
expedient first to prove the following Lemma. The lemma is proved
for a two-<limensional region, as that fonu of it will be required later.
• Bromvieh. Tktory of In^nltt Serin, | I.
** It mu pointed oat bj Heine. VrrUe'a JomrmU, vol. Lxxi (1870). p. 801 and
vol. ucxnr (1879), p. 186, ttwt it is not obvloos that continuity impliM uniform
(>nntif)ftitv
20 COMPLEX INTEGRATION [CH. II
Godrsat's Lemma". Given (i) a function of position of two point* P\ P,
which will he written {P', P], and (ii) an arbitrary positive number t; let a finite
tioo-dimensional closed region^^ It have the property that for each point P of R
voe can choose a positive number d {depending on the position of P), such that
\[P\ P)\^* whenever the distance PP' is less than or equal to i, and the
point P' belongs to the region.
Then the region, R, can he divided into a finite number of closed sets of
points such that each set contains at least one point P^ such that the cotidition
I {/*', /*,} I < f is satisfied for all points P' of the set under consideration.
If a set of points is such that for any particular positive nmnber <, a point
P, can l>e found such that
\{P\P^}\<^
for all p(jints P' of the set, we shall say that the set satisfies condition (A).
A set of points which satisfies condition (A) will be called a suitable set.
Let R~ he the continuum formed l)y the interior of R ; take any {xjint
of R~ and draw a square, with this point as centre, whose sides are jmrallel
to the axes, the lengths of the sides of the square being 2Z, where Z is so
large that no ix)int of R lies outside the square.
If every j)oint of R satisfies condition (A), what is required is proved.
If not, divide the square into four equal squares by two lines through its
centre, one parallel to each axis. Let the sets of }.x)ints of R which lie either
inside these scjuai-es or on their boundaries be called oi, oj, 03, 04 respectively
of whidh oj, a.^ are above 03, 04 and ai, 03 are on the left of 02, 04.
If these sets, aj, 02, 03, a^, each satisfy condition (A), what is required is
proved. If any one of the sets, say a, , does not satisfy condition (A), divide
the square '3 of which oj forms part into four equal squares by lines parallel to
the axes ; let the sets of points of R which lie inside these squai-es or on their
b<;)undaries be called /Sj, ^2» ^3 ("* the figure one of the .squares into which
ai is divided contains no jx>int of R).
If condition (A) is satisfied by each of the sets, we have divided oj into
sets for which condition (A) is satisfied ; if the conditit)n (A) is not sati.sfied
by any one of the .sets, say ^3, we draw lines dividing the square (of side iZ),
of which ^3 fonus j«irt, into four etpial squares of side ^L.
This i)rocess of .sulxlividing scpiai-es will either terminate or it will not ;
if it docH terminate, R has been divided into a finite number of closed sets of
IK)ints each satisfying condition (A), and the lemma is proved.
Suppose that the process dt)e8 not terminate.
A closed set of jKunts R' for which the process does terminate will Ije said
to siitisfy condition (B).
Then the set R does not satisfy condition (B) ; therefore at least one of the
" This form of the statement of Goursat's Lemma is due to Dr Baker.
" Consisting of a continuum and its boundary.
'-'' A square which does satisfy condition (A) is not to be divided ; for some of
^be subdivisions might not satisfy condition (A).
\t]
COMMJa IICTVIIUTIOJI
S7
daw not
Tlw |wo0Mi of dividit^ tiir «(tMrr, ill wtiicti ftUs Ml Iki^ Mo imr oqoal
farta gIvM Ai muat fc«ir «!• uf putuu, «if wUeb •! UMt oim «i dotw not
Mtuify r-^k*"" (BX T«ke the ftnA of tlMNn whieb daw nol, and oaotiiuM
thtii tiiimuM of dhi«i«i (u»d wekeiian, Tlw rMttk «f Um pfoowa b to gh*
•n aiiMMling MK|ti(itioe nf aqoMwi mMyia§ Um ftJlowiitg ooodMow :
irtb* wqwnw be calM v <i« «ft. -^ thM**
I
»l
/I
/
!^
N,
/
^
ti
-*
CiC
)
A
<-.
^
y
-
-
'^ *»,
\
/
^
>
e»
■^
'
1
V
A
J
f
"~"
—
^
(i) Th0«tdeaf «. iMuf lenKth 2-"iL.
(U) No point of «• . I Ihm odtitidc «..
(iii) Two MideH of «, , , lie along two sidcM of «..
(ir) i^ outitAtna at lea«t oiw [loiiit of R.
(t) The Mi of pointji of H which arc iiutiilc ur on «, do not mHitij
oondttion (A).
Let the wMirdinatcM of tlio oonteiw of «, be called
where x,<'> < x,«», jr,<'» < y,».
Then (Xa<*i) t« a non-decTBaidiig iie({i>etK« and (x.A) is « nun-increaaing
■equenoe ; and x,«-r,<'>— S""/, ; thcwfore tlw m^ihmicoh (x,n»), (x,«) hare
a ooiuiuon limit ( wuch that x.(»<{<x.(^: lumilaHy tlie Mequencoi (jr.('*X
(jr^W) have a cvfiuiHiii limit ly MtK>h tlutt jrai''<i| (y.*^-
Oanaeqtiently ({, i)) lim iiutiJe or on tite UHiiKlaricM of all tlie Nquarm of the
wnuancw (v) '• furtlier, ((, ^) licw imudc or on the UiiUKlarA- of the region R ;
** W« take the Jint poMihU aqiuue of aaoh group of four ao as to gvt a d^miu
MqoMwe of ■quana.
>• Cp. Branwiah, Tkmtr^ of Im/mitt Stru; | lAO.
28 COMPLEX INTEGRATION [CH. II
for since «„ contains at least one point of R, the distance of (f, i;) from at
least one jxjint of /f is less than or equal to the diagonal of »„, i.e. 2~"X ^2-
Hence, corresponding to each square, «„, there is a point P^ such that
where IT is the point whose coordinates are (^, i;) ; this sequence of points
(/*„) obviously ha.s n for its limiting point ; and since the region R is closed,
the limiting jMiint of any sequence of points of A is a ix>int of R. Therefore
n is a point of R.
Then | {P', 11} | < e when P' is a point of R such that P'U < d», where d»
is a positive number dej«nding on 11.
Choose n so that 2~"Z ^/2 < 8„ ; then all jxiints, P', of »„ are such that
P'n < 8n ; and therefore «« .satisfies condition (A) ; which is contrary' to
condition (v).
Consequently, by assuming that the process of dividing squares does not
terminate, we are led to a contradiction ; therefore all the sequences terminate;
and consequently the nmnber of sets of points into which R has to be divided
is finite ; that is to say, the lemma is proved.
[The reader can at once extend this lemma to space of n dimensions.]
In the one-dimensional case, the lemma is that if, given an arbitrary
positive number f, for each point /* of a closed interval we can choose d
(depending on P) such that | {/*', P}\< t when PP < 5, then the interval
can be divided into a ^nite number of sub-intervals such that a point Pj of
any sub-interval can be found such that | {P, P^} \ < * for all points P of that
sub-interval ; the proof is obtained in a slightly simpler manner than in the
two-dimensional case, by bisecting the interval and continually bisecting any
sub-interval for which the condition (A) is not satisfied.
The proof that a continuous function of a real variable is uniformly
continuous is immediate. Let /(x) be continuous when a'^ a^%b ; we
shall prove that, given c, we can find So such that, if w', x" be any two
points of the interval satisfying \x' —x"\<. So, then \f{x') -f{x") \ < c.
For, given an arbitrary positive number c, .since /(a*) is continuous,
corresponding to an)^ x we can find 8 such that
j f{x') -f{x) I < \t when | ^' - .r | < 8.
Then, by the lemma, we can divide the interval a to b into a finite
number of closed sub-intervals such that in each sub-interval there is
a point, a-,, such that \/(x')-/(xi) | < ^« when x' lies in the interval
in which ;r, lies.
Let 8„ be the length of the smallest of these sub-intervals ; and
let x', x" be (i}ii/ two points of the interval a -^ x ^ b such that
Ix'-x'l-cBo;
IS- 18] ooMPLBi umniuTioM
than M, j^ li« in the imm or in M^ivn^ tab-inlenrttb ; if je\ y li« in
Um auM •ub-intMTtl* Umo wo ou ftad &% m that
If y, jT lie in m^*''^' *ub-int«nraU let ^ be their common end-
point; thin «• c«n find a puint x, in the first tub-interval and a
point j^ in the eeoond «uch that
|/(*')-/(x.)|<J*. !/(«-/(*,) !<K
l/(^)-/(A)l<i«. i/U)-/(*.)l<K
•o that
l/(y) -/(X-) - ; {/(*) -/(x.)) - 1 /(o -y (*.)}
-|/('")-/(^)} + {A«-/(A)}l
In either oaae l/ijO - /{r") \ < * whenever I ^r' - y ; < 8,. where 8,
is indfpmdtmt of y, x" ; that in to say, /{x) is uniformly continuout.
18. Prmifii «>f thr fiillowing theorems niay lie left to the rtMuier.
I. It AB )« A Himple cune with htuitcd viuiatiinui mid if /(<) be con-
tfawiw on the oune AB, then
!'/(«) «i.--|^/(«) A.
TbAt i» to MV, cbanging tbe orientation of the {nth of int4>gnition chaiigw
Um aifii of the integnU of m given functi<Hi.
II. If C be • puint on tlie aiiuple oun'e AB, and if /(<) be oonttnuouH on
(heourve, then
III. If t« and Z t« the coniplei coordinaten of A and B raipactively, and
if wl A l« a stniple cttr%'e joining A, B, then
IV. With the noUtion uf TlicirenM I and II of § II, hy Uking <p- V'*
and Zp in turn equal to <y<*i and t^ , ,<**, it followH that
^'tds'^ lin. i [(«,.,«•» -!,«"») */•>]
/:
/:
• ••• r-»
-H>n> » [(«r.l«)'-('r'"'>'l
CHAPTEE III
CAUCHYS THEOREM
§ 14. The value of an integral may dej)end on the path of integration.
— § 15. Analytic functions. — § 16. Statement and proof of Cauchy's
Theorem. — § 17. Removal of a restriction introduced in § 14.
14. Let Ci, Co be two unclosed simple curves with the same end-
points, but no other common points, each curve having limited varia-
tions. If ;So, Z be the end-points and \i f{z) be a function of z which
is continuous on each curve and is one-valued at z^, and Z, then
\^/{z)dz, \^/{z)dz
both exist.
\i f{z) = z, it follows from Theorem IV of § 13 that these two
integrals have the same value. Further, if Cj, (7, be oriented so that
Zo is the first point of Cx and Z the first point of Co, and if Cj, C^ have
no points in common save their end -points, Ci and C^ taken together
form a simple closed curve, C, with limited variations, and
/<
zdz = 0.
c
This result suggests that the circumstances in which
Jiz)dz^O,
I
(where C denotes a simple closed curve with limited variations' and
/(z) denotes a function of z which is continuous on C) should be
investigated.
' A regular closed curve, satisfying this condition, regarded as a path of
integration, is usually described as a closed contour.
14-15] oauciiy'm thkorem 81
The iaveiitifpitioo appMn «U the more ueccwHuy from the fiurt'
that if C be the unit circle |«|« 1. deacribed oouiiterolockwiae, mad
/(<) • z'\ (to that < « ooe I •*> I an /, - V < K v), it cati be iihewn that
/<
c
Con<ittiuiiM fur the truth of the equation
/(a)d!t»0
/c
were titHt iuv6itiffttid by Cauchy*.
It iM N44 nffhitmt that/(«) sbouUi be oontinaoos and one-valued
on t)ie re^uUr closed curve C, as is obvious from the example cited, in
which /{z) = c"' ; and, further, it is not mffident that/(«) should be
continuous at all points of C and itM int^'riur.
A sufficient condition for the truth of the equation is that, given a
function /(;) which exists and is continuous and one-valued on Uie
curve C, it should be possible to define a function*, /(z), which exists
and is continuous and is one-valued at all points of the closed r^OQ
formed by (' and its interior, and which possesses the further property
tliat the unique limit
should exist at every point z of this closed region, it being supposed
that z is a point of the closed region. The existence of this limit
implies the continuity of/(r) in the region.
It is, further, convenient, in setting out the proof, to lay a restriction
on the contour C, namely that if a line be drawn parallel to Ox or to
Oy, the portions of the line which are not points of C form a finite
number of segments. This restriction will be removed in § 17.
15. Dbfixition. Analytic /unctiotui. The one- valued continuous
function /(z) is said to be analytic at a |K>int ; of a continuum, if
a number, /, can be found satisfying the condition that, given an
' Hardy. A Cour»e »f Purt M,ithrmatict, f :104.
* itfmoirt tur U0 inUgraltt (///i'miV* prtMt entrt dei limiu* imtaffimaim (1835);
this nMaoir U reprinted in i. vti. »nd l. riii. of ths BulUttH tU» Scimert UmtM-
* Up to tb« prMent point « function, /(<), of the complex variftble t. luu mouit
OMTtly a faneiion of the two real variablw x and y.
32 cauchy's theorem [ch. hi
arbitrary positive number c, it is possible to find a positive number 8
(depending on e and z) such that
\/(z)-/(z)-l(z'-z)\%.\(z'-z)\,
for all values of z such that \z' - z\%8.
The number / is called the differential coefficient, or derivate, of
/(z) ; if we regard z as variable, I is obviously a function of z ; we
denote the dependence of / upon z by writing l=/'(z).
So far as Cauchy's theorem, that I /(z) dz = 0, is concerned, it is
not necessary that/(2;) should be analytic at points actually on C; it
is sufficient that f{z) should be analytic at all points of the interior of
C and that for every point, z, of C,
\f{z)-f{z)-f{z).{z'-z)\%.\z-z\,
whenever \z -z\<.8 (where 8 depends on c and z\ provided that z is a
point of the closed region formed by C and its interior.
In such circumstances, we shall say that/(«) is semi-analytic on C.
It is not difficult to see that analytic functions form a more re-
stricted class than continuous functions. The existence of a unique
differential coefficient implies the continuity of the function ; whereas
the converse is not true ; for e.g. | « | is continuous but not analytic.
16. It is now possible to prove Cauchy's Theorem, namely that :
If f{z) be analytic at all points in the interior of a regular closed
curve with limited variations, C, and if the function he continuous
throughout the closed region formed by C and its interior, then
I
^f{z)dz = 0.
The theorem will first be proved on the hypothesis that/(«) is
subject to the further restriction that it is to be semi-analytic on C.
In accordance with § 8, let the orientation of C be determined in
the conventional manner, so that if the (coincident) end-points of the
path of integration be called z^ and Z, with parameters ^o and T, then,
as t increases from tf, to T, z describes C in the counterclockwise
direction.
The continuum formed by the interior of C will be called R~ ; and
the closed region formed by R' and C will be called R.
Let L be the sum of the variations of x and y as s describes the
curve C ; take any point of R~, and with it as centre describe a square
of side 2L, the sides of the square being parallel to the axes ; then no
1&-Iflj CAUCHT'S THBOIBM 83
point of /f Uei oaUid« Uiii Miiure ; for if (a. ft) be Um omtn of Um
•quMV aud j^, j% tlM mraat ViloM of x un //, th«o
^ < j^ i JV. 0 ^ X, - X, ^ /,,
•0 that X, •*• J^ > J4 4- £r > X, ; i.e. the riffht-haotl aide of the aqtuue ia
on the right of H ; a|>|UyiiiK «iiuilar reaaooing to t)ie other three ddea
of the iquare, it in apparent that no point of /f is outxide the M|uare.
Lei < be an arbitrary poaitive number ; then, since /(z) i» analytio
inade C and •eni'analjrtio on C, eorraipooding to any point, z^ of H
«e can find a positive number i such that
l/(«')-/(«)-(«'-*)/'(«)|i:«l«'-»|.
whenever !«'- 1 1 < S and s' is a point of H.
Hence, by Goumt's lemma (^ 12), we can divide B into a finite
number of aeta of points such tluit a point, s, , of each set can be found
each that
!/(«')-/(«.)-(*'-«,)/' («.)l^«l»'-s.l.
where z' is any member of the set to which t^ belongs.
Suppooe tliat /{ m divided into such setii, as in the proof of Goursat's
lemma, by the pruceM uf dividing up tite 8i|uare of side 2L into four
equal squarea, and repeating the process of dividing up any of theae
■loares into four eijual stiuares, if Huch a process is necessary.
The effect of bisecting the square of side 3J^ is to divide /i' into
a ^mit^ number of continue, by Theorem W of § 6 combined with the
hypothesis at the end of § 14 ; the boundarie/i of theee continuH are C
and the straight line which bisects the Mjuarc ; tlie prooeas of dividing
up the aquare again is to divide these eontinua into other oontinua ;
and finally when /f lias been divided int<) Huitahle hct^ Il~ has been
diviiicd into a finite number uf eontinua whoMe btjundariea are portions
of C and portions of the sides of the squares.
The squares into which the square of side '2L luu been divided fall
into the following three claases :
(i) Squares auch that every point inside them is a point of /i.
(ii) Squares such that some points inside tliem are points of /f,
but other points inside them are not points of It
(iii) Si|uares such tluit no {loint inside them is a point of If.
The points inside C which are inside any iMirticular miuare of
claas (i) form a Mmtinuum, namely the interior of the square ; the
points inside C which are in.<ide any {tarticular i«{uare uf cUss (ii)
form one or more eontinua.
w. c. I. 3
34 cauchy's theorem [CH. Ill
Let the squares of class (i) be numbered from 1 to iNT and let the
oriented boundary of the kth. of these squares be called C*.
Let the squares of class (ii) be numbered from 1 to iV'. Let the
set of oriented boundaries of the continua formed by points of I(~
inside the Arth of these squares be called Ck'.
N r N' r
Consider 2 f{z)dz+ % /{z)dz;
we shall shew that this sum is equal to / /(z) dz.
The interiors of the squares of class (i), and the interiors of the
regions whose complete boundaries are C^, are all mutually external.
The boundaries formed by all those parts of the sides of the squares
which belong to R~ occur twice in the paths of integration, and the
whole of the curve C occurs once in the path of integration. By
Theorem II of § 8, each path of integration which occurs twice in the
sum occurs with opposite orientations; so that the integrals along
these paths cancel, by Theorem I of § 13.
Again, the interiors of all the regions whose boundaries are Ck and
Cjfe' are interior to C ; so that the orientation of each part of C which
occurs in the paths of integration is the same as the orientation of C ;
and therefore the paths of integration which occur once in the summation
add up to produce the path of integration C (taken counter-clockwise).
Consequently
2 \ f{z)dz+'k f /{z)dz=[ /{z)dz.
Now consider I f{z) dz ; the closed region formed by the square
Cfc and its interior has been chosen in such a way that a point Zi of the
region can be found such that
\f(z) -/(%) -{z- z,)f' {z,) \<.\{z-z;)\,
when z is any point of the region.
Let f{z) -f{z,) -{z- z,)f' (z,) = v{z- z,),
when z=¥Zi.
When z=Zi let v=0 ; then v is a function of z and Zi such that | v| <e.
It follows that
16] CAUCHv's THEOREM 85
But by Theorem III of § 13, jdz^Z-Zo, where z,, Zare the end-
points of the path of integration ; since C* is a closed curve, Z~z,,ao
tliat / rfc = 0 ; so also, by Tlieorera IV of § 13, f ^ zdz = 0.
y(c») j(Ci,)
Therefore [ /(«) dz = L . (« - «i) vdz.
Therefore*, since the motlulus of a sum b less than or equal to the
8inn of the moduli,
^ ( \(z-Zi)vdz\
<j^^ l,j2€\dz\
^ UAt J2,
where Ik is the side' of C^ and At is the area of Ct, so that Aii = lk \ it
is obvious by the lemma of § 9 that / \dz\ does not exceed the peri-
meter of C*.
We next consider / , /(«) dz ; if the region of which C*' is the
j(Ck)
total boundary consists of more than one continuum (i.e. if Ck consists
of more tlian one regular closed curve), we regard Cu as being made up
of a finite number of regular closed curves ; and since the interior of each
of these lies wholly inside C, any portion of any of them which coincides
vrith a portion of C has the same orientation as C ; and the value of
\dz^ jzdz round each of the r^ular closed curves which make up Cu
is zero.
Hence, as in the case of C*, we get
• The expnwion [\/{x)(U\ means lim 2 | (tn-i^^^-'r'^O/C'r'*')! i *!»*» *he
notation of Chapter 11; arguments similar to those o( Chapter II shew that the
limit exists.
* The sqoarea C^ are not necessarily of the same sixe.
S— 2
36 cauchy's theorem [ch. hi
where 4' is the length of the side of that square of class (ii) in which
Ck lies.
Let the sum of the variations of x and y, as z describes the portions
of C which lie on C^, be L^ ; so that
JV'
/^' .
( 2 Xjfe' will be less than L if part of C coincides with a portion of a
side or sides of squares of class (i). ]
Now L,J<^^l^^' + 44';
for, by the lemma of § 9, / , | «fe | is less than or equal to the sum of
j{yk)
the variations of x and y && z describes the various portions of C*'.
Therefore I { ,^ / {z) dz I ^ {L^ + 44') « 4' ^2
\J\yk) I
since 4' ^ 2i/ ; -4 ft' is the area of the square Cu.
Combining the results obtained, it is evident that
^ 2 4^ft e ^2 + 2 (4ylft' €J2 + 2L( L^ J2).
k=\ k=\
N N'
But it is evident that S Ak+ 2 -4fc' is not greater than the area
k=\ k=\
N'
of the square of side 2L which encloses C ; and since 2 L^ ^ L, we
k=\
see that
/ /(«) dz ^4.^ {2Lf X c 72 + 26X» V2
• $18cZV2.
Since Z is independent of c, the modulus of / f{z) dz is less than
a number which we can take to be arbitrarily small. Hence / f{z) dz
is zero, if /(c) be analytic inside C and semi-analytic on C.
16-17] CAirilYs TMKUH&M 87
17. The rmultii of tho fuUowing two tbeocwiMi maka it po«ibls to remove
the iwthcuoii Uid on C in j^ 14, dmmI/ tlut if a line be drawn pemllel to
(Ar or to Oy, th<if« {Mirtioim of the liite which are not pointo of C form
» finite iiuni W uf liogtueuU ; aliio it will follow that the aaMimption made
at tho bogtiining of % 18, that/(i) ia aemiatMlytio on C, b onneoeeMry.
Thborkm I. Oiven^ a nffular cUmd cmtm C and a potUiwt mmb$t ^ a
elottd polggim D can b0 drawn twAtAat tmy point of D i* iaaidt C and muA
that, giren at^ point PomC, a point Q on Dean be/tmnd $yek thai P<i<h.
TBBoaKM II. If /{,») bt eontinuoui tkromgkont C and its interior, thtn
f /{i)dt- I /(«)A can be wuuU arbitrarily tmall by taking h $uffieimUy
It ia obvioua that the oooditioo of § 14 ia aatiafted for polygoua, ao that if
/(«) be cuntinuoua throughout C and ita interior and if it be analytic inaide C,
( /(f) A-0, and therefore \ /(*)<ii-0.
Trbohbii I. Let the elementary ourveit which form C be, in order,
y^9\{x\ x-A,(y), y-^t(x), x-A,(y), ... y-y.(x), 4?-A,(y),
and let the interior of C be called S~.
Let * < lira sup \PQ,
where P, ^ are any two {X)int8 on C
Each of the elementary curvee which form C can be divided into a finite
number of aegmenta such that the sum of the fluctuations of x and y on each
aegmeot doe* not exceed ^ so that Urn sup PQ < \h, where /*, (^ are any two
pointe on one aegmeot. Let each elementary curve be dinded into at leaat
three such segments and let the segments taken in order on (? be called
o^, 9t, ... (r,«i, their end-iK)int» being called /*,» Ai ••• ''■♦i ("'^j)*
Choose JT <a BO that lira inf PQ > y, where P, ^ are any two points of
C whioh do not lie on the same or on adjoining segments*.
Oover the plane with a network of squares whoae aides are parallel to the
axes and of length ^ ; if the end-|)oint of any segment o-, lies on the skle of
a square, ahift the squares until thin is no longer the ease.
Take all the squares which have any point of a, inaide or on them ; these
squares form a aingle dosed region S, ; for if v, be on y«-^(«X the squares
fonning 8r can be grouped in oolumna, each column abutting on the column
on ita left and also on the column on its right Let the boundary and interior
of Sr be called C, and S,- respectively.
Then S, posseaaea the foUowiuK propertiea :
(i) Sr contains points inside C and points ouUdde C.
f This result will be obUined by lbs methods of de la ValUe Ponssin. CoMr«
i'Amtitm It^mMdwuU (1914). H S48-S44.
• 8ss note 15. p. 10.
38
CAUCHYS THEOREM
[CH. Ill
(ii) Sr~ has at least one point P,. (and therefore the interior of one
square) in common with iS'~^ + i.
(iii) Sr, Sr+2 have no point in common; for if they had a common
point P, points Q^, Qr + 2 could be found on o-^, 0-^ + 2 respectively, such that
PQv^k^'s/^, ^^r+2<i8V2, and then QrQr+2^i^' s/^ < 8', which is im-
possible.
(iv) Since Sr-i, S^+i have no common point, Sr consists of at least
three squares.
(v) 1{ i/=g(x) has points on m squares which lie on a column, the siun
of the fluctuations of x and y as the curve completely crosses the column is at
least (to — 1)8', (or 8' if m = l); in the case of a column which the curve does
not completely cross, the sum of the fluctuations is at least (to - 2) 8', (or 0 if
TO=1). The reader will deduce without much diflBculty that the ratio of the
perimeter of S^ to the sum of the fluctuations of x and 7/ on o-^ cannot exceed
12 ; in the figure, the ratio is just less than 12 for the segment orr + i-
If (Tr-i, (Tr, o-r+i be all on the same elementary curve, it is easy to see that
a point describing Cr counter-clockwise (starting at a jwint inside C and
outside C,._i, C,.+ j) will enter S~r-i, emerge from S~r-i outside C, enter
S~r + i outside Cand then emerge from S~r+i-
If, however, o-^-i, o-;. be on adjacent elementary curves, a point describing
Cr niay enter and emerge from S~r-i more than once; but it is possible to
take a number of squai-es forming a closed region <S/, whoso boundary is Ef,
consisting of the squares of S^ and Sr-\ together with the squares which lie
in the regions (if any) which are completely surrounded by the squares of
Sr and Sr-i- Then, as a point descrilxis Ef counter-clockwise, it entera and
emerges from S~r-i and S~r + i only once. If we thus modify those i-egioub
Sr which correspond to end segments of the elementary curves, we get a set
of ?«-!- 1 (< n) closed regions T,,, with boundaries Dp and interiors Tp~, such
17] OAUCBT't THEORBM
that D, mMto Dp, I but nap-uunwontif rafkna mn whoDjr «ittrml to do*
•ooUMr.
Now ooMkier the ato of each polygon D, which Um ontakfe T'^.i uid
r-y.i butinaider; tha« ovvrUppinf atoi fonn • oIomkI poljfaa J> which i*
wh«4l7 toiiife ^* v*^ ^'B" of At A* •••« owivrinf oo ii in ordar. Aim, if P
be aitjr point of #„ therD ia « |Kiint <^ of r^ or rr*i «biob b inaida a aqnaiw
which abuta on D, and therafore the diirtanoe of P from aoma point of D doaa
not exceed /V-*^K < i<-»-K < ^•
Theiirrai I ie thcnlara oomplatdljr proved.
Thkomui II. Let « be an arbitrary poattire namber.
(i) Chooae a ao amall that
l/(0-A*)l<A'^-'.
whenever jj'— ai<8d and «, / are any two |iuinta on or inside C, whUe
Li^Min where L ia the aum of the fluctuAti«iiiN of x and y uu C.
(ti) Cbooao such a {mrauieter t for the cune C that
i/(0-/(«)l<A«^"'.
whenever |f-l|<a: tht« i« obvioualy poaatble, for, if the inequality were
only true when f* - / ; ^ XA, where X ia a poaitive tnuuber lew than unity and
indepeiKlcut of <, we uliould take a new parameter r«X~'<.
It is evident fWran (i) that
i/(0-/(«)l<!^«^-'.
whenever 1 1' - 1 j <8 and s, x* are any two {loiiitM on C.
Draw the polygon D fur the value of d under oou«iderati<>ii, as in Thetwem L
Take any uno of the ciir>'es Dp ; if it wholly c«>ntAiiw more than one of the
regions S,, let thctu be .SV - 1. a^r- Then there in a )M>iiit x^ of »,. i or v, in one
f>f the nquareH <if Dp which abuts on Z> ; let (^ be a point on the aide of thia
aqiuuv which im |Mirt of D.
Then Sp^i in on tr, • t , aihI hence \tp^i-tp[ duos not exceed the aum of the
fluctiutioiut of X and y on <rr _ i , <rr« tr^ « t ; i.e. c^ «. | — <^ | < |d < d.
Alw> the arc of D joining (p to ^^^ i doea nut exceed 12 times the stun of
the fluctuations i^x and y on the atVM <r^_ ,, v,, v,,i and so doea not cxoeeil
3d : and the sum of the fluctiiatioiut of ^, if lut ( dewrribes />does not exceed the
sum of tlie perimeteni of the curves C„ i.e. it does not exceed Li^liL.
Take as the {lanimeter r, of a |H>iiit ( on />, the arv of D measured (V«im
a ftxed |>oint to (.
We can now omisider the value of I /{s)di.
By oiHKlitioiiM i) ajul (ii) ompled with The«in*ni II of § 11, we see that,
MKl ^ \^j^/{,)di- M*,.,-«,)/(«,j}|<|».
40 cauchy's theorem [CH. Ill
But If nz)dz-f fiodc
\J C J D
+|{//(0<^c-J^(fp.x-(rp)/m
I Ip— 0 p=o ; I
Write
so that
Then
ip=o I
/(Cp)==/(2p)+Vp. fp = 2p + »7p,
2 (2p + l-2p)A'2p)-(fp + l-Cp)/(fp)
p=0 I
2 [('?p-';p+i)/(^p)-(Cp+i-Cp){/(Cp)-/(2p)}]
p=0 I
I "* I
= 2 hp+l{/(2p + l)-/(2p)}-(Cp+l-^p)vp]|
m m
< 2 hp + ,{/(2p + i)-/(^p)}|+ 2 KCp + i-Uvpl.
p=0 p=0
Now
by condition (ii), while
2 |'7p+i{/(^p + i)-/(^p)}|<^(»i+l)S'eZ-S
p=0
2 |(Cp + i-Cp)vpl<3V*A-' 2 |fp + i-fp|
p=0 p=0
Therefore, collecting the results and noticing that (»i + l)S'<Z, we
see that
jj{z)dz-jj{z)dz
<f.
If now, in addition to the hypothesis of the enunciation of Theorem II,
that f{z) is continuous throughout Cand its interior, we assume that /(?) is
analytic in the interior of C, then /(«) is analytic throughout D and its
interior, and so I f{z)dz=0, by §16; and then, bv the result that
J d'
I f('^)dz\ < €, we infer that / f (z) dz^sQ. The residt stated at the be-
\J c I J c
ginning of § 16 has now 'been completely proved.
CHAPTER IV
MISCELLANEOUS THEOREMS
I 18. Change of variable in an int4*graL— § IB. Diflerentiation of an
integral with regard to one of the Umita.— § 20. UnifiHin differentia*
Ulity ittiplioH a utntiDUoua diflerential coefficient, and the ooa\
18. Changt t^ tfonable in an integral. Let C be the complex
coordinate of any point on a simple curve AB, with limited variationjf.
Let £ • jT (0 be a function of C which lta« a continuous differential coeffi-
deoi, g'iCi, at all points of the curve, so tliat, if C be any particular
point of the curve, given a positive <, we can find 5 such tliat
li/(n-^(C)-({'-C)^'(0!^«IC'-C!.
when \f -t\^h\ it being supposed that t, t' are the parameters of C C-
If /„ T be the parameters of .<4, B, suppose that z describes a
simple curve CD as / increases from <, to 7^.
Then the equation
/,„/(j(0)^(0<«=j^„/W*
%$ tru€ i//(z) be a continuotu function on the curve CD,
By Theorem II of § 11, K'iven any {M)Kitive number «, it is possible
to find a positive number S such tliat if any r numbers /|, <^, ■■■ t,
are taken so that 0 < <^, - 1, -> 5', and if TV be such that /, ^ T, ^ <,. „
then
l/<
Given the same number «, wo can find fi" such that if any r numbers
'i, <ti •.. tp are taken so that 0^<,.,-/,^«", and if T^ be such that
tp^ Tp^tf^i, then
\L^^
(0)?-(0J{- s ((,.,-(,) AX,) g{tt%)
pm9 I
42
MISCELLANEOUS THEOREMS
[CH. IV
where Wp, Zp are corresponding points on AB, CI) ; we take 8 to be
the smaller of 8', 8" and choose the same values for ^i, ^2, ... t, in both
summations, where 0 %tp^.i-tp%8, and we take Tp the same in both
summations.
Now divide the range ^0 to T into any number of intervals each
interval being less than 8 ; and subdivide each of these into a number
of intervals which are ' suitable ' for the inequality
Then taking the end-points of these intervals to be ^o> *i>
and, taking Tp to be the point of the joth interval such that
\9{0-9{^Vp)-a-Wp)g'{Wp)\%.\C-Wp\
at all points C of the arc CpCp+i of AB, we have
2 {zp,^-Zp)/{Zp)- 2 ap..-QAZp)g'{Wp)
p=0 p=0
K,T,
2 /{Zp) {g (^p,0 - g {Q - (^p., - Q g' ( Wp)}
j> = 0
2 AZp) [{g (Cp.,) -g{lVp)- (^., - Wp) g' ( Wp)}
p=0
-{9ar>)-9(Wp)-ap-Wp)g'{Wp)}]\
^ 2 \/{Zp)^{\Cp.,-Wp\^\Wp-(p\}\.
p=Q
Let L be the sum of the fluctuations oi $, t) on AB and let ML~^
be the upper limit of \J(z)\ on CD; M exists since /(z) is con-
tinuous.
Then, by the last inequality,
2 {zp,,-Zp)f{Zp)- 2 {;p.,-Qf{Zp)g{Wp)
p=0 p=0
iM.
Therefore
\\^j^n9{0)9'{^)di:-\^^f{z)dz
\l
,J{9{0)g{0dK- 2 {Cp^.-Qf{Zp)g{Wp)
AB p=o
-/<
/{z)dz+% {zp^,-Zp)f{Zp)
CD p=o
+ 2 {i:„^.-Qf{Zp)g{Wp)- 2 (^^-c„)/(Zp)
< (2 + M) € ;
18-19] ummLLkvmo^ thboiicmm 4S
■inoe M b fisad, c m vbitnmly lunall aud the two iuiegnht aasl» we
inf«r Uuit
j,,y(»(f))»'<o*-/,^/<.)A.
CaroUanf. Tftkiiig/(«)» 1, we tee that
thin is the fonuttk for the integml of a contitiuotui difleroiitul ooefli>
citMlt.
19. Djfinmiiaiitm t/an imUgral tcitA regard to one qfths limit$.
Let .4 A be * regular unclosed curve such that if any point P on it
be taken, and if Q be any other iwint of it, the ratio of the sum of the
ranations of the cur\'e between P and (^ to the length of the chord
PQ haa a finite upper limit', k.
iM /{s) b» eomtimmmg on th* curve and let z^t Z, Z-^k he any
tkree points on it ; tkm if z^ be fixed, I /{t) dz it a /unetion qf Z
onijf, sajf ^{Z); and
!-•• n
t ietke difference qftke parameter* qf Z, Z -^ k.
We can find S lo that /(Z+k) -/{Z) [ < « when t < B, where < is
arbitrary.
Now
k-'\Mz*k)-^(zn'k-' j'^''Az)dz
= A- lim 2 /(Z * kn . (Aro<'» - ir*"X
where ^••"•^O, i..,'"» = i; it being suppcMcd that the pointe V are
chosen in the itame way as the points £,**' iu § 10 of Chapter II.
Therefore
\k-'\Mz^k)^^{zn-AZ)
~\k-' lim i/(Z^krn.(K,r-kr*'*)'k-'/{Z) i (il,..<-»-A,«"»)
- \k-* lim J \AZ^Kn"AZ)\.(K,r -A.''')
1 — -^
' Thb eoodiUoQ U Mlirtad bjr most eurt— which oeeur in fnetict.
44 MISCELLANEOUS THEOREMS [CH. IV
^\h-'\ lim ^\{/{Z+hrn-/(Z)}\.\hrJ'^)-hr^*)\
n-»oo r=l
<|A-M.«2 |A,,,(-)-A,(»)|.
r=l
But, by the lemma of § 9,
\h-'\ 2 |A,+i(»)-A,(")|<A:,
r=l
80 that I h-' {i>{Z + h)-<f> (Z)} -/(Z) I < k€,
since « is arbitrary and A; is fixed, it follows from the definition of a
limit that
20. Uniform differentiability implies a continuous differential
coefficient, and the converse.
Let/(s) be uniformly differentiable throughout a region ; so that
when c is taken arbitrarily, a positive number S, independent of z,
exists such that
\f{z')-f{z)-{z'-z)f'{z)\%^^\z-zl
whenever \zf — z\%h and z, z are two points of the region.
Since \z — z'\^^, we have
\f{z)-f{z')-{z-z')f'{z')\%\.\z-z'\.
Combining the two inequalities, it is obvious that
\{z-z){f'{z)-f'{z)\\^^^\z-z\^\.\z-z'\,
and therefore I/' {z) -/' («) I ^ «,
whenever \z -z\%^; that is to say, /' (s) is continuous.
To prove the converse theorem, let/'(2;) be continuous, and there-
fore uniformly continuous, in a region ; so that, when c is taken
arbitrarily, a positive number 8, independent of s, exists such that
\f'{z)-/'{z)\^h,
whenever \z' — z\ ^8.
Consider only those points z whose distance from the boundary of
the region exceeds 8 ; take '^ z- Z\ "$8.
Then since /(c) is differentiable, to each point (, of the straight
line joining z to Z there corresponds a positive number 8^ such that
whenever | {' -f j ^ 8^ and C is on the line zZ.
lO-SO] maClLLAWIIOUa TUBUIIDIIi i^
By (}<MirMt'ii bmna, we nay divMia the line tZ into a ^iu
nnmber of inUmraU, ny %t Um pointo {.(*<)• Ci» {>•••{•. d.i(*i?)*
•ooh that there is • point z, in the rth intervkl which i« Kuch that
!/(0-/<«r)-(C-«r)/'(«r)l<4«l«-«r!.
for all pointe C of the intenral.
Therefoie /(t)-/(«r)-(C-«r)/'(«r)-«v(Cr-«r).
AIms since |Sr-«l<^
/'(«r)-/'(*)^1»..
Therafore /(C) -/(t-,) - (Cr - C-,)/' («r)
Taking r • I, S, ...«•*■! in tarn, and summing we get
/(Z)-/(s)-(Z-x)/'(.-)
= 'aV(Cr-Cr-,)*"a K(t-«r)-«'/(Cr-,-*r)}.
r-l r-l
But« tinoe the points C(=<)> ^i Ci. <••••{.. £.. {•*i(-Z) are in
order on a atimigfat line,
M-l
r«l
audio |/(Z)-/U)-(^-«)/'(«)l^i«!^-«''^i«'^-«!.
wheMTW I Z - s I < S and the dirtance of z from the boundary of the
region does not exceed K Therefore, if/' (;) is continuous throughout
a rcgkn, /(<) is uniformly differentiable throughout the interior of
th«r«gioik
The reader will find no difficulty in proving the corresponding
tbaorems when /(;) is uniformly differentiable or when/'(r) is con
tioooiis, and ; is, in each case, restricted to be a continuous function of
a real variable t.
CHAPTER V
THE CALCULUS OF RESIDUES
§ 21. Extension of Cauchy's Theorem. — § 22. The diflferential coefficients
of an analytic function. — § 23. Definitions of pole, residue. — § 24. The
integral of a function round a closed contour expressed in terms of the
residues at its poles.— § 25. The calculation of residues. — § 26.
Liouville's Theorem.
21. Let C be a closed contour and let f{z) be a function of z
which is continuous throughout C and its interior, and analytic inside
C. Let a be the complex coordinate of any point P not on G. Then
the extermon of Cauchy's theorem is that
-^ f i^dz = Q if P be outside c]
2Tn jc z-a J- .
= /(«) if P be inside C J
The first part is almost obvious ; for if P be outside C it is easily
proved that f{z)/{z - a) is analytic at points inside C and continuous
on C. Therefore, by the result of Chapter III,
2W
iirt Jc Z-a
Now let P be a point inside C.
Through P draw a line parallel to Ox; there will be two^ points
Qi, $2 on this line, one on the right of P, the other on the left, such
that Qi, Q,, are on C, but no point of QiQ^ except its- end-points lies on
C. [The existence of the points ^i , Q. may be established by arguments
similar to those in small print at the foot of page 11.]
1 Points on the line which are sufficiently distant from P either to the right or
left are outside C. Since a straight line is a simple curve, the straight lines
joining P to these distant points meet C in one point at least.
21] TiiK CAUrULua or RBIOUW 47
Vi. <A divide C luto two iMirt« <r,. <r, with tlie mim oheaiftliont m
(' ; l«t <r|. If, b« ch<Men m tliat (/, . V, are the eiulpotnli of the orienlad
etirre v,, aod (/„ (^, are tlie end-pointJi of tlis oriented curve «r,.
Let the ahortest dietanoe of pointu on C from Z' be S, ; cbooMe B to
that ScSi. S<1 and
/*(»)-/(a)-(«-a)/'(a)|<«|»-«|
whan |«-ai < £» where « it an arbitrary positive number; draw a circle
with eentre /' and radius r ( <; ^S).
liOt QiQt meet this circle in Pi, l\ ; let tlie upper lialf of the circle,
with the oneutation (a- rya-¥r) of it« end-point«, be called Bu and
let the lower half of the circle, with the orientation (a -f r, a - r) of ita
end-pointa, be called lit.
Let the circle, properly oriented, be called C„ ao that tlie orienta-
tions of Bi and /A are opposite to that of Ci.
Proofii of the following theorems are left to the reader :
(i) ^u QtPu Bu PiQi form a closed contour, C^ properly
oriented
{>») »«. ^Z*!. Bu PtOt form a closed contour, C,, properly
oriented.
<iii) P is outside C, and Cf
(iv) /i:)l{z - a) 'i» analytic inside Ci and (7, ; and it is continuous
Uiroughout the regions formed by 6\, C, and their interiors.
Now consider
JCiZ-a Jc,s-a
48 THE CALCULUS OF RESIDUES [CH. V
The path of integration consists of the oriented curves o-j, o-j,
P^Qu QxPu P2Q,, Q,P„ Bu B,.
The integrals along the oriented curves o-j, o-j make up the integral
along C. The integrals along the oriented curves P\Qi, QiPi cancel,
and so do the integrals along the oriented curves P^Qi, QtPi ; while the
integrals along the oriented curves Bi, B^ make up minus the integral
along the oriented curve C3, since the orientations of B^, B, are
opposite to the orientation of Cg.
Hence
jCyZ — a jc^z-a Jc z-a Jc^ z-a
Now f{z)l{z - a) is analytic inside Cj and C^ and is continuous
throughout the regions formed by Ci, C^ and their interiors. Hence, by
§ 17, the integrals along C^ and G^ vanish.
Hence' i^^ dz= l -^dz.
JCz-a Jc^z-a
Let /(z) -/{a) -{z- a)/' (a) = v{z- a),
so that, when 5; is on C3, | v | $ c
Then
[ fM dz^fia) I — +/'(«) [ dz+ [ vdz.
Jc^z-a ''^'jc^z-a •" ^ ' JC^ JCs
But, since C3 is a closed curve, I dz = 0, by Theorem HI of § 13.
r dz
To evaluate / , put z = a + r (cos 0 + i sin 6); 6* is a real
JC3 z — a,
number and is the angle which the line joining z to a makes with
Ox. Consequently, since a is inside C,, 0 increases by 27r as s de-
scribes C3.
Hence, by the result of § 18,
f dz _ /■»+2» - sin ^ + i cos 0
Jc^ z — a~Ja cosO + i sin 6
= 27r/.
dd
Therefore f P^ dz-2m/{a)= j vdz,
JC2 z a JCf
* This result maybe stated "The path of integration may be deformed from
C into C3 without affecting the value of the integral."
SI-SS] THB CAVOVhVM Of RniDUKH 40
Putting s»« ^r(o(Ml 4- lainlX ««g«t'
when i;**<<r*i«. ^••^-<.. «.♦,••♦ -<; + J»».
Heooe /c 0i ^- " ^''■•^<'') |
is leas than 2«rc, where r < I and € iN^rbitmnly Huiall. ('onseiiueiitly
it must be leru ; that in to nay
Jc s-a ''^ '
22. Let ^' be a closed contour, and let /(s) be a function of z
which iM analjTtic at all pointa in»ide (' and continuous throughout C
and '\U interior ; let a be the complex coordinate of any |)oint inaide C.
Then /{:) pmmmM mmi</«^ di^errntial ctt*>fficiemU qf' aU ordtn at a ;
ttttd
d'Aa) H}^ [ /(*)
da* 2w» Jc iz - «!•*•
All points sufficiently near a are inside C ; let S be a positive
number «uch that all points Natisfying the inei|uality z-a < 25 are
inxide C ; and let i be ciajr complex number such tliat , A I < £.
Then, by 8 21,
* The BOtalioo of Chapter II i» hrin« etnplojrvd.
w. c, L 4
50 THE CALCULUS OF RESIDUES [CH. V
Therefore
/(a+A)-/(«)
2m jc (z - af 2m jc (z - af (z-a - h)
h
Now when z is on C,
\z-a\^2h, \z-a-h\^h, and* \f{z)\<K,
where A' is a constant (independent of h and S).
Hence, if L be the sum of the variations of x and y on C
lc(z-ay{z-a-hy^\^ }c
{z-aY{z-a-h) \ jc\{z-af{z-a-h)\
Therefore
f{a + h)-f{a) 1 r J>)
A 2mjc{z-aY '
where | v | < | A | ZX/(87r83).
Hence, as ^ ^ 0, u tends to the limit zero.
Therefore li„ /("-tiiz/W
has the value ^p-. I . ^^'vjO?^; that is to say,
^TTI JC \Z — CI/)
da-^^""' 2m}c{z-af'^^'
The higher differential coefficients may be evaluated in the same
manner; the process which has just been carried out is the justification
of ' differentiating ^^^th regard to a under the sign of integration ' the
equation
'^ ^ ' 2m jc Z-a
If we assume that —j\ exists and
{a)_n\ ( f{z) ,.
d^f{a)_n\ f f{z)
da
* On G, the real and imaginary parts of f{z) are continuous functions of a real
variable, t; and a continuous function is bounded. See Hardy, A Course of Pure
Mathematics, § 89, Theorem I.
23- 24] TUI CAVCVtV* or UBMDinHI 51
m •imiltf proeaM will jiuttfy diffenrntutiBg Uiii equation with ragard
to • onder the aign of int<»frmti<»»i <k> that
But (6) in true when « -^ I ; hence by (6ri), (6) in true wlien n ■ 9 ;
and hence, by induction, (6) i« true for all puiitive integral value* of n.
