Basic Complex Analysis, Third Ed.

519 Pages·2007·53.69 MB·English

by  Jerrold E. Marsden

BASIC
COMPLEX
ANALYSIS

Third Edition

JERROLD E. MARSDEN
California Institute of Technology

MICHAEL J. HOFFMAN

California State University, Los Angeles

W. EL Freeman
New York
Acaiscnms itr: Richard J. Bonacel
Marketing Manager: Kimberly Manzi

Project Editar Penelope Hull

Text Designer Jerrold F: Marsden

Cover Designer: Michael Minchillo

Tastrator: Fina Line Studio

Prxduction and

IUkestration Coordinator: Susan Wen

Compositer: Jerrold F Maren
Manuaorurer 2 R Donnelley & Sons Company

Library of Congress Cateloging-in-Publication Data

Marsdea, Jerold E.
asic complex analysis / Jerold E. Mareden, Michael J Moiinan. — 3rd ed.

pcm.
Includes index.
ISBN 0-7167-2877-X (EAN: 9780716728771)
Functions of eumples variables, | Hotfman, Michuel J. 11. Title
QA331.7. M36 90
5151.944e21 9841459

cP

© 1999, 1987, 1973 by WH. Freeman anu! Company: AN rights reserved,

No part of this book may be reprduced hy uny mechanical, photographic, or eivetronic
process. or inthe form af'a phanogeaplic recording, nor may it be stored in a retrieval
syslers, Iresmitted or otherwise copied for public or private use without written
permission rom the publisher.

Printed in ie United States of America

Fifth printing

1.11 Freeman and Company
41 Madison Avenue, New York, NY 10010
Houndm's. Basingstoke RG21 6XS, Eoulund
Contents

Preface

1 Analytic Funetions
2 Introduction to Courplex Nuuubers

2. Progerties of Complex: Numbers

13° Some Klementary Aunetfons

14 Continsous Fimetione

1.5 Basie Properties of Anslytic Funstions

LG Tiiffeventiation of the Blementaey Fuetions

2 Canehy’s Thenrenn
2.1 Contour Inzegrals
22 Cauchy's Theorem A Mirst Loa:
2.3 A Clows Look ai Cauchy’s Theorem
A. Catchy's Tntagral Forel.
25 Maxiioum Modulus ‘Theorie end Harsranie Functions

Representation of Analytie Functions
3.1 Convergent Series of Analstic Munetions,

3.2 Dower Series aud ‘Layler's ‘Theorem

3.3. Laurent Series and Clhscification of

4 Caleulus of Residues

1 Csleuletion of Resictues

42 Residue Lbepeca

13 Evaluation of Definite Integrals

44 Hvaluation of Lofinite Series and
Partial-Frastion Expansions

Conformal Mappings

5.1 Thasic ‘Theory of Coulocinal Mappings

5.2 Fractional Linear and
Schware-Christoffel Trensformacions
5.3. Applications of Conformal Mappings ta
Laplace's Equation, Heat Condetion,
Bleerrostaties, and Hydrodynamies

@ Further Development of the Theory

GL Analytic Continuation and Elementary
Riemann Surfaces

6.2 Rouché’s ‘Theurem and Principle of tie

Argmmont *

6.8 Mapping, Propertis

of Analytic Functions

7 Asymplotic Methods
7.1 Tofinite Products aad the Gaununa Function
7.2 Asymptovle Expansions and the

Method of Stevpest Deseont
Ta Stirhing’s Formala ane Bescel Functions

8 Laplace ‘ansform and Applications
S.L Basic Properties of Laplace Transforms
82 Complex Inversion Forrauls
88. Applivatioa of Laplaee Transforms to

Ordinary Diferential Kquations

Auswers to Odd-Numbered Exercises

Index

Conteris
ais in mathematics, the physical sciences

‘ap through calenlns of sevoral variables ad Gruea’s theorem, are adequate
preparation for the comic. lke lext contains sone references to liawar algebra and
basic facts about e-8 analysis, but tlhe extent to which they ate emphasized cxat be
adjueted by the insteuctoe depending on the background and need af the cl

