Complex Analysis

577 Pages·2013·5.08 MB·English

by  Joaquim Bruna

EMS Textbooks in Mathematics
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Tammo tom Dieck, Algebraic Topology
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Joaquim Bruna
Julià Cufí
Complex Analysis
Translated from the Catalan
by Ignacio Monreal
Authors:
Joaquim Bruna and Julià Cufí
Department of Mathematics
Universitat Autònoma de Barcelona
Campus de Bellaterra
08193 Cerdanyola del Vallès, Barcelona
Catalonia, Spain
E-mail: [email protected]
[email protected]
2010 Mathematics Subject Classification: 30-01, 31-01
Key words: Power series, holomorphic function, line integral, differential form, analytic function,
zeros and poles, residues, simply connected domain, harmonic function, Dirichlet problem, Poisson
equation, conformal mapping, homographic transformation, meromorphic function, infinite product,
entire function, interpolation, band-limited function
ISBN 978-3-03719-111-8
The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography,
and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch.
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© European Mathematical Society 2013
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∞ Printed on acid free paper
9 8 7 6 5 4 3 2 1
Preface
Our original purpose in writing this book was to provide a brief manual, perhaps
moreaptlycalledaguidebook,thatwouldcoverthecontentsofabasicone-semester
course in complex analysis as described in most university curricula. The result,
however,hasbeenamoreextendedtextthatdoesnotfitintoasemestercoursebutis
ratherappropriateforavarietyofadvancedcourses. Italsocontainssomematerial
that is not usually found in the textbook literature of complex variables. For this
reason we hope it will prove to be a good complement to many of the references
thatarecommonlyusedbybothstudentsandteachers.
Wewrotethisbookbecausewewantedtoprovidesomethingnew,notonlyin
presentationbutalsoincontent,whencomparedwiththelongandstillgrowinglist
ofcomplexvariabletextbooks,manyofwhichhavebecomeclassics. Thestarting
pointwastoframecomplexanalysiswithinthegeneralframeworkofmathematical
analysis.Althoughitispossibletopresent–asmanytextsdo–thecomplexvariable
asanisolatedbranchofstudyinanalysis,wehavechosenadifferentoption,namely
toseekamaximumnumberofpointsofcontactwithotherpartsofanalysis. This
has resulted in the inclusion of some sections that are not common in other texts
andanewformulationofsomeclassicalresults. Wehighlightafewofthembelow.
In Chapter 3 we give a real version of the theorems of Cauchy and Cauchy–
Goursat. The result is a version of Green’s formula with very weak regularity
assumptions, which serves also for classical theorems of vector calculus. In the
samechapter,thepresentationofCauchy’stheoreminthecontextofvectoranalysis
allowsustoformulateanapproachtotheconceptofaholomorphicfunctionfroma
realvariableviewpoint,intermsoffieldsthataresimultaneouslyconservativeand
solenoidal. Theconceptofaharmonicfunctionthennaturallyappears.
Chapter6providesahomologicalversionofGreen’sformulathatcanbeinter-
pretedasaGreen’sformulawithmultiplicities. Withthehelpofthisformulaanda
standardprocessofregularization,aquestionbyAhlforsisansweredaffirmatively,
aboutthepossibilityofmodifyingtheproofofCauchy’stheoremtocoveralsothe
caseofanylocallyexactdifferentialform.
Chapter7systematicallystudiesharmonicfunctionsandtheLaplaceoperatorin
thecontextofrealvariablesinRn,withemphasisonthespecialcaseofdimension
2 and the relation with holomorphic functions. The study includes in detail the
propertiesoftheRieszpotentialofameasureanditsimportanceinsolvingPoisson’s
equationandtheDirichletandNeumannnon-homogeneousproblems.
Chapter 9 examines the relationship between Green’s function and conformal
mapping,whichallowsonetoproveRiemann’stheoremusingthesolutionofthe
Dirichletproblem;wealsopresentKoebe’sproofbasedonthepropertiesofnormal
vi Preface
families. TheexistenceofsolutionstotheDirichletproblemisprovedbyPerron’s
method,whichisgeneralizabletoanydimension.
In an analogous way to the Poisson equation, which is the inhomogeneous
caseoftheLaplaceequation,Chapter10dealswiththeinhomogeneousCauchy–
Riemannequations. ThesolutioninthegeneralcaseisobtainedusingtheRunge
approximation theorem and is applied to study the Dirichlet problem for the @N
operator.
Chapter 11 is devoted to the study of zero sets of holomorphic functions, and
clearly shows the relationship between this topic and the Poisson equation. This
allowsustoanalyzethedistributionofzerosofaholomorphicfunctionintermsof
theirgrowth.
Finally,thelinkbetweenrealandcomplexvariablesalsoappearsinChapter12
with the complex Fourier transform or Laplace transform. We provide a proof of
theShannon–Whittakertheorem,wellknownininformationtheory,usingmethods
inChapter10onthedecompositionofmeromorphicfunctionsinsimpleelements.
To read this text, it is sufficient to have a good knowledge of the topology
of the plane and the differential calculus for functions of several real variables.
Fromthere,thebookisself-containedandgivesrigorousproofsofallstatements,
includingafewissuesthattendinmanybookstobetreatedsomewhatsuperficially.
Inthisregardweemphasizethestudy,inChapter1,ofplanedomainswithregular
boundary,includingatreatmentoftheorientationoftheborder. Thisstudyallows
us to formulate a precise version of the classic theorems of complex analysis for
domainswithregularboundary,whicharethemostusedinapplications. However,
in Chapter 6, we also give the homological version of the fundamental theorems
alongthelineinitiatedbyAhlfors,whichismoregeneralandrelatestotopological
propertiesofthedomain.
Thelengthandstructureofthetextallowsthereadertopursueavarietyofpaths
throughit,andtofollowarouteatdifferentlevels. Forexample,onecanfollowa
basic course in complex variables with Chapters 1 and 2, Chapter 3, Sections 3.1
to 3.5 and Chapters 4, 5 and 8, without the later Sections 8.8 and 8.9. Another
possibilityistouse, totallyorpartially, thecontentsofChapters9, 10, 11and12
foranexpandingcourseincomplexvariables.
Given the initial goal of providing maximum interconnection with other parts
ofanalysis,wehaveputgreatemphasisontheroleofharmonicfunctions. Wehave
devoted Chapter 7 to them, which is the longest chapter of the book and can be
usedasanintroductiontopotentialtheory. Thischaptercanbereadindependently
knowing only the content of Chapter 3; on the other hand, Sections 7.7 to 7.12
may require a level of maturity in mathematics a little higher than the preceding
chapters.
Ingeneralwehavedevotedmuchattentiontothedetailsoftheproofs. However,
insomesectionsofChapters7, 10, 11and12thelevelofprecisionislowerthan
formostofthechaptersandthiscanmakereadingthemalittleharder.
Preface vii
Eachchapterisdividedintosections, eachsectionintosubsections. Allstate-
ments (theorems, propositions, lemmas and corollaries), and also examples, are
numberedconsecutivelywithineachchapter,onlyobservationsarenumberedsep-
arately. Thelastsectionofeachchaptercontainsstatementsofexercises.
Needlesstosay,inpreparingthisbookwebenefitedfromtheworkandexperi-
enceofpreviousauthors. WeexpressourdebttoAhlfors[1],Burckel[3],Gamelin
[7],andSaks–Zygmund[11]. WearegratefultoJuanJesúsDonaireforhisreading
of the original, to Lluís Bruna and Miquel Dalmau who read various parts and to
MarkMelnikovwhoprovideduswithsomeexercises. Theyallhavemadevaluable
suggestions. WealsothankIgnacioMonrealforthetranslationintoEnglishofthe
Catalanoriginaltext.
Finally, the book would not exist without the excellent typographical work of
RaquelHernández,MariaJuliàandRosaRodríguez. Ourthankstoallofthem.

Contents
Preface v
1 Arithmeticandtopologyinthecomplexplane 1
1.1 Arithmeticofcomplexnumbers . . . . . . . . . . . . . . . . . . 1
1.2 Analyticgeometrywithcomplexterminology . . . . . . . . . . . 8
1.3 Topologicalnotions. Thecompactifiedplane . . . . . . . . . . . 12
1.4 Curves,paths,lengthelements . . . . . . . . . . . . . . . . . . . 17
1.5 Branchesoftheargument. Indexofaclosedcurvewithrespect
toapoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.6 Domainswithregularboundary . . . . . . . . . . . . . . . . . . 28
1.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2 Functionsofacomplexvariable 39
2.1 Realvariablepolynomials,complexvariablepolynomials,
rationalfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.2 Complexexponentialfunctions,logarithmsandpowers.
Trigonometricfunctions . . . . . . . . . . . . . . . . . . . . . . 41
2.3 Powerseries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.4 Differentiationoffunctionsofacomplexvariable. . . . . . . . . 59
2.5 Analyticfunctionsofacomplexvariable . . . . . . . . . . . . . 69
2.6 Realanalyticfunctionsandtheircomplexextension . . . . . . . 76
2.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3 Holomorphicfunctionsanddifferentialforms 85
3.1 Complexlineintegrals . . . . . . . . . . . . . . . . . . . . . . . 85
3.2 Lineintegrals,vectorfieldsanddifferential1-forms . . . . . . . 88
3.3 Thefundamentaltheoremofcomplexcalculus . . . . . . . . . . 94
3.4 Green’sformula . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.5 Cauchy’sTheoremandapplications . . . . . . . . . . . . . . . . 107
3.6 Classicaltheorems . . . . . . . . . . . . . . . . . . . . . . . . . 111
3.7 Holomorphicfunctionsasvectorfieldsandharmonicfunctions . 124
3.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
4 Localpropertiesofholomorphicfunctions 136
4.1 Cauchyintegralformula . . . . . . . . . . . . . . . . . . . . . . 136
4.2 Analyticfunctionsandholomorphicfunctions. . . . . . . . . . . 140
4.3 Analyticityofharmonicfunctions. Fourierseries . . . . . . . . . 145