Complex analysis

332 Pages·6.384 MB·English

by  Bak, Joseph;Newman, Donald J

Joseph Bak
Donald J. Newman
Complex
Analysis
Third Edition
1 3
Undergraduate Texts in Mathematics
Editorial Board
S. Axler
K.A. Ribet
For other titles Published in this series, go to
http://www.springer.com/series/666
Joseph Bak • Donald J. Newman
Complex Analysis
Third Edition
1 C
Joseph Bak Donald J. Newman
City College of New York (1930–2007)
Department of Mathematics
138th St. & Convent Ave.
New York, New York 10031
USA
[email protected]
Editorial Board:
S. Axler K. A. Ribet
Mathematics Department Mathematics Department
San Francisco State University University of California at Berkeley
San Francisco, CA 94132 Berkeley, CA 94720
USA USA
[email protected] [email protected]
ISSN 0172-6056
ISBN 978-1-4419-7287-3 e-ISBN 978-1-4419-7288-0
DOI 10.1007/978-1-4419-7288-0
Springer New York Dordrecht Heidelberg London
Library of Congress Control Number: 2010932037
Mathematics Subject Classification (2010): 30-xx, 30-01, 30Exx
© Springer Science+Business Media, LLC 1991, 1997, 2010, Corrected at 2nd printing 2017
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Preface to the Third Edition
BeginningwiththefirsteditionofComplexAnalysis,wehaveattemptedtopresent
the classical and beautiful theory of complex variables in the clearest and most
intuitiveformpossible.Thechangesinthisedition,whichincludeadditionstoten
ofthenineteenchapters,areintendedtoprovidetheadditionalinsightsthatcanbe
obtainedbyseeingalittlemoreofthe“bigpicture”.Thisincludesadditionalrelated
results and occasional generalizations that place the results in a slightly broader
context.
The Fundamental Theorem of Algebra is enhanced by three related results.
Section1.3offersadetailedlookatthesolutionofthecubicequationanditsrolein
theacceptanceofcomplexnumbers.Whilethereisnoformulafordeterminingthe
rootsofageneralpolynomial,weaddedasectiononNewton’sMethod,anumerical
technique for approximating the zeroes of any polynomial. And the Gauss-Lucas
Theorem provides an insight into the location of the zeroes of a polynomial and
thoseofitsderivative.
A series of new results relate to the mapping properties of analytic functions.
A revisedproofofTheorem6.15leadsnaturallytoa discussionofthe connection
between critical points and saddle points in the complex plane. The proof of the
SchwarzReflectionPrinciplehasbeenexpandedtoincludereflectionacrossanalytic
arcs, which plays a key role in a new section (14.3) on the mappingpropertiesof
analytic functionson closed domains. And our treatmentof special mappingshas
beenenhancedbytheinclusionofSchwarz-Christoffeltransformations.
Asingleinterestingapplicationtonumbertheoryintheearliereditionshasbeen
expanded into a new section (19.4) which includes four examples from additive
numbertheory,allunitedintheiruseofgeneratingfunctions.
Perhapsthemostsignificantchangesinthiseditionrevolvearoundtheproofof
theprimenumbertheorem.Therearetwonewsections(17.3and18.2)onDirichlet
series. With that background,a pivotal result on the Zeta function (18.10), which
seemed to “come out of the blue”, is now seen in the contextof the analytic con-
tinuationofDirichletseries.Finallytheactualproofoftheprimenumbertheorem
has beenconsiderablyrevised.The originalindependentproofsby Hadamardand
de la Vallée Poussin were both long and intricate. Donald Newman’s 1980article
v
vi PrefacetotheThirdEdition
presentedadramaticallysimplifiedapproach.Stilltheproofreliedonseveralnontriv-
ialnumber-theoreticresults,duetoChebychev,whichformeda separateappendix
in the earlier editions. Over the years, further refinements of Newman’s approach
have been offered,the most recentof which is the award-winning1997 article by
Zagier.WefollowedZagier’sapproach,therebyeliminatingtheneedforaseparate
appendix,astheproofreliesnowononlyonerelativelystraightforwardresultdue
ofChebychev.