‘The buok lia: a generous number of examples, exercises, and applications. We
have snede a special effort to motivate students by making ths book readable for
self study std have provided plonty of material to bela studems gain an
itive understanding of the eubjees. Our arrangement enables wplication-ariented
sindeals tu skip the mote technical parts without serifcing an unde:standing of
the main theorstical peiuls. Applications inelnde electric potentials, heat onnduc
Lion, ldrodynamics (studied with the aid of hermonie funetioes and confornal
mappings), Laplace tsansforas, asymp'otie expansions, the Gamma fimetion, and
Bessel functions,

The core of Chapters 1 Wy 6 ean be taught in a ouesemescer courye for
ematies majors. In applied mathematics cous, if some of the vechnival p
Chapter 2 aad pats of Chapter 6 are omitced, then pasts of Chapters and 8 can
be covered im onc semester, It is healthy for teathematics majors to wee as mony
of the applications us possible, for they are an integral part of the enlsural and
Listorieal horitage of rathematies.

ols used i Us text are, for the mos part, standard. “The set
vwlile © denolos the set of complex camubers, “IA
stands for “if and ouly if? (except in definitions. where we wiite only WP}. ‘The
fend of a pronf is rmnrkes Ml, tue end of the proof of a Jemuma in the middle of a
proof of a Ueoretn is marked ¥ ane occasionally, Le end of an example in the Lex
is marked @. ‘The notation |a,0| represents the apan interval consisting of all real
uiubers satisfying a <2 <4, ‘This is to avoid eorfusion with she onder! pair
notation (2,6). ‘The notation f: ACC + C mas that the mopping F maps tbe
domain A, which is a subset of C, tuto ©, and we wrive z+ f(z) for the effect
fa che poist 2= A, Ovrsionally, > is used to wean “implies”. ‘The sel chieuretic
difference of the seta A ard 4 is denoteel by AB. while Geir union and interseotie
are deaoted by ALJ? and A”B. The definitions, theorem 7
nd (amples are aumbered oonecentivaly for easy cross reforoneo; for exarule,
A relies bo Lae thied \tera In Section 6.2,

Symbols The «
of seal nurabers is donot

iropesitions, leram
vil Preface

spite she Tangy: uaunbere of texts waitten in: recent genes, some of
She older elassits remain the best. & few that are worth looking ak are A. Tarwi
Td Tt. Courant, Vorlesungen der allgereine Punktioneniteurie wail eltiptiche
Tinabtionen [Beelin: Julas Speiager, 1925); FT, Whittalos and GN. Warson,
Course of Modern Anatyeia, Fourth Edition (London: Cambridye Universiy U'ess,
EY harels, Ther Theory of Fumetions, Yd od. (NewYork: Oxford
us 'y Proms. 1939, reprinted 1985}; and K. Knopp, Theory of Functions (New
York: Dover, 1947). The ceader who wishes furtive information on verions of the
howe adhnnced topics can sroftably cousull F. Lille, Analyte Runetiun Theory,
J mobaaes, (Basen: Glu, 1950}; L. V- Ablfurs, Compier Analysis (New Yorks
MoGraw-Hill, 1960); W. Rudin, Real and Campier Analysis (New York: MeGrar
Titi, 1969); and 12M. Morse wad H. Reshbach, MCthads af Theovotien! Physics
Now Vark: McCraw Hill, 1978), Some additional zeferenees are giver: dhroughont
the leat

‘The moder troalasent of complex analysih di uot evolve rapidly or smoothly.
Tho mmierous cxeators of tls area of raathiamatios traveled over rary Tough tual
find euevunterot many blind alleys before the superior routes were found. An a
preciotion of she hiscory uf wiathernaties aud its futiiate connection to the pliysical
Nijences is imporats to every student's ecucetion. We: recontsuesel fooking at ML
Klein's Matheiaatica! Thought frum Ancient to Modern Times (Lonlon: Oxford
University Pre, 1972).

