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COMPLEX VARIABLES AND APPLICATIONS Eighth Edition James Ward Brown ProfessorofMathematics The UniversityofMichigan–Dearborn Ruel V. Churchill LateProfessorofMathematics TheUniversityofMichigan COMPLEXVARIABLESANDAPPLICATIONS,EIGHTHEDITION PublishedbyMcGraw-Hill,abusinessunitofTheMcGraw-HillCompanies,Inc.,1221Avenueofthe Americas,NewYork,NY10020.Copyright2009byTheMcGraw-HillCompanies,Inc.Allrights reserved.Previouseditions2004,1996,1990,1984,1974,1960,1948Nopartofthispublicationmay bereproducedordistributedinanyformorbyanymeans,orstoredinadatabaseorretrievalsystem, withoutthepriorwrittenconsentofTheMcGraw-HillCompanies,Inc.,including,butnotlimitedto,in anynetworkorotherelectronicstorageortransmission,orbroadcastfordistancelearning. Someancillaries,includingelectronicandprintcomponents,maynotbeavailabletocustomersoutside theUnitedStates. Thisbookisprintedonacid-freepaper. 1234567890DOC/DOC098 ISBN978–0–07–305194–9 MHID0–07–305194–2 EditorialDirector:StewartK.Mattson DirectorofDevelopment:KristineTibbetts SeniorSponsoringEditor:ElizabethCovello DevelopmentalEditor:MichelleDriscoll EditorialCoordinator:AdamFischer SeniorMarketingManager:EricGates ProjectManager:AprilR.Southwood SeniorProductionSupervisor:KaraKudronowicz AssociateDesignCoordinator:BrendaA.Rolwes CoverDesigner:StudioMontage,St.Louis,Missouri ProjectCoordinator:MelissaM.Leick Compositor:LaserwordsPrivateLimited Typeface:10.25/12TimesRoman Printer:R.R.DonnellyCrawfordsville,IN LibraryofCongressCataloging-in-PublicationData Brown,JamesWard. Complexvariablesandapplications/JamesWardBrown,RuelV.Churchill.—8thed. p.cm. Includesbibliographicalreferencesandindex. ISBN978–0–07–305194–9—ISBN 0–07–305194–2 (hardcopy:acid-freepaper)1.Functionsof complexvariables.I.Churchill,RuelVance,1899-II.Title. QA331.7.C5242009 515(cid:1).9—dc22 2007043490 www.mhhe.com ABOUT THE AUTHORS JAMES WARD BROWN is Professor of Mathematics at The University of Michigan–Dearborn.HeearnedhisA.B.inphysicsfromHarvardUniversityandhis A.M. and Ph.D. in mathematics from The University of Michigan in Ann Arbor, where he was an Institute of Science and Technology Predoctoral Fellow. He is coauthor with Dr. Churchill of Fourier Series and Boundary Value Problems, now in its seventh edition. He has received a research grant from the National Science Foundation as well as a Distinguished Faculty Award from the Michigan Associa- tionofGoverningBoardsofCollegesandUniversities.Dr.BrownislistedinWho’s Whoin the World. RUELV. CHURCHILL was,atthe time of his deathin1987, ProfessorEmeritus ofMathematicsatTheUniversityofMichigan,wherehebeganteachingin1922.He receivedhisB.S.inphysicsfromtheUniversityofChicagoandhisM.S.inphysics and Ph.D. in mathematics from The University of Michigan. He was coauthor with Dr. Brown of Fourier Series and Boundary Value Problems, a classic text that he firstwrotealmost70yearsago.HewasalsotheauthorofOperationalMathematics. Dr. Churchill held various offices in the Mathematical Association of America and in other mathematical societies and councils. iii TotheMemoryofMyFather GeorgeH.Brown andofMyLong-TimeFriendandCoauthor RuelV.Churchill TheseDistinguishedