Complex Variables and Analytic Functions: An Illustrated Introduction

373 Pages·2019·17.785 MB·English

by  Bengt Fornberg, Cécile Piret

Complex Variables and
Analytic Functions

Complex Variables and
Analytic Functions
An Illustrated Introduction
Bengt Fornberg
University of Colorado
Boulder, Colorado
Cécile Piret
Michigan Technological University
Houghton, Michigan
Society for Industrial and Applied Mathematics
Philadelphia
Copyright © 2020 by the Society for Industrial and Applied Mathematics
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Library of Congress Cataloging-in-Publication Data
Names: Fornberg, Bengt, author. | Piret, Cécile, author.
Title: Complex variables and analytic functions : an illustrated
introduction / Bengt Fornberg (University of Colorado, Boulder,
Colorado), Cécile Piret (Michigan Technological University, Houghton,
Michigan).
Description: Philadelphia : Society for Industrial and Applied Mathematics,
[2020] | Series: Other titles in applied mathematics ; 165 | Includes
bibliographical references and index. | Summary: “This book is the first
primary introductory textbook on complex variables and analytic
functions to use predominantly functional illustrations”-- Provided by
publisher.
Identifiers: LCCN 2019030487 (print) | LCCN 2019030488 (ebook) | ISBN
9781611975970 (paperback) | ISBN 9781611975987 (ebook)
Subjects: LCSH: Functions of complex variables--Textbooks. | Analytic
functions--Textbooks.
Classification: LCC QA331.7 .F67 2020 (print) | LCC QA331.7 (ebook) | DDC
515/.942--dc23
LC record available at https://lccn.loc.gov/2019030487
LC ebook record available at https://lccn.loc.gov/2019030488
is a registered trademark.
Contents
Preface ix
1 ComplexNumbers 1
1.1 Howtothinkaboutdifferenttypesofnumbers . . . . . . . . . . . . . 1
1.2 Definitionofcomplexnumbers . . . . . . . . . . . . . . . . . . . . . 2
1.3 Thecomplexnumberplaneasatoolforplanargeometry . . . . . . . 8
1.4 Stereographicprojection . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5 Supplementarymaterials . . . . . . . . . . . . . . . . . . . . . . . . 11
1.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 FunctionsofaComplexVariable 17
2.1 Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Someelementaryfunctionsgeneralizedtocomplexargumentbymeans
oftheirTaylorexpansion . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3 AdditionalobservationsonTaylorexpansionsofanalyticfunctions . . 33
2.4 Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.5 Multivaluedfunctions—BranchcutsandRiemannsheets . . . . . . . 43
2.6 Sequencesofanalyticfunctions . . . . . . . . . . . . . . . . . . . . . 51
2.7 Functionsdefinedbyintegrals . . . . . . . . . . . . . . . . . . . . . . 54
2.8 Supplementarymaterials . . . . . . . . . . . . . . . . . . . . . . . . 56
2.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3 AnalyticContinuation 71
3.1 Introductoryexamples . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.2 Somemethodsforanalyticcontinuation . . . . . . . . . . . . . . . . 73
3.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4 IntroductiontoComplexIntegration 93
4.1 IntegrationwhenaprimitivefunctionF(z)isavailable . . . . . . . . 95
4.2 Contourintegration . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.3 Laurentseries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.4 Supplementarymaterials . . . . . . . . . . . . . . . . . . . . . . . . 112
4.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5 ResidueCalculus 119
5.1 Residuecalculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.2 Infinitesums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
v
vi Contents
5.3 Analyticcontinuationwithuseofcontourintegration . . . . . . . . . 152
5.4 WeierstrassproductsandMittag–Lefflerexpansions . . . . . . . . . . 160
5.5 Supplementarymaterials . . . . . . . . . . . . . . . . . . . . . . . . 165
5.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
6 Gamma,Zeta,andRelatedFunctions 173
6.1 Thegammafunction . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
6.2 Thezetafunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
6.3 TheLambertW-function . . . . . . . . . . . . . . . . . . . . . . . . 181
6.4 Supplementarymaterials . . . . . . . . . . . . . . . . . . . . . . . . 184
6.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
7 EllipticFunctions 191
7.1 Someintroductoryremarksonsimplyperiodicfunctions. . . . . . . . 191
7.2 Somebasicpropertiesofdoublyperiodicfunctions. . . . . . . . . . . 191
7.3 TheWeierstrass℘-function . . . . . . . . . . . . . . . . . . . . . . . 194
7.4 TheJacobiellipticfunctions . . . . . . . . . . . . . . . . . . . . . . . 197
7.5 Supplementarymaterials . . . . . . . . . . . . . . . . . . . . . . . . 203
7.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
8 ConformalMappings 209
8.1 Relationsbetweenconformalmappingsandanalyticfunctions. . . . . 211
8.2 Mappingsprovidedbybilinearfunctions . . . . . . . . . . . . . . . . 212
8.3 Riemann’smappingtheorem . . . . . . . . . . . . . . . . . . . . . . 214
8.4 Mappingsofpolygonalregions . . . . . . . . . . . . . . . . . . . . . 215
8.5 Someapplicationsofconformalmappings . . . . . . . . . . . . . . . 219
8.6 RevisitingtheJacobiellipticfunctionsn(z,k) . . . . . . . . . . . . . . 222
8.7 Supplementarymaterials . . . . . . . . . . . . . . . . . . . . . . . . 226
8.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
9 Transforms 231
9.1 Fouriertransform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
9.2 Laplacetransform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
9.3 Mellintransform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
9.4 Hilberttransform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
9.5 z-transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
9.6 Threeadditionaltransformsrelatedtorotations . . . . . . . . . . . . . 264
9.7 Supplementarymaterials . . . . . . . . . . . . . . . . . . . . . . . . 266
9.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
10 Wiener–HopfandRiemann–HilbertMethods 273
10.1 TheWiener–Hopfmethod . . . . . . . . . . . . . . . . . . . . . . . . 273
10.2 AbriefprimeronRiemann–Hilbertmethods . . . . . . . . . . . . . . 286
10.3 Supplementarymaterials . . . . . . . . . . . . . . . . . . . . . . . . 288
10.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
11 SpecialFunctionsDefinedbyODEs 291
11.1 Airy’sequation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
11.2 Besselfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
Contents vii
11.3 Hypergeometricfunctions . . . . . . . . . . . . . . . . . . . . . . . . 299
11.4 ConvertinglinearODEstointegrals . . . . . . . . . . . . . . . . . . . 303
11.5 ThePainlevéequations . . . . . . . . . . . . . . . . . . . . . . . . . 307
11.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314
12 SteepestDescentforApproximatingIntegrals 317
12.1 Asymptoticvs.convergentexpansions . . . . . . . . . . . . . . . . . 317
12.2 Euler–Maclaurinformula . . . . . . . . . . . . . . . . . . . . . . . . 319
12.3 Laplaceintegrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
12.4 Steepestdescent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
12.5 Supplementarymaterials . . . . . . . . . . . . . . . . . . . . . . . . 346
12.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350
Bibliography 355
Index 359