Classic texts Di

Third edition The third edition fewtuses an Instructor's Stypplerensé ax well es
2 Siadent Guide, Answers to Ute o¢d omubered problems sxe in the fnek of the
Fook aud exercises with sulutions it the Student Guide are marked with a bullet
(*) in the

We have sreewulined © ntraber of featires in the vext, suit se Ube srearment of
Cauchy's ‘Mhauem. We have ebstamially zewritten Chapcer 4 on the evaluation
tof intoerads, making the trensment Toss encyeiopedie, Ar filernet Supplanent ts
fivallable frew from htty:/; varw.whfrveman.con:/ (look it the mathemalice section}
tr foots herp,/fedacoltce!ela/“marscen ook wuniler “oooks"}, Te contains ad
Sitioual information for (Lose who wan to dolve tio sO tnpies in a lntle more

depth

Acknowledgments We ore grateful to the may rders who supplied correc
(ions and comments for this edition. ‘There arr too nany vo bo thanked divide
nally, but wr would like especially we mension (more or Irs chronolegicaliy) ML
Buchner {who helpod significantly wih the First Bulition), ©. Risk, P. Roeder, W
Barker, ©. Vil, J. Seitz, J. Bodoweki, H. O, Cordes, M. Chai W. T. Stallings, E
seen. R. Utis, N. Stare, D. Fow'er, L. L. Campbell, I. Goldschmiddi, T. Kato, I.
Neanoe, P. Kenshait, KL. ‘feo, G. Bergmann, J, Herrin and C, Daniels, Finally
Parhara Marsden for her accurate typsectiny of this new edition
Chapter 1

Analytic Functions

I tits chapter the baste ideas about caraples embers and aaalytie Cauctions «
introduces. “The onguuiealion of the text ts araloguns f© Ubat of an elementary
cealeulus textboals, which heyinss bye ateodnoing, RB tao sev of rea nnaliers. and
fmecions fc] o” a zeal vaziable x One then slucties the thon
nlffersntiation anid integration off
iv we beggin by ‘ntroduring C, the ext of eeuuplex numbers = We thea study
2) of a comple varishle 2, wrica are dilferemtiable in a ennnples seas
ues axe called analytic fanetions.
‘The awalegy Betwrvn row! nel complex ver‘ables is, however, w HsLle deveptive

faxesions

because couples ausdysis is ¢ strprisingly rieber theory: a lor
about an ana‘ytie finesion Una about @ eiferentiable Junction of». rea! variable,
fae ill be fully developed in subecquent chapters.

Tn addition to becoming familiar wiil: the choory, the winder should strive ro
agnin facility seth the standard (or Yelementary") funelious such a polyuria
in = us in celenlns, Llaese lunctioas are studied ia LS and appear
‘requently throughout the text.

1.1. Intreduction io Complex Numbers

ste Zamniintity with che wneite properties of
oul murtiers. ‘The real number system resulted fort the senich fara sysbenn (ak
abstract set together with certain cules) thet inched tue rational bit thar also

provided eolutious to s 2

‘The following disensson will assume

polynomial xjantions 6

Historical Perspective Historically, a similay vonsiderasion gave tise to an ex
tension of the real inaabers, Ax early es Hie sixteenth eensmy, Geronimo Carcany
ouside quadrate (and enbie) equations stich a8 2% which is sat
fsfied by uo real uuzser e, Tho qnabatic Zormails (b+ v2 ade} (2 vivleis
formal" expressions for the caw solations of the equation wx? tiie ¢— 0. Lt this
2 Chapter 1 Anulylie Functions

fornia may involve square rats oF megative muaoens for exaeple, =
the equation 2? —2e +2— 0. Cardano noticed that if these “oomplex numbers
were trestet ax urdinary nuaibere wilh Ube arided rule that: yT-y=T = —1, vaev
dirk inex] sale the enations,

The importa expression v

is now xiven the widely accepted desiguation

4 YT. {An altemative convention is followed hy many electrical engineers
who prefer the symbei j = y=T since they wish ta reserve the symbol for electric
erent.) However, i the past ws fell that uo meaning could actually be nssigneed