23. Dsnytnova. PoU, Rtmduf. Let /(«) be continaotu
throughout a cloeed contour C and itji interior, except at certain poinu
(I,, a,, ... a., inmd* C, and anal3rtic at all poiutii in«ide C except at
a„ a„ ... a..
Let a function ^ (s) exist which DatiMfien the following conditions :
(i) ^ (z) is continuous throughout C and its interior, analytic at
all points inside C.
(ii) At pointM on and inside C, with the exception of a„ a,, ... a.,
/(s) = ^(c)+ 5 ^,(5) (7X
Then /(c) is said to have a poh of order Mr at the point a, ; the
coefficient of (z -ar)'\ viz. f\,, is called the residue of /(*) at a,.
It is evident by the result of Chapter III tliat
/.
^^(z)di = 0;
m that, by (7), j^./(^)^- = ^^ \c^r{z)dz.
24. TluN hut eiiuation enables us to evaluate {/(z)dz\ for
consider \^,(z)dz. The only point inside 6^ at which ^(c) is not
analjrtic ii* the point z^a^. With centre a^ draw a circle C of positive
radius p, lyinx wholly inside ('\ then by reasoning precisely similar to
that of li 'i\, we can defomi the path of inte^p^tion (' into C without
aflfecting the value of the integral, so tlmt
4-J
52 THE CALCULUS OF RESIDUES [CH. V
To evaluate this new integral, write
z = ar + p (cos 0 + i sin 6),
so that 6 increases by 27r as z describes C" ; as in § 21, if a be the
initial value of 0,
r fa+tir
j ^<f)r(z)dz= j <f>r(z)p(-smO + icosO)dO
^r fa+2ir
= 2 6«,r/o^-*'^/ {cofi{n-l)0-lsm{n-l)e\dO.
Now it is easily proved that
'a+2n gQg
/:
m6de = 0,
sm
if m is an integer not zero.
Therefore I , ff>r (z) dz= \ bi^ridB = 2 W 6i, r . '^
Therefore finally,
^^f{z)dz^lj^<i>r{z)dz
m,
r=\ '
This result may be formally stated as follows :
Xf f{z) he a fwmtion of. z analytic at all points inside a closed
contour C with the exception of a number of poles, and continuous
throughout C and its interim- (except at the poles), then I f(z)dz is
J ^
equal to 2Tri multiplied by tJie sum of the residues of f{z) at its poles
inside C.
This theorem renders it possible to evaluate a large number of
definite integrals ; examples of such integrals are given in the next
Chapter.
25. In the case of a function given by a simple formula, it is
usually possible to determine by inspection the poles of the function.
To calculate the residue of f{z) at a pole z = a, the metliod
generally employed is to expand f{a + t) in a series of ascending
powers of t (a process which is justifiable** for sufficiently small
values of |*|), and the coefficient of t'^ in the expansion is the
required residue. In the case of a pole of the first order, usually
" By applying Taylor's Theorem (see § 34) to {z-a)'*f(z).
24-26] THB CALOULUtt OF ElSlOUn 58
rallMl a simple \n>U\ it U fMMnUjr sboiter, in pnotioe, to eTaliutto
liiii (t - «)/<«) by the rules toft daterminiiig limito ; • oomideimtioo of
the 6Xpree«ioii for/(s) in the iioiKhbourhood of a pole ahawt that the
reRitiuc {m this limit, providod tliat the limit exista.
26. LaouviLLB'8 Tiikokkm. /W /(z) tf0 anaiytic /or all vahm qf
z ami tt^ iy(s) ! < K trAiTf A' i/» <i cuiittant. TAm/(z) in a cntutant.
l^et c, z be any two points and lot C be a contour «uch tluit :. z'
are inside it ; then by g 29
take 6' to be a circle whose centre is z and whose radius is p > 2 1 s' - : |.
(>n C write C= s + p<^; since ( C- s' I > ip when C ia on C, it follows tluit
<2\z-z\Kp-\
Tliis is true for all values of p > 2 1 r* — s i.
Mttke p-*- X , keepinjj z and z fixed ; then it \a obvious tliat
/(2')-/(*)-0 ; tliat is to say, /(c) is constant.
CHAPTER VI
THE EVALUATION OF DEFINITE INTEGRALS
§ 27. A circular contour.— § 28. Integrals of rational functions.— § 29. In-
tegrals of rational functions, continued. — § 30. Jordan's lemma. —
§ 31. Applications of Jordan's lemma. — § 32. Other definite integrals. —
§ 33. Examples of contour integrals.
27. If/(^, y) be a rational function of w and y, the integral
r.
Jo
/(cos e, sin B) dS
can be evaluated in the following manner :
Let z^cosd + i sin 0, so that z~^ = cos6- i sin 6 ; then we have
wherein the contour of integration, C, is a unit circle with centre at
z = 0. If/ (cos 0, sin 6) is finite when 0 < ^ $ 27r, the integrand on the
left-hand side is a function of s which is analytic on C ; and also
anal3rtic inside C except at a finite number of poles. Consequently
L
J (cos e, sin &) dO
0
is equal to 2tri times the sum of the residues of
«-'^-v{^(^+^-o, ^-(^-^-o}
at those of its poles which are inside the circle ls| = 1.
„ , r^'' dO 27r , . , . .
Lxample. j . — - = ^/7 i"^ji\i t"^^ **5"^ being given to the
radical which makes \a - J{a^ — ^01 < l^i) *^ being supposed that a/b is
not a real number such that^ - 1 $ afb % 1.
' Apart from this restriction a, h may be any numbers real or complex.
S7-S8] THE EVALUATIOH OF DBTINITB nrnBOmiLS 55
MiikiiiK th« above •nbcUtatioD,
iiUen % fi an th6 roou oi b:* -*■ iaz ■*■ f> = n. The polm of the intogruid
are the iiointe s * «, £ ■ /8.
Since <i/9 1, of the two numborh a , /9' one in greater than 1
and one leiw tliaii 1, unlesM both are equal to 1. If both are eqnal to 1,
put •oooey-ficiny, where y U real; then /8 = a-*«oo§y-i«iny, so
tlint Si^5 « - « - /J « - 2 coe y, i.e. - 1 < a/^ < I, which ii» contrary to
liyiM)llieiii.
Let..-.ty(«'_r*5. /g.r£r.>^«lz*5,u«it«gn being given
til thenulical which inakee la- ^(a*-ft^l <I6! ; so that :a|<l, |/9|>1;
then s - a is the only pole of the int^^nd inside C, and Ute reeidue of
K«-«)(5-i9)}-atc = «i«(«-^)-'.
TTieiefcie f » ^ = 2»* ^ .. x — «
/o a -»- fr cos B lb a- fi
28. 1/ P (x\ Q (x) U polffnomiaU in x nieh that Q (jt) ktu no rtal
liHfiir J'ttcior* and tin degree itf' P (x) it Um than the d«grm4/Q{x) by
(* P(x)
at Uad % tk*n * | vr-W dx i» e*fual to 2-ri time* the $um qf tkt rmdmm
</ P(:)/Q(i) at tko9e qf iU poles which He in the ha{f plamabow tkt
real axit.
liCt P(j') = a.j* + a,jr»-' -»• ...+a,,
Q (x) = b,jr + 6, J'-* ^ ..."». 6.,
where « - n > 2, fi, 4i 0, A. 4' 0 ; choose r m Urge that
r r r*
and -i- + ^ + ..,* —■ ^ J A,' ;
* TIm nmiet will rtmMnbnr that »n Inflnilc inU^tnU U de6n«d in tb« following
: If l/ij)4s^0(lt),ihm j /{x)dx dmuu litn 0{n).
56
THE EVALUATION OF DEFINITE INTEGRALS
[CH. VI
P(.^\=
a, Oh
«,, + — +...+^
^1 1 l«ll l«lil
'5 2|aol.
Q{^)\^\K\-
>
\K\-
\b,\ \b,\
r r^
\b„,\
and
so that, \i\z\^r, then Ic™"" P (c)/Q (~)l ^ 4 iao^o"'!-
Let C be a contour consisting of that portion of the real axis which
joins the points — p, +p and of a semicircle, r, above the real axis,
whose centre is the origin and whose radius is p; where p is any number
greater than r.
f P(z)
Consider | ^/ ( dz ; this integral is equal to
Jc WAS)
/:
Piz)
,Q{z)^''' hQ{z)
L
^i(^h^
dz.
Now F (z)/Q (z) has no poles outside -the circle |s| = »* ; for outside
this circle \P {z)/Q (s)\^A |ao^o~' I r""'".
Therefore f ' ^l dz + [ ^. dz
J -p (^ {z) Jt V W
is equal to 2in times the sum of the residues of P (z)/Q (z) at its poles
inside C ; i.e. at its poles in the half plane above the real axis.
. Further, putting z = p (cos 6 + i sin 6) on r,
\ f P(z) I I /■"■ P(z)
/ 77/ \<^-h= I ttM P (cos ^ + «■ sin ^) ^rf^
\Jr H{z) I |./o ^(c)^
f' I P (z')
^ r ^\aoh-'z'-'"\pde
Jo
Since w -»i + 1 ^- 1, lini / ,, y( dz ^ 0.
p-» Jt Q (-)
:;m) thb bvaluatiox or DtrufiTB iwnottAUt tl
..«^
liu> /'
where Sr^ nittiiui the Mum uf the raudnet of P(:)!Q(s) at iU polet
ill the hulf plane above the real axU.
Sbea it luM ht<en lihewii tliat lini / .. ^'Vc-O, thiit is the theorem
fitateiL
Kmmpl,. (/«>,,. j'^ y^^ . r . f ^ ■ . ^
The Talue of the integral \» twi tintos the randue of (•'•l-a')'' •( a' :
The randue ia therefore - 17(40*) ; and hetioe the integral is equal to
If Q(jr) haM non-repeated real linear factors, the priitnpai mJmf*
of the integral, which is written /' i ^/''dLr, exist*. //* raim^ m
iwi time$ the turn </ tkf rrmdHen of P(:)!Q(z) at tkote qf it* polm
wkirk tie in tJkt> ka(/ plane aU>rr tkr retil axig plus iri tiaiet tk» turn qf
tka rtmdm$* at tkute qf the polet trkirk lie im tke real axis.
To prove thiM theorem, let <i/ be a real root of Q (x). Modify the
contour by omitting the {MirtA of the real axis between a,' - &, and
a,' *■ I, and inserting a semicircle r^, of radius i^ and centre a,, above
the real axis ; carry out this proceKN ftir each real root. When a
contour is modified in this way. ho tliat its interior is diminished, the
contour in said to be imtmted.
The limit of the integral along the surviving parts of tlie real ax\*
' * If f \
when the numbers 4- tend to lero in P \ .,,-( Jjr.
' Mine* lim I tsisu. it U equal to Itm I
A*"** .'-# 0 -»• } #
* Bromwieh, Thtnr^ of ImUmiU S*rin. p. 41A. Tbe u*r of lb« l«Mrr /' tn Iwo
will nol autm eoofusion.
58 THE EVALUATION OF DEFINITE INTEGRALS [CH. VI
If Vp be the residue of P{z)IQ{z) at ap', the integral along the
semicircle y^ is - / q¥^ {z - o,p) idd, where z = a^ + ^pS*', and this
tends to - -jrirp as Sp-^O; hence
7
^ > .da^-iril r„' - 2wi 2 r„ .
29. A more interesting integral than the last is / n7~i ^^*
where P {x), Q (a:) are polynomials in a: such that Q (.r) does not vanish
for positive (or zero) values of x, and the degree of P (.r) is lower than
that of Q (x) by at least 2.
The value of this integral is the sum of the residues of
\og{-z)P{z)iq{z)
at the zeros of Q {z); where the imaginary part of log (- z) lies
between ±eV.
[The reader may obtain the formula for the principal value of the integral
when Q{x) = 0 has non-repeated positive roots.]
f P(z)
Consider / log (- z) ^ yi dz, taken round a contour consisting of
the arcs of circles of radii ** R, 8, and the straight lines joining their
end-points. On the first circle -z = Re^^ {- ir ■^ 6 % it) ; on the second
circle —z = he^^ {-tt^B %Tr). And log (- z) is to be interpreted as
log l^l +earg(-c), where — tt ^ arg (— c) $ tt ; on one of the straight
lines joining S to R, arg (- z) = tt, on the other arg {- z) = - ir.