Preface
The topics of Complex Variables and Analytic Functions are of fundamental impor-
tancenotonlyinpuremathematics,butalsothroughoutappliedmathematics,physics,and
engineering. Their theorems and formulas also simplify many results from calculus and
forfunctionsofrealvariables. Thereisalreadyawidechoiceoftextbooksavailable,rang-
ingfromtimelessclassicssuchasbyWhittakerandWatson[41],Copson[8]andAhlfors
[2],1 tomanylaterones,raisingthequestionofwhyanyonewouldwanttoseeyetanother
one. Our main motivation lies in the evolution that has occurred in other fields. For the
last half century or so, it has been unthinkable to use introductory text books for calcu-
lus that do not graphically illustrate the basic elementary functions, such as f(x) = x2,
√
f(x) = x,f(x) = sinx,f(x) = logx,etc. Whenwefirsttaughtacourseoncomplex
variables and analytic functions, we became puzzled about why complex variables texts
shouldnotalsovisuallybuildontherealcases,familiartoallstudents,andthenliberally
illustratehowthesesamefunctionsextendawayfromtherealaxis. Whileformulasalone
forsomestudentsmightprovideafeasiblealternativetovisualimpressions,weareaiming
this text at students that find the latter to be helpful for gaining at least their initial intu-
itivefeelingforthesubject. Althoughwehaveincludedanabundanceofillustrations(and
givebriefcodetemplatesfordisplayinganalyticfunctionswithMATLABandMathemat-
ica),thisbookisanintroductiontotheclassicaltheoryofcomplexvariablesandanalytic
functions. It contains enough materials to support a two-semester course, but has been
structuredtomakeiteasytoomitchaptersorsectionsasneededforaone-semestercourse
(offeringalotofflexibilityincourseemphasis). SIAM’swebsiteforthisbook, available
fromwww.siam.org/books/ot165,contains“NotestoInstructors,”withsuggestionsfordif-
ferentone-andtwo-semestersyllabi, ideasforsupplementarystudentprojects, etc. Once
asolutionmanualforalltheexercisesinthetexthasbeendeveloped, itwillalsoprovide
informationforhowinstructorscangetaccesstoit.
Textbooksoftendifferwithregardtotheorderinwhichtopicsarecovered.Onestrategy
is to make sure each step follows rigorously from previous steps. While that can have
someappeal,itmightnotnecessarilybethebestorderfordevelopinganinitialconceptual
understanding,anditalsodoesnotcorrespondtohowmathematicalproblemsolvingand
researchiscarriedout.2 Introducingkeyideasearlyonmightrequirecertainissuestobe
revisitedlater,onceadditionaltoolshavefalleninplace. Ineithercase,theendknowledge
1Supplementingthesetexts,JahnkeandEmde’s“TablesofFunctions”[28]hasexcellent(precomputerera)
illustrations,butlackstextbook-typematerials.
2TheeminentmathematicianPaulHalmoswritesinhisautobiography[24,page321]:“Mathematicsisnota
deductivescience-that’sacliché.Whenyoutrytoproveatheorem,youdon’tjustlistthehypotheses,andthen
starttoreason. Whatyoudoistrialanderror,experimentation,guesswork. Youwanttofindoutwhatthefacts
are,andwhatyoudoisinthatrespectsimilartowhatalaboratorytechniciandoes.”
ix