Gradnal’y, especially
tas 4 estll of the work of Leonkaed Kuler in Use eighleeuth cexsury, Uhese imaginary

to such expressions, which wore rherrécre termed Simuagina

quactities came to pley an ‘mmportant role. For example. Enfer's formula eM —
vos — isin revealed the existence of « profound relationship eelwees vomples:
numbers and tae trigenomecrie functions. The rule e*:'*+! — 640% gras found
to sumenatize Lie rules for expanding sine and rasine of sim af two ales in
near ving, and this result lone indicated that soute mewting should bw artachod va
these “maggcuary” men

However, i louk uourly Vinee huadued years until the wurk of Casper Wesel
(ea. 1797), Jenn Roberts Argand (1806), Kurl Briediei Gunns (188), Sir Woliaan R,
Heaniltow (3
imate mathematieal objecis, anel i was realized tual here ix nochinge Sn
bout then wal (althousgs Uhis teu is sl used

‘The complex ausiysis that is the subject uf Us boos was developed in the
ninelomeh centue, mainly hy Augustin Cauchy (1789-1857). Laer his Giewry
igorous and extended Ly such, vuathematicians as Peter Disichle
eoyg Fedtich Berane Riera

auc others, ven “imaxinaty” oumbers were recuguiz as leg
uginary

wos male more
180 1859), Karl Weiorstrass (1815-1897), anc
1326-1866),

Thesenrel fur a meth ta destribe beat cunduetion iafhyenced the dexelopmest
cof the theory, which ans found many ases onrsicie marhematies. Subsequent cas
tess ll isc some of these applinntions to problems in physics and ecygiewaring,
such as hydiodyramies and electrostatics. The thenry also has mathematical

plications to problems (at at inst do not eee: wo involve complex numbers, Bor
sexomiple, the punt that

re

suientary calculus, buat these

nay be diffesult o¢, in gome
of complex varinbles.

‘lead ities can be veadily proved -wing the tech
Ld Introduction to Complex Numbers 4

‘The Complex Number System Complex analysis bas beeome wn indispensatsle
and standard tuo! of the working mathematician, physicist. and engineer. Nogloet
oft cnm prove to he a secre haadivay in anost arras of zesearch and applicasioa
involving mnthematical idea nel techniques. ‘The drst nijeative of this eeetion will
he tw dedive complex mmmbers and w slow chat rhe uaa. elgebraie manipttarons
hold. ‘Ty hegin, zecall that the ay plane, denoted by R4, cousists of all ordered
its (,y) of 22a mbes

Definition 1.1.1 The system of complex numbers, drwated C, is the set ®
agether with the woul rales of vector addition aut sealar reubtiplication by

mer a, namely

(ent) + toate) = Ga beda tad

ake) = (ae.ay)

owl with the operation of complex multipticution, defined by

= (aya matte ban

We will uced so explain where this strange rule of multiplication comes froma!
Rather than using (7. y! to tepresent a cotuplex ouenbes, we will find it more come
lent to rotuen to more siuxlurd notation xs follows Tet ns identity wat namabers
with points on the = axis: thas aud (2.0) stad fur the seme pons (2,0) in
2, The y axis will be called the imaginary aizis, and te unis pois (0,1) will
be deuoved i, This, by dedinision, + ~ (0,1). Then

feag=ate

bbecance thi righe side of Le equntion stanly for
2.0) + (0,1) = (2) Oy} = Qaag he
Using y = (9.011 wend Definition 1.1.1 of eompiex muleiptcation, we get

fy = OG.) Oy 1 O,y-1 40-0) = (0.0) =

so-we eur also wsite (2,9) <a |i. A single symbol wack as 2 — a 44 is generally
used 9 indicate a complex uumber. The notation 2 ¢C moans tat + belongs to
{the set of complex mambers.

Note that

eis) (01)

1,0) = =1,

0-11 0401)

0 we do lave the property we want