COMPLEX ANALYSIS COMPLEX ANALYSIS A Modern First Course in Function Theory Jerry R. Muir, Jr. TheUniversityofScranton Copyright(cid:2)c2015byJohnWiley&Sons,Inc.Allrightsreserved PublishedbyJohnWiley&Sons,Inc.,Hoboken,NewJersey PublishedsimultaneouslyinCanada Nopartofthispublicationmaybereproduced,storedinaretrievalsystem,ortransmittedinanyform orbyanymeans,electronic,mechanical,photocopying,recording,scanning,orotherwise,exceptas permittedunderSection107or108ofthe1976UnitedStatesCopyrightAct,withouteithertheprior writtenpermissionofthePublisher,orauthorizationthroughpaymentoftheappropriateper-copyfeeto theCopyrightClearanceCenter,Inc.,222RosewoodDrive,Danvers,MA01923,(978)750-8400, fax(978)750-4470,oronthewebatwww.copyright.com.RequeststothePublisherforpermissionshould beaddressedtothePermissionsDepartment,JohnWiley&Sons,Inc.,111RiverStreet,Hoboken,NJ 07030,(201)748-6011,fax(201)748-6008,oronlineathttp://www.wiley.com/go/permissions. LimitofLiability/DisclaimerofWarranty:Whilethepublisherandauthorhaveusedtheirbesteffortsin preparingthisbook,theymakenorepresentationsorwarrantieswithrespecttotheaccuracyor completenessofthecontentsofthisbookandspecificallydisclaimanyimpliedwarrantiesof merchantabilityorfitnessforaparticularpurpose.Nowarrantymaybecreatedorextendedbysales representativesorwrittensalesmaterials.Theadviceandstrategiescontainedhereinmaynotbe suitableforyoursituation.Youshouldconsultwithaprofessionalwhereappropriate.Neitherthe publishernorauthorshallbeliableforanylossofprofitoranyothercommercialdamages,including butnotlimitedtospecial,incidental,consequential,orotherdamages. Forgeneralinformationonourotherproductsandservicesorfortechnicalsupport,pleasecontactourCustomerCare DepartmentwithintheUnitedStatesat(800)762-2974,outsidetheUnitedStatesat(317)572-3993orfax(317)572- 4002. Wileyalsopublishesitsbooksinavarietyofelectronicformats.Somecontentthatappearsinprint maynotbeavailableinelectronicformats.FormoreinformationaboutWileyproducts,visitourwebsiteatwww.wiley.com. LibraryofCongressCataloging-in-PublicationData: Muir,JerryR. Complexanalysis:amodernfirstcourseinfunctiontheory/JerryR. Muir,Jr. pagescm Includesbibliographicalreferencesandindex. ISBN978-1-118-70522-3(cloth) 1. Functionsofcomplexvariables.2. Mathematicalanalysis.I.Title. QA331.M852014 515–dc23 2014035668 PrintedintheUnitedStatesofAmerica 10 9 8 7 6 5 4 3 2 1 To Stacey, proofreader, patron, and partner CONTENTS Preface ix 1 TheComplexNumbers 1 1.1 Why? 1 1.2 TheAlgebraofComplexNumbers 3 1.3 TheGeometryoftheComplexPlane 7 1.4 TheTopologyoftheComplexPlane 9 1.5 TheExtendedComplexPlane 16 1.6 ComplexSequences 18 1.7 ComplexSeries 24 2 ComplexFunctionsandMappings 29 2.1 ContinuousFunctions 29 2.2 UniformConvergence 34 2.3 PowerSeries 38 2.4 ElementaryFunctionsandEuler’sFormula 43 2.5 ContinuousFunctionsasMappings 50 2.6 LinearFractionalTransformations 53 vii viii CONTENTS 2.7 Derivatives 64 2.8 TheCalculusofReal-VariableFunctions 70 2.9 ContourIntegrals 75 3 AnalyticFunctions 87 3.1 ThePrincipleofAnalyticity 87 3.2 DifferentiableFunctionsareAnalytic 89 3.3 ConsequencesofGoursat’sTheorem 100 3.4 TheZerosofAnalyticFunctions 104 3.5 TheOpenMappingTheoremandMaximumPrinciple 108 3.6 TheCauchy–RiemannEquations 113 3.7 ConformalMappingandLocalUnivalence 117 4 Cauchy’sIntegralTheory 127 4.1 TheIndexofaClosedContour 127 4.2 TheCauchyIntegralFormula 133 4.3 Cauchy’sTheorem 139 5 TheResidueTheorem 145 5.1 LaurentSeries 145 5.2 ClassificationofSingularities 152 5.3 Residues 158 5.4 EvaluationofRealIntegrals 165 5.5 TheLaplaceTransform 174 6 HarmonicFunctionsandFourierSeries 183 6.1 HarmonicFunctions 183 6.2 ThePoissonIntegralFormula 191 6.3 FurtherConnectionstoAnalyticFunctions 201 6.4 FourierSeries 210 Epilogue 227 A SetsandFunctions 239 B TopicsfromAdvancedCalculus 247 References 255 Index 257