Graduate Texts in Mathematics 245 Graduate Texts in Mathematics SeriesEditors: SheldonAxler SanFranciscoStateUniversity,SanFrancisco,CA,USA KennethRibet UniversityofCalifornia,Berkeley,CA,USA AdvisoryBoard: ColinAdams,WilliamsCollege,Williamstown,MA,USA AlejandroAdem,UniversityofBritishColumbia,Vancouver,BC,Canada RuthCharney,BrandeisUniversity,Waltham,MA,USA IreneM.Gamba,TheUniversityofTexasatAustin,Austin,TX,USA RogerE.Howe,YaleUniversity,NewHaven,CT,USA DavidJerison,MassachusettsInstituteofTechnology,Cambridge,MA,USA JeffreyC.Lagarias,UniversityofMichigan,AnnArbor,MI,USA JillPipher,BrownUniversity,Providence,RI,USA FadilSantosa,UniversityofMinnesota,Minneapolis,MN,USA AmieWilkinson,UniversityofChicago,Chicago,IL,USA GraduateTextsinMathematicsbridgethegapbetweenpassivestudyand creativeunderstanding,offeringgraduate-levelintroductionstoadvancedtopicsin mathematics.Thevolumesarecarefullywrittenasteachingaidsandhighlight characteristicfeaturesofthetheory.Althoughthesebooksarefrequentlyusedas textbooksingraduatecourses,theyarealsosuitableforindividualstudy. Forfurthervolumes: http://www.springer.com/series/136 Rub´ı E. Rodr´ıguez (cid:129) Irwin Kra (cid:129) Jane P. Gilman Complex Analysis In the Spirit of Lipman Bers Second Edition 123 Rub´ıE.Rodr´ıguez IrwinKra FacultaddeMatema´ticas DepartmentofMathematics PontificiaUniversidadCato´licadeChile StateUniversityofNewYorkatStonyBrook Santiago,Chile StonyBrook,NY,USA JaneP.Gilman DepartmentofMathematics andComputerScience RutgersUniversity Newark,NJ,USA ISSN0072-5285 ISBN978-1-4419-7322-1 ISBN978-1-4419-7323-8(eBook) DOI10.1007/978-1-4419-7323-8 SpringerNewYorkHeidelbergDordrechtLondon LibraryofCongressControlNumber:2012950351 ©SpringerScience+BusinessMediaNewYork2007,2013 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’slocation,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer. PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.Violations areliabletoprosecutionundertherespectiveCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. While the advice and information in this book are believed to be true and accurate at the date of publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with respecttothematerialcontainedherein. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) ForVictor,EleanorandBob andtothememoryof Maryand LipmanBers Preface to Second Edition The second edition contains significant new material in several new sections. We have also expanded sections from the first edition (improving, we expect, the exposition throughout) and included new figures and exercises for added clarity, aswellas,ofcourse,correctederrorsandtyposinthepreviousversion. Amongthemostimportantchangesare: (cid:129) Wehaveexpandedandclarifiedseveralsectionsfromthefirstedition. (cid:129) We have significantlyenlargedthe exercisesections. Someof the problemsare routine,otherschallenging,andsomerequireknowledgeofothersubjectsusually coveredin variousfirst-year graduate courses. The problemsare listed in more orlessrandomorderasfarastheirdifficulty. (cid:129) In both editions of this text, we use the approach to integration based on differentialforms.Inanalternativeapproachdifferentialformsareaby-product ofworkonintegrationoffunctionsmotivatedbyideasfromstandardtreatments ofintegralcalculus.ThatistheapproachthatBerstookincoursesthathetaught; itisalsotheapproachusedbyAhlfors.Ofcourse,eitherofthetwoapproaches are equallyvalid and lead to the same major results. In this second edition,we provideanappendixthatoutlinesthisalternativepathtothemainresults. (cid:129) New sections on Perron’s method for solving the Dirichlet problem, Green’s function,analternativeproofoftheRiemannmappingtheorem,andadescription of the divisor of a bounded analytic function on the disc via infinite Blaschke productsareincluded. (cid:129) We prove the Bers theorem on isomorphisms between rings of holomorphic functionsonplanedomains. (cid:129) AsectiononhistoricalreferencespreparedbyRanjanRoyhasbeenadded. Inaddition,thefollowingitemsrelatedtoourworkmightbeofinteresttothose readingthisvolume. (cid:129) An answer manual for the exercises prepared by Vamsi Pritham Pingali is available,toinstructorsusingthebookforacourse,fromthepublisher. (cid:129) Anelectronicversionofthebookisavailablefromthepublisher. vii viii PrefacetoSecondEdition (cid:129) One of the authors(IK) has created and maintainsa section aboutthe book on hiswebsite.Itcontains,amongotherthings,updatedinformationanderrata. http://www.math.sunysb.edu/(cid:2)irwin/bookcxinfo.html Theotherauthorsmayalsohaveinformationaboutthebookontheirwebsites. ItisourpleasuretothankRanjanRoyforproducingandallowingustoincludein thisvolumehishistoricalnote.Wearegratefultoourcolleaguesandstudentswho pointedoutplacesfor improvementin the first edition and in drafts of the second one. Among them: Bill Abikoff, Robert Burckel, Eduardo Friedman, Bryna Kra, Peter Landweber, Howard Masur, Sudeb Mitra, Lee Mosher, Robert Sczech, and JacobSturm.Itisstilltrue,ofcourse,thaterrorsandshortcomingsmayremainin thefinalversionofthiseditionandtheseareentirelyourresponsibility. Spring2012 NewYork,NY,USA IrwinKra Santiago,Chile Rub´ıE.Rodr´ıguez Newark,NJ,USA JaneP.Gilman Preface to First Edition This book presents fundamental material that should be part of the education of everypracticingmathematician.This materialwill also be of interestto computer scientists,physicistsandengineers. Becausecomplexanalysishasbeenusedbygenerationsofpracticingmathemati- ciansworkinginanumberofdifferentfields,thebasicresultshavebeendeveloped and redeveloped from a number of different perspectives. We are not wedded to any one viewpoint. Rather we will try to exploit the richness of the development andexplainandinterpretstandarddefinitionsandresultsusingthemostconvenient toolsfromanalysis, geometryand algebra.Complexanalysishasconnectionsand applicationstomanyothersubjectsinmathematics,bothclassicalandmodern,and toothersciences.Itisanareawheretheclassicalandthemoderntechniquesmeet andbenefitfromeachother.Wewilltrytoillustratethisintheapplicationswegive. Complexanalysisisthestudyofcomplexvaluedfunctionsofacomplexvariable and its initial task is to extend the concept of differentiability from real valued functions of a real variable to these functions. A complex valued function of a complexvariable that is differentiableis termedanalytic, and the first part of this book is a study of the many equivalent ways of understanding the concept of analyticity.Theequivalentwaysofformulatingtheconceptofananalyticfunction are summarized in what we term the fundamental theorem for functions of a complexvariable. In dedicatingthe first part of this book to the very precise goal of stating and proving the fundamentaltheorem we follow a path in the tradition of Lipman Bers from whom we learned the subject. In the second part of the text we then proceed to the leisurely exploration of interesting consequences and applicationsofthefundamentaltheorem. We are gratefulto LipmanBersforintroducingusto the beautyofthe subject. ThebookisanoutgrowthofnotesfromBers’soriginallectures.Versionsofthese notes have been used by us at our respective home institutions, some for more than20years,aswellasbyothersatvariousuniversities.Wearegratefultomany colleaguesand students who read and commentedon these notes. Our interaction ix