* (The path of integration is not, strictly speaking, wliat has been
previously defined as a contour, but tlie region bounded by the two arcs
' In future, the letter 11 will not be used to denote ' the real part of.'
S8-t9] THE EVALUATION OP DEFINITE INTEORALH 59
und Um ftnigbt linM in olmonaly one to which Qoarwfs Imnnui
Mild the Mudjnu of Chaptor III can be applied.)
n'iU ikt omwmiimt asto]og(-g\ log(-£) P(t)IQ{z) is ammiftie
imtitU tMt txmtomr ^jtnpi at tk§ unt qfQ{x).
Aa ill S 2K we can chooM >?« and ^ ao that \t^P(s)/Q(z)\ doea
not exoeed a lixeti number, JC, when \t\<'B>B; and no that
\P(*)/Q (^)\* ^ whc" - ^ < ^; where IC in independent of S and //.
Let the circle of rmliuH It be called r, and the circle of radiiu i be
<-allo<i y ; let c,, r, be the linea arg (- :) *• - ••, arg (- x) = ».
Then tlie integral round the contour is 3W times the sum of the
nwiduw of Utg (- z) P (z)/Q (z) at the reroe of Q{z) (theee are the
••nly \to\e* of the integrand witliin the contour).
But the integral round the contour =/ -fl -t-l -*-|>
which -»0 as R~^x> , gince /i~* log /f -►O as /f-»x .
< [' logS + i^ATWtf,
which tends to U as 6-^0, since SlogS-«>0 as S-»0.
Put - s - jtt-** on C| , - 5 - xe*' on c,. Then
nnd / / (#«• + l«>g J-) ^r-T-N <'•'*.
Hence 2W tiine« tlie «um of the residues of log (- z) P {i)IQ(i) at
the leroH of <J(z)
which proves the proposition.
60 THE EVALUATION OF DEFINITE INTEGRALS [CH. VI
The interest of this integral lies in the fact that we ai)ply Cauehy's
theorem to a particular value (or branch^) of a many valued function.
If P {x), Q (x) be polynomials in a* such that Q (.r) has no repeated
positive roots and Q (0) + 0, and the degree of P (.r) is less than that
of Qix) by at least 1, the reader may prove, by integrating the branch
of (- z)"'^ P {z)IQ {z) for which | arg (- c) | ^ tt round the contour of
the preceding example and proceeding to the limit when S-»-0, 7?-^* ,
that, if 0 < a < 1 and af^~^ means the positive value of a^~^, then
r p (x)
P \ ^'^ Tr)\ dx = Tr cosec (air) "Xr — Tr cot {air) 2r',
.'0 H K^)
where 2r means the sum of the residues of (- c)""' P (z)/Q (z) at those
zeros of Q(z) at which z is not positive, and 2r' means the sum of the
residues of x^~^P(x)IQ(x) at those zeros of Q.(^) at which x is
positive. When Q (x) has zeros at which x is positive, the lines Ci, c,
have to be indented as in the last example of § 28.
Examples. If (i<a<l,
r xf-' . TT „ r ^-' ,
I . ax = -. , P I dx = tr cot air.
jQ l+x sm aw J a \-x
30. In connection with e.xamples of the type which will next be
considered, the following lemma is frequently useful.
Jordan's Lemma". Let f{z) be a function of z 2vkich satisfies the
folhiinng conditions ichen \z\>c and the imaginary part of z is not
negative (c being a positive constant) :
(i) f(z) is analytic,
(ii) \f(z) I -^ 0 uniformly as\z\-*'cc.
Let m be a positive constant, and let T be a semicircle of radius
R (> c), above the real axis, and having its centre at the origin.
Then lim ( I e""'f(z) dz) = 0.
ii-*.» \.'r /
If we put z = R (cos 6 + i sin 6), 0 increases from 0 to x as c de-
.scribes r.
Therefore / e'"^'f(z) dz = (' ze"'''f(z) idO,
Jv Jo
• Forsyth, Theory of Fiinctioim, Chapter vni.
'' Jordan, Coi(r» (VAtuilyse, t. ii, § 270.
29-ai) THK KVALt'ATlOJf Of DRriNITK IXTBOEAUI 01
<jfjit#-/(«)|^
< r !»/?#-"«*'• d».
wliere i| iit th« grafttott value of i/(s) ; wheti s.^ B.
Ill the iMt iiit«gnil, divide the range of integration into two parts,
vii. from o to |v and from |r to v ; write v - 0 for 9 in the taoond
part; then, noting that, when* Q^$^\w, nn0>90/w, lo that
, ■ tjita* < 4 -teM^, «« Me that
^''^Z.
<^(l-'-*)<f«»-».
= 0.
But If -»0 a« /^-^ X ; and thenjfore
lim I / <r*-/(s) d:
K^» ' Jv
3L 'riio following theorem may be proved, by the uoe of Jordan's
lemma :
LM P(jf), Q(jt) b« pofymtmial* in x fttrk that Q{x) ktu ho r«*ii
liimtr/aeior*, amd tk» d»grm (^ P (x) does wd extxtd the dsgm qf Q (<r) ;
and Utm>Q.
"- /;{^-8--»-}^
M eiimU to wiP (0)/<^ (0) ptn* 2*i timet the $um qf the rmdttee qf
P(z)^
,j4-( -^ at the term <if Q (s) in the ha{f' phne aborm the real axie,
[The reader msy obuiii the funutiU for the prindpiU valu« of Um integnU
wh«m ^(x)«0 ban iM>ii>re|iMU«d real r«M>tA.]
Consider / ttt-I^^ — taken aloii*{ a contour C cotutisting of the
stfaiglit line joining - ft to -^ a semicircle, y, of radius 5, alxtve the
real axi«, and with it^ centre at the origin, tlie utraight lino joining; £ to
* It w«di»w Uw Rimphs yacinx. ystU/r, UiU inequality appHU* obviotu; il
■in 0
may be proved by •hewing that —^ diwwiiii ae 0 inewawi from 0 to |v.
62 THE EVALUATION OF DEFINITE INTEGRALS [CH. VI
R, and the semicircle r, where 8 < 80, i? > i<:,„ and 8 and It are to be so
chosen that all the poles oiP(z)/Q (z) lie outside the circle is| = 80 and
inside the circle |5;| = ^o-
~^. e""" -^ is equal to 2m times the sum of the residues
P (z) e"*^"
of Qj^. — at the zeros of Q (z) which lie in the upper half plane, as
may be shewn by analysis similar to that of § 28.
.dz
Then /
Jc
„»II2
JcQ{z) z \)-R Jy Js hJQ{z) z
Put 5 = - a? in the first integrand, and z=x\\\ the third ; then
\)-R^ Jb )q{z)' z h \Q{xY gpio It-
Since lim P(z)/Q(z) is finite, /" ^^^--^^o as 7?-x, by
Jordan's lemma.
Also, putting ;:; = 8«'* on y,
where \f{z)\ does not exceed a number independent of c when 8 < 8„.
Hence lim / fi!-).-^ = -„-^).
5*0 Jy Q{Z) Z q (0)
Making i2-.-x, 8-^0 in the above formula for f TrP^e'"" — , the
JC (/ (5) z
result stated follows at once.
Carollaiy. Put P (z) = Q(z)=l; then, if m>0,
sin ma:
1:
dx = W.
/o x
By arguments similar to those used in proving Jordan's lemma, we
may shew that
1:
^acosto gJn (^ gjjj ^^) j:^^ ^ 1 ^ (^^e-». _ j)^
'0
if a > 0, 6 > 0, r > 0.
f zdz
Consider / ^«^ -* ^^^ , where the contour C consists of the straight
line joining the points - H, E and of the semicircle r, where R > 2r.
The only pole of the integrand inside the contour is at z = ri;
and the residue of the integrand is easily found to be !<?«*"*'.
HI ) THE eVAIA'ATloN OF DKKINITB INTIOIULN
III the fimt iiiioKml put s * -x, in the lecond put s » x.
Tlieii
Now <»«^- I +<M»*-^ — o^»** + ...
wfcfre l«'l"l**Tr'*""8T ■*■'
<I +
a#^
and ou T, |^| = ^-*««»n» < i, where z==B(cM0*Uin0); so that
Conseriuently, putting c = /f (cob ^ + i sin $) on T,
9:^+1^ Jo s" + r*
But |s«^-r'i>!s«|-|r»i>J/ir»,
and therefore / . — , < r-ss.
■ |./# 5" -♦■HI 8/f»
whiK- r,ii^ te .^ i(/^| K r I a«^' ip --'* _, i\/0 1
.'0 z'+ r I /o I c* + r* 1
.'0
< ^ <r. , as in Jordan's lemma.
lV)ii«e.jueiitly /.''***' r« /r" ' " ^ **'
where hin «k»0; and therefore, by the detinition of au infinite
inti'K'ral,
64
THE EVALUATION OF DEFINITE INTEGRALS [CH. VI
32. Infinite integrals involving hyperbolic functions can fre-
quently be evaluated by means of a contour in the shape of a rectangle ;
an example of such an integral is the following :
/■" cosh ax , , , ,
; — -, ax = hsec ha, wmn -■7r< a <Tr.
Jo coshirar
Consider I — ^ — dz taken along the contour r formed by the
Jr cosh TTZ
rectangle whose corners have complex coordinates -R, R, Ii-*-i,
-B + i, where R>0; let these corners be A, B, C, D. The zeros rf
-R-f i
I
R + t
D
'¥
C
A
B
cosh TTZ are at the points s = (w + 1) i, where n is any integer; so that
the only pole of the integrand inside the contour is at the point z-^i.
li z = \i + t, then
cosh irz i sinh irt
_ ^ {l+at + ha'f + ...)
-^' (l+^7r2>+...) '
so that the residue of e*" sech -rrz at U is e^^^Kvi).
Therefore
2e^
Jr cosh TTZ
Now I ^^ dz=(( + I + / + f ),:^„J^-'
Jv COSh-n-Z \JaB Jbc J CD J DA / ^OSn TTZ
on A B we may put z = a: where x is real ; on CD we may put c = / + ^
where x is real ; on BC we may put 5 = ijJ + iy where y is real ; and on
DA we may ipntz=-B + iy where y is real. Therefore
, vi2 .R ax f-R ^(i + J)
jr cosh its; ; _ij cosh rr.v Jr cosh tt (2 + x)
■" jo coshVpT^/'^^'' A cosW(^^>) "^
= (l+f^"')/
cosh ttX
dx + <^,
32-3.')] THE EVALUATION OP DEFINITE INTEORALH 65
«1ko / — p — Jjfm I — r dje*l —-r — dm
" ]% cosh vj* /# oathvar '
on writing - jr for x in the iiecond integral.
Therefore 2#***-2(l ■^o/'^^^ + «Jl•
Now I..I ^ I j^ ^^^ ^ ^^^ ^ ^^^ .^y I . /^ ^fi^OR^) '^ I
jo lcoi»li»i^/t'i- »y)| Jo looehrC^-ryJl'
A1m> |2ooA»(/?±i»i«j<r<«*'r)-f«— (**<»)|
/•I ^R fl g-att
Tlierefore |«iii^/ T-r — ndy+l . v — ^</y
' *' Jo Binh irR ' Jo 8inh»/f ^
^2 cosh (o^)
sinh (irH) '
But, if - ir<a<», lim 2 . ,) ui = 0: therefore, if -»<a<»,
«♦• sinh (irJi)
lim «j| - 0.
But / — , — (ir is equal to -r-r, 3\ ; *nti therefore
Jo com wx ^ 2 (1 -»• «r)
[ COAll fill*
/ L <ir=i8ecla, (-«"<a<»).
33. Solutions of the fulKiwing exainplcK, of which the earlier ones
are taken from recent College and rniven<ity Kxaininatiou Pa{)er8, can
be obtained by the methods <ievelupe<I in X\\\* cliapter.
1. Shew that / , ~ de = ,7 (« - Ja^ - W\
la tt ■*■ It COM if fr
sin'tf
.'0
when a> b> 0. Give reasons why tliis equation Khotild litill l>e tnte
when rt - 6. (Math. Trip. 1'.m»4.)
w. c. I. ii
66 THE EVALUATION OF DEFINITE INTEGRALS [CH. VI
2. Evaluate j^ ^^ ^ j^j ^^ ^ ^y when/> 0, gr > 0.
(Trinity, 1905.)
3. Shew that j_^ (^ ^-^,j (^.:, ,.). = 2a-6(a + 6)- ^^^^ «"^' *"«•
(Whittaker, Modern Analysis.)
4. Prove that, if a > 0, 6 > 0, c> 0, b--ac>0, then
i, «'T2iV.^%«„^(,,,«)- (Trinity, .908.)
5. Evahiatej^ (.^ ^ a^ {a- + 6^ (^ + <ff ^^"" ^' ^' " ^'^ ^^^-
(St John's, 1907.)
6. Shew that, if a > 0,
Cw^-'-^^Vli- (Trinity. 1902.)
7. Shew that I ^— ; (?^ = ^ tan ha when - tt < a < tt.
Jo smrnra;
8. Shew that I -^^dx = '^. (Clare, 1903.)
9. By integrating I e-"^ <fo round a rectangle, shew that
[ e-*'cos2a^.«?^=e-«'r(^), [ e-«'sin2a*.<?* = 0.
10. Shew that i ^^ dx = ^ir t&uYi^ir. (Clare, 1905.)
f /t* COS /Z^ 7r ^^^'^
11. Shew that | -^-, dx = jz —x-,, when a is real
Jo smha? (l+e"*")^
(Math. Trip. 1906.)
12. Evaluate I — -4 taken round the ellipse whose equation is
sc'-xy + f + x^y = (i. . (Clare, 1903.)
13. Shew that, if m > 0, « > 0,
[ xsinrnx J TT -^2 .ma /m • -i. mA/? \
8S] TIIK EVALUATION OF UEKIMTE IXTCUHAL.H 67
14. Shew that j *"^y^<faf |y. (Pet«riioaM, 190A.)
lA. Shew that, if « > 0, a > 0,
(Peterhouae, 1907.)
15. Shew that // (Y7^*f = |(»+»^').
(Petarfaotue, 1907.)
17. Shew thftt, if a > 0,
(Math. Trip. ItKni.)
la Shew that, if m > 0, a > 0,
(Trinity, 1907.)
19. Shew that, if m > 0, a > 0,
r" sin' our w .
j. **(a'*x«)'^ = W^' -**^*">-
(Trinity, 191«.)
20. Shew that, if a > 0,
(CUre, 1902.)
21. Shew that f* **'~?^'<ir = a (Trinity. 1 90S.)
22. Shew timt, when n is an even positive integer,
/ 3 — i • —(fj-^ryj t\ .-I- (Jesus, 1908.)
23. By taking a 4uatlrant of a circle indented at ai as contour,
shew that, if m > 0, a > 0, then
/* or cos 11U-* a sin nu- , •--*•/ ••-v /on- 1 u v
I ^ i rfj-^-*****/! («•*■■). (SciUomilch.)
[/•(#••) is defined by Bromwich, lt\finit^ Srrief, \\. 3*i5.]
5—2
/:
68 THE EVALUATION OF DEFINITE INTEGRALS [CH. VI
24. Shew that, if m>0, c> 0 and a is real,
/ ^^ <- -z — 7, = -; — o i 1 (c COS ma -a sin ma) \ .
;_«e x-a ar» + c'' a* + tr' I c ^ ^J
(Trinity, 1911.)
25. Shew that, if m>n>0 and a, 6 are real, then
sin m{x-a) sin w (x-b) , _ sin yt (a - b)
(tX — T r ,
a — 0
(Math. Trip. 1909.)
26. Shew that, if 0 < a < 2, then
["sin^xsmax, , , „ /t j n
/ -J cte= *7ra-i7r«-'. (Legendre.;
Jo ^
27. Shew that PI dx = ^Tr. (Legendre.)
Jo X
28. By using the contour of § 29, shew that, if - 1 <j»< 1 and
— ?r < X < TT, then
/■* x'Pdx TT sinoA. ._, , .
I , — t: c 5=^ .-", . (Euler.)
;o 1 + 2a: cos k + ar sm pv sin a
29. Draw the straight line joining the points ± i, and the semi-
circle of I « I = 1 which lies on the right of this line. Let C be the
^ontour formed by indenting this figure at —i, 0, /. By considering
I z""^ (z + z"^)"* dz, shew that, if n>m>-l, then
[ ' ^^ cos™ ^ (f^ = 2 ^-'" sin ^ (» - m) TT f r-""' (1 - O"* <^^-
J-iir ' Jo
Deduce that*
/.
^ /, m/,7/1 irr(m+l)
COS nO cos™ ^ rf^ =
/o ^"'''^''^"° "" 2"^^^(^w^ + i7l+l)^(^w-i« + l)'
and from the formula cos (n + 1) ^ + cos (n - 1) 6* = 2 cos ^ cos nd, estab-
lish this result for all real values of « if w > - 1.
30. By integrating I e~'^ dz round a rectangle whose corners are
0, B, R + at, ai, shew that
f e-" 8m 2 at. dt^e-'^ I ^dy.
» The result T (a) T (1 - a) = ir cosec air, which is required in this example, may
be established by writing x = //(l - t) in the first example at the end of § 211, when
0<fl<l, and making use of the value of the first PJulerian integral; it maybe
proved for all values of a by a use of the recurrence formula T (« + 1) = a r(a).
88] THE IVALUATIOV OP DinNITI IXTIOBAUI 69
81. iM Q(») be • polynomUI mkI let the red pert of « be
namerioelly Ioms Uiaii w. By integreting f X ^ " — ^- d» roood e
/ ooelK-f eoiA
reotengle, »hew tlut
Deduce thet
32. liet r, be e contour consisting of the pert of the reel axis
joining; Uie |x>iutM ± /i, and of a itemicircle of radius /t above the reel
axis, the contour being indented at the points nw/'b where « tekes ell
integral values and /> > 0 ; also let bUjw be half an odd integer. Let
r, be the reflexion of F, in the real axis, profjeriy oriented.
Shew that, if -b<a <b and if P {z\ Q {z) are polyuomiahi such
that Q (z) has no real fiustors and the degree of Q {z) exceeds that of
/»(s), then
^r ^ P{x). ... //• «- /»(»).
jr. «n 6z Q («)*•/ •
where the limit is taken by making A -»ao and the radii of the
indentations tend to lero.
Detlucethat pC -f, f'.^'^dU' = «(Sr-Sr'X
where Sr means the sum of the residues of the integrand at its polae
in the upiwr lialf-plane and Sr' the sum of the residues at the poles in
the lower half-plane.
33. Shew that, if - 6 < a < &, then
., /"Hinrtj" </x , sinha ^/"cosar xi/x , cosh a
/o »in /«x 1+x* ' sinh^' .'o «m Ax I •♦■ x* ' sinh6'
p /* sinaj* ^^ , sinh a „ r* cos ax djr _y ooah e
;. oos&rx(l-^j«)*«'coSli* ./, cos«wl*> *'oiihA-
(Legentlre, Cauchy.)
70 THE EVALUATION OF DEFINITE INTEGRAI^ [CH. VI
34. If (2m-l)b<a<(2m+ l)b and w is a positive integer,
deduce from Example 33 that
„ /■" sin ax dx _^ cosh {a - 2mh) - g""
io sin 6a: 1 + ar* ^ sinh b '
and three similar results. (Legendre.)
/■" dx
35. Shew that I tz — Zs\ xTn. — s = log2.
(Math. Trip. 1906.)
[Take the contour of integration to be the square whose corners
are ±N, ±N+ 2Ni, where N is an integer ; and make N-*- » .]
The results of Examples 36 — 39, which are due to Hardy, may be
obtained by integrating expressions of the type
f g" dz
j 1 + 2p^ ±e^' z + ia
round a contour similar to that of Example 35. In all the examples
a and 8 are real ; and, in Examples 36 and 38, — ir < 8 < tt.
Jo cosh x + cos8a^ + af a sin S „=o { (2» + l)'ir + a}^-8^'
Deduce that
1 <^ 1 1
Jo
/o cosh x + co8 8tt^ + a^ 8 sin 8 4 sin^ i 8 '
Jo cosh ;r + cosh 8 a'^ + ar' sinh8 n=o{(2» + l)T + aP + 8«'
r cosh I a? dx w « ( - )" { (2w + 1) tt -4- g}
Jo cosh X + cosh 8 a^ + ^ ~ flj cosh i 8 „=o { (2w + 1) tt + a}* + 8**
r cosh^a? cga? ^ TT I (-)"{(2w+l)7r+a}
' Jo cosh ar + cos 8 a* + a:^ aco8|8 ,1=0 {(2n+ l)7r + ap- 8* "
Deduce that
Jo cosh (I a;) . (a'' + a^) a Jo 1 + ^' "
39 rC ^ dx ^ 1 ( 8 ,n_.4_.
j_oo sinhar-sinh 8 Tr' + a:* cosh 8 l8* + 71^ 8j sinh 8
Sa] THE EVALUATION OF DtnifflTB nfTBORALi 71
40. Shew that, ifa>0, M>Oi,-l<r<l,
xdLr aiaSMr A
/.
/• 4rd[r nnor itr^
Shew tluit, if the {irinciiMiI nduaii of the integmb are taken, th«
ramltii are true when r« 1. (Legmdre.)
41. By integrating \ ^^ dz round the rectangle whoee oomen
are 0, R, l( •»■ i, i, (the rectangle being indented at 0 and t) shew that,
if a be real, then
f ^ ^'j ^ = 1 coth (i a) - i a'\ (LegeDdrei)
42. By employing a rectangle indented at \ i, ahew that, if a be
real, then
/ ^v',<"^=t«"'-ico«®ch(}a). (Ugendre.)
43. By int^rating \«-*^ 2"-' dz round the sector of radios R
bounded by the lines arg s » 0, arg c = a <: | «■, (the Hector being in-
dented at 0), shew tliat, if X > 0, « > 0, then
I x«-»# *'«^*ooe(Aj-8ina)dlr-X— r(ii)ooiiNK
/ X-- » «-*■«•• sin (Aj* sin o) (ir X— V (») sin no.
.'0
TlieHe results are true when a = | «• if n < 1.
Deduce that
rcoe(y)</y= r«in(y»)i/y -(;»)*. (Baler.)
.'• /o
44. Tlie contour C starU from a point R on the real axis,
encin'les the origin unce counterciiH^kwise and returns to R. By
defonuiitt; the cunt«iur into two straight linoM and a circle of radius S
(like the figure of !i29 with the lnr;;i> circle omitted), and making
h — o, i»hew tliat, if ^ > 0. (where C - ^ ♦ »1/. «»«d -» ^ arg (- s) ^ «• on C\
then
lim (( 2)f »i.-'«/s--2i«n(rC).r(C).
(Hankel, Math. Amn. M I.)
72 THE EVALUATION OF DEFINITE INTEGRALS [CH. VI
45. If r (0 be defined when ^ < 0 by means of the relation
r (^+ 1) = ^r (0, prove by integrating by parts that the equation of
Example 44 is true for all values of C
46. By taking a parabola, whose focus is at 0, as contour, shew
that, if a > 0, then
r (0 = ^^ f e-'^t' (1 + fy- i cos {2af + (2^- 1) arc tan t} dt.
(Bourguet.)
47. Assuming Stirling's formula", namely that
{logr(«+ l)-{z + ^)\ogz + z-Uog27r}->^0
uniformly as \z\^>^ cc, when - ir 4- 8 < arg « -= tt - 8, and 8 is any positive
constant, shew that, if - ^ ir < arg (- 0 < ^ '^j t'hsn
where a > 0, and the path of integration is a straight line. (The
expressions may be shewn to be the sum of the residues of the second
integrand at its poles on the right of the path of integration.)
(Mellin, Acta Soc. Fennicae, vol. xx.)
48. Let C be a closed contour, and let
f{z)= n {z-arT'<l>{z)
r=l
where the points ar are inside C, the numbers Wr are integers (positive
or negative), while ^ {z) is analytic on and inside C and has no zeros on
or inside C. Shew that, if /' {z) be the derivate of / {z), then
±.j'(^dz= 2 ur. (Cauchy.)
2-n Jc /{Z) r=l
49. By taking the contour C of Example 48 to be a circle of
radius R, and making ^ -*- ao , shew that a polynomial of degree n hits
n roots. (Cauchy.)
50. With the notation of Example 48, shew that, if il/{z) be
analytic on and inside C, then
-1-. [ ^{z/l:^^dz= 1 7,,^ (a,). (Cauchy.)
"* Stieltjes, Liouville's Journal, t. rv.
CHAPTER VU
EXPANSIONS IN SBRIBi
I 34. Taylor'* Tbeoreni.— g S5. LiiirMii'M Theoram,
34. Tayu)r'8 Tukuekm. LH f{t) b« a /itmetiom <^ z wkitA is
anafytic at aU poiitU intids a eireU q/* radius r whose cfntrs is tks
poimi ttkose complex eoordinats is a. Lti C ^ f>*!f point umde tkit
drds,
Tktm /{O f*tf* f** sjrpand^ into tkf nrnvergent seriss :
/(o -/(a)> (c-«)/'(«) * ^-^u-ayr (<>)*'" + ^.(c-«r /^-»(«)+ ....
wksrt/*^ (a) dsnctes ^'^^jf^ •
Let|{-a|.»^, M>t)iAtO<0< 1.
Let C be the contour fonuetl by the circle |s-a{<"^r, whers
9<tt <\ ; let $',9 m»^, m that C-oi-^ |£-a| " ^i < 1 ; then by | SI.
%wt Jc t-a\ t-aJ
1 - y f{z)ds I / /(O(C-a)-'
where **- ^^r.^/c(^-ar•'
/<•• (a), by | 2tl.
74 EXPANSIONS IN SERIES [CH. VII
On C, \f(z)\ does not exceed some fixed^ number K, 8ince/(«) is
analytic and, a fortiori, continuous on C.
Therefore
2ire
It^^l
Or
<jsri9,«+v(i-^i),
since \z-V\ = \{z-a)-{t, — o)\>\z — (i\-\l-a\.
Now lim ^^i"*V(l - ^1) = 0, since 0 ^ ^1 < 1 ; and therefore
,. 1 f /(z)a-ar-^
(;2-a)»+^(z-0
"a ....
consequently /(^)= lim 2 —^,(C -«)"*; since this limit exists it
00 ^
follows that the series 2 — ^(^-a)"* is convergent; and it has
m=0 Wil
therefore been shewn that / (0 can be expanded into the convergent
series :
/(o=«o+^,a-a) + |^(c-a)^+...+^a-ar-f...
where a -f^^)(a)-— f ^^''^^'^
35. Laurent's Theorem. Z«# f{z) he a function of z which is
analytic and one-valued at all points inside the region bounded by two
oriented concentric circles (F, T'), centre a, radii r^, r/ where r/ < r^.
Let C be any point inside F and outsids F '; th&nfijC) can he expressed
as the sum of two convergent series :
/(0=«o + «i(C-a) + «2(4-a)-+ ••• + «m(^ -<*)"'+-• •
+ h,a-a)-' + h,{^-a)-'+...+h^(^-a)-'^+,..,
*- ^'^ = i^lc0^- ^"• = 24e-l(-«)""'^«^^^
the circles C, C are concentric with the circles F, F' and are of radii
r, r such that r/ < r' < | ^ — a | < r < rj .
* See note 4 on p. 50.
S4-<'i5] EXPASHIONM IS MKKIEH 7$
Dr»w a dijunetar A BCD of cbe eiiolat C, C, uot pMnni; thrmifb C
Ii0t 6',, CV bo the Mmicirclat od om ade of thin duuneter, while O,, Ci
are the fletnicircle* on the other ndt.
Umii C|, AB, Ci, CD can be oriented to form a contour T, ; and
Ctt DC, C,', BA can be oriented to fonii a contour F,; it h eaitily seen
that AB, BA have opposite orientationn in tlie two contours, as do
CD, DC; C,, r, have the name orieiitatiun as C; C,', C,' hare
opposite orientations to C ; and /(z) is analytic in the closed regions
formed by T,, l\ and their interiors,
Th,«fo« /• ^^<^..| ^^d,, f-^-«A- I ^-^<U:
Jt\s~i JvtS-i Jcs-i Jcs-i
the integrals along BA , A B cancel, and so ilo thoxe along CD, DC.
But, by 8 21, /' '^J^hz * /' -{^'hz = -i^i/CC) ;
.'r, z~ Q Jr, z~i
for C is inside one of the contours T,, T, and outsiile the other.
76 EXPANSIONS IN SERIES [CH. VII
By the arguments of § 34, it may be shewn that the last integral
tends to zero as n^c»: so that t—-. ( -^^dz can be expanded into
00
the convergent series 2 a,„ (C - «)".
m=0
In like manner
_ 1 I m i, = _L. f m u - ^— «)- d.
1 /• /(r)(c-a)"
Since, on C",
U^ — a
< 1, the arguments of § 34 can be applied to
\i-a
shew that the last integral tends to zero as w-*qo ; so that
_± if{z)
2-iri
Jcz-i
can be expanded into the convergent series 2 6,„(«-a)~'"; that is
7>l = l
to say
m=0 m=l
each of the series being convergent.
CHAPTER VIII
HISTORICAL SUMMARY
§ S6l DeflnitionH of Aniilytic functuMui.— )| 37. Pnnih of (*«uchjr'H thearmL
36. The earliwt saggestion of the theorem to which Ckachy's
name haA been given is contained in a letter' from (Sauiu) to Beaael
iiateil Dec. IK, 1811 ; in thi.s letter (lautw }X)intA out that the value of
I X'* dr taken along a complex path depends on the path of integration.
The earliest investigation of Cauohy on the subject is contained in a
memoir' dated 1814, and a formal proof of the complete theorem is
given in a memoir' publi8hed in 1825.
The proof contained in this memoir consists in proving that the
Tariation of | f{z)dz^ when the path of integration undergoes a
small variation (the end-points remaining fucetl), is zero, provided that
/{z) has a unique continuous differential coefficient at all |Mints oo
the })ati) A B.
The following is a summary of the various assumptions on which
proofs of Cauchy's theorem liavc been based :
(i) The h>'pothe«is of Goursat': /"(i) exists at all {wints within
and on C.
(ii) The hypotJiesis of Cauchy*: /' (:) exists and is timtimmms.
I Brie/»etctu<l tritckrm Gum** umd llf$t*l (IHK)), pp. IM^ 157.
* Ofurrn compltUs, t^r. I. I. 1, p. Wi *t »*H-
* itimtoire tur U» int/finiUt tWnnirt pritet fntrt ttfi timtiU* iima^maim.
Ii«f«r0nc«« to Cauobj'ii iiul>MN]urnt rf>«<<«rcli«>« mrv i;ivrn 1>t I.in«iri6f, CaU-ml de
Rltidtu.
* TramMUtUmt of the Amtfrirat itathrmuttictit .^wiriVfy. >ul. i (ItfOO), pp. 14-16.
* 6m Um memoir cited above.
78 HISTORICAL SUMMARY [CH. VIII
(iii) All hypothesis equivalent to the last is : f{z) is uniformly
differentiable ; i.e., when e is taken arbitrarily, then a positive S, inde-
pendent of z can be found such that whenever z and z' are on or inside
C and \z' -z\ % 5, then
\f{z')-f{z)-{z'-z)f'{z)\^.W-z\.
In the language of Chapter II, this inequality enables us to take
squares whose sides are not greater than 8/^2 as 'suitable regions.'
(iv) The hypothesis of Riemann* : f(z) = P + iQ where P, Q are
real and have continuous derivates with respect to a; and y such that
cP_SQ ^_Q__^P
dx dy ' dx dy'
These h3rpotheses are effectively equivalent, but, of course, (i) is
the most natural starting-point of a development of the theory of
functions on the lines laid down by Cauchy. It is easy to prove the
equivalence'' of (ii), (iii) and (iv), but attempts at deducing any one
of these three from (i), except by means of Cauchy's theorem and the
results of §§ 21-22, have not been successful; however, it is easy
to deduce from § 22, by using the expression for /' {z) as a contour
integral, that, if (i) is assumed, then (ii) is true in the interior of C.
The definition of Weierstrass is that an analytic function f{z)
is such that it can be expanded into a Taylor's series in powers oi z -a
where a is a point inside C. This hj^othesis is simple and funda-
mental in the Weierstrassian theory of functions, in which Cauchy's
theorem appears merely incidentally.
37. A proof of Cauchy's theorem, based on hypothesis (i), requires
Goursat's lemma (which is a special case of the Heine-Borel theorem)
or its equivalent ; the apparent exception, a proof due to Moore*,
employs, in the course of the proof, arguments similar to those by which
Goursat's lemma is proved.
The hypotheses (ii) and (iii) are such as to make it easy to divide
G and its interior into suitable regions.
The various methods of proof of the theorem are the following :
(i) Goursat's proof, first published in 1884 [this, in its earliest
form*, employs hypothesis (iii)], is essentially that given in this work.
* Oeuvres inathimatiquex (1898), Dissertation iaaugurale (1851).
" The equivalence of (ii) and (iii) has been proved in § 20.
8 Transactions of the American Mathematical Society, vol. i (1900), pp. 499-506.
8 Acta Mathematica, vol. iv, pp. 197-200 ; see also his Cours d' Analyse, t. n.
36-37] H18T01UCAL 8UIIMAKY 79
(U) Cauchy't proof luui alroady bean deioribed.
(Hi) Riemann'N pruuP* coimmU in tran«formiii|;
jiPdo' - Qdy) ^ i jiQdj- ^ Pdy)
into a double integral, by using Stukeit' theorem.
(iv) Moore'it proof oon^iittM in aKMuniing that the integral taken
round the A\de» of a square is md xeru, but luis luudulus tf, ; the square
is divided into four equal sqnares, and the modulus of the integral
along at least one of them; must be > ^i^; tiie process of subdividing
squiU'es is cuntiiuied, giving rise to at least one limiting point { inside
•very square S, of a sequence such that the modulus of the int^^l
along ^V i" not less tlian rfji". Assuming tliat /(z) has a derivate at
C it is proved tliat it is possible to find y, such tliat, when k> !>«, the
modulus of the integral along IS, is less than y}^'4'. This is the contra-
diction neede<i to complete the proof of the theorem. The deduction
of the theorem for a closed contour, not a squ&re, may then be
obtained by the methods given above in § 17.
Finally, it should be mentioned that, although the use of Cauchy's
theorem may afford the simplest methixl of evaluating a definite
integral, the result can always be obtained by other methods ; thus
a direct use of Cauchy's theorem can always be avoided, if desired,
by transforming the contour integral int^) a double integral as in
Kiemann's prouf. Further, Cauchy's theorem cannot be employed to
evaluate all definite integrals ; thus / e'*' dr has not been evaluated
eacoept by other methods. £tc (SfrU/w^f Utl T\ ,
*" See the diaserUtion cited abore.
PBINTED BY JOHN CLAT, H.A.
AT THB UNIVEBSrnr PRESS
nr*^ / *•
QA Watsony Georf^e Neville
331 Conplex Integration and
W35 Cauchy*8 theoraa
PlQFMCal &
Applied Sci.
PLEASE DO NOT REMOVE
CARDS OR SLIPS FROM THIS POCKET
UNIVERSITY OF TORONTO LIBRARY
Live Music Archive
Librivox Free Audio
Metropolitan Museum
Cleveland Museum of Art
Internet Arcade
Console Living Room
Books to Borrow
Open Library
TV News
Understanding 9/11