The first edition contained no solutions to the exercises. In the second edition,
responding to many requests, we included solutions to all exercises. This edition
contains66newexercises,sothattherearenowatotalof300exercises.Onceagain,
in response to instructors’ requests, while solutions are given for the majority of
the problems,each chapter contains at least a few for which the solutions are not
included.Thesearedenotedwithanasterisk.
Although Donald Newman passed away in 2007, most of the changes in this
edition were anticipated by him and carry his imprimatur. I can only hope that
all of the changes and additions approach the high standard he set for presenting
mathematicsinalivelyand“simple”manner.
Inanearliereditionofthistext,itwasmypleasuretothankmyformerstudent,
PishengDing,forhiscarefulworkinreviewingtheexercises.Inthisedition,it is
anevengreaterpleasuretoacknowledgehiscontributiontomanyofthenewresults,
especiallythose relatingto themappingpropertiesofanalyticfunctionson closed
domains.Thiseditionalsobenefitedfromtheinputofanewgenerationofstudents
atCityCollege,especiallyMaxwellMusser,MatthewSmedberg,andEdgerSterjo.
Finally, it is a pleasure to acknowledge the careful work and infinite patience of
ElizabethLoewandtheentireeditorialstaffatSpringer.
JosephBak
CityCollegeofNY
April2010
Preface to the Second Edition
One of our goals in writing this book has been to present the theory of analytic
functions with as little dependence as possible on advanced concepts from topol-
ogyandseveral-variablecalculus.Thiswas donenotonlyto makethe bookmore
accessibletoa studentintheearlystagesofhis/hermathematicalstudies,butalso
to highlightthe authenticcomplex-variablemethodsand argumentsas opposedto
those of other mathematical areas. The minimum amountof backgroundmaterial
requiredispresented,alongwithanintroductiontocomplexnumbersandfunctions,
inChapter1.
Chapter2offersasomewhatnovel,yethighlyintuitive,definitionofanalyticity
asitappliesspecificallytopolynomials.Thisdefinitionisrelated,inChapter3,to
the Cauchy-Riemannequations and the concept of differentiability.In Chapters 4
and5,thereaderisintroducedtoasequenceoftheoremsonentirefunctions,which
arelaterdevelopedingreatergeneralityinChapters6–8.Thistwo-stepapproach,it
ishoped,willenablethestudenttofollowthesequenceofargumentsmoreeasily.
Chapter5alsocontainsseveralresultswhichpertainexclusivelytoentirefunctions.
The key result of Chapters 9 and 10 is the famous Residue Theorem, which is
followed by many standard and some not-so-standardapplicationsin Chapters 11
and12.
Chapter13introducesconformalmapping,whichisinterestingin itsownright
andalsonecessaryforaproperappreciationofthesubsequentthreechapters.Hydro-
dynamicsisstudiedinChapter14asabridgebetweenChapter13andtheRiemann
Mapping Theorem. On the one hand, it serves as a nice application of the theory
developedin the previouschapters, specifically in Chapter 13. On the other hand,
it offers a physical insight into both the statement and the proof of the Riemann
MappingTheorem.
In Chapter 15, we use “mapping” methods to generalize some earlier results.
Chapter16dealswiththepropertiesofharmonicfunctionsandtherelatedtheoryof
heatconduction.
Asecondgoalofthisbookistogivethestudentafeelingforthewideapplicability
of complex-variable techniques even to questions which initially do not seem to
belong to the complex domain. Thus, we try to impart some of the enthusiasm
vii
viii PrefacetotheSecondEdition
apparent in the famous statement of Hadamard that "the shortest route between
two truths in the real domain passes through the complex domain." The physical
applications of Chapters 14 and 16 are good examples of this, as are the results
of Chapter 11. The materialin the last three chaptersis designed to offer an even
greaterappreciationofthe breadthofpossibleapplications.Chapter17 dealswith
thedifferentformsananalyticfunctionmaytake.ThisleadsdirectlytotheGamma
and Zeta functionsdiscussed in Chapter 18. Finally,in Chapter 19, a potpourriof
problems–again,some classical andsome novel–ispresentedand studiedwith the
techniquesofcomplexanalysis.
The material in the book is most easily divided into two parts: a first course
coveringthematerialsofChapters1–11(perhapsincludingpartsofChapter13),and
asecondcoursedealingwiththelatermaterial.Alternatively,oneseekingtocover
thephysicalapplicationsofChapters14and16inaone-semestercoursecouldomit
someofthemoretheoreticalaspectsofChapters8,12,14,and15,andincludethem,
withthelatermaterial,inasecond-semestercourse.
The authors express their thanks to the many colleagues and students whose
comments were incorporatedinto this second edition. Special appreciation is due
to Mr. Pi-ShengDing forhisthoroughreview ofthe exercisesand theirsolutions.
WearealsoindebtedtothestaffofSpringer-VerlagInc.fortheircarefulandpatient
workinbringingthemanuscripttoitspresentform.
JosephBak
DonaldJ.Newmann
Contents
PrefacetotheThirdEdition ......................................... v
PrefacetotheSecondEdition........................................ vii
1 TheComplexNumbers ......................................... 1
Introduction.................................................... 1
1.1 TheFieldofComplexNumbers .............................. 1
1.2 TheComplexPlane......................................... 4
1.3 TheSolutionoftheCubicEquation ........................... 9
1.4 TopologicalAspectsoftheComplexPlane ..................... 12
1.5 StereographicProjection;ThePointatInfinity .................. 16
Exercises ...................................................... 18
2 FunctionsoftheComplexVariable z ............................. 21
Introduction.................................................... 21
2.1 AnalyticPolynomials ....................................... 21
2.2 PowerSeries .............................................. 25
2.3 DifferentiabilityandUniquenessofPowerSeries................ 28
Exercises ...................................................... 32
3 AnalyticFunctions ............................................. 35
3.1 AnalyticityandtheCauchy-RiemannEquations................. 35
3.2 TheFunctionsez,sinz,cosz................................. 40
Exercises ...................................................... 41
4 LineIntegralsandEntireFunctions.............................. 45
Introduction.................................................... 45
4.1 PropertiesoftheLineIntegral ................................ 45
4.2 TheClosedCurveTheoremforEntireFunctions ................ 52
Exercises ...................................................... 56
ix
x Contents
5 PropertiesofEntireFunctions................................... 59
5.1 The Cauchy Integral Formula and Taylor Expansion
forEntireFunctions ........................................ 59
5.2 LiouvilleTheoremsandtheFundamentalTheoremofAlgebra;The
Gauss-LucasTheorem ...................................... 65
5.3 Newton’sMethodandItsApplicationtoPolynomialEquations .... 68
Exercises ...................................................... 74
6 PropertiesofAnalyticFunctions ................................. 77
Introduction.................................................... 77
6.1 ThePowerSeriesRepresentationforFunctionsAnalyticinaDisc.. 77
6.2 AnalyticinanArbitraryOpenSet............................. 81
6.3 TheUniqueness,Mean-Value,andMaximum-ModulusTheorems;
CriticalPointsandSaddlePoints.............................. 82
Exercises ...................................................... 90
7 FurtherPropertiesofAnalyticFunctions ......................... 93
7.1 TheOpenMappingTheorem;Schwarz’Lemma................. 93
7.2 The Converseof Cauchy’sTheorem:Morera’sTheorem;The
SchwarzReflectionPrincipleandAnalyticArcs................. 98
Exercises ...................................................... 104
8 SimplyConnectedDomains ..................................... 107
8.1 TheGeneralCauchyClosedCurveTheorem.................... 107
8.2 TheAnalyticFunctionlogz.................................. 113
Exercises ...................................................... 116
9 IsolatedSingularitiesofanAnalyticFunction ..................... 117
9.1 ClassificationofIsolatedSingularities;Riemann’sPrincipleandthe
Casorati-WeierstrassTheorem................................ 117
9.2 LaurentExpansions......................................... 120
Exercises ...................................................... 126
10 TheResidueTheorem .......................................... 129
10.1 WindingNumbersandtheCauchyResidueTheorem............. 129
10.2 ApplicationsoftheResidueTheorem.......................... 135
Exercises ...................................................... 141
11 ApplicationsoftheResidueTheoremtotheEvaluationofIntegrals
andSums ..................................................... 143
Introduction.................................................... 143
11.1 EvaluationofDefiniteIntegralsbyContourIntegralTechniques ... 143
11.2 Application of Contour Integral Methods to Evaluation
andEstimationofSums ..................................... 151
Exercises ...................................................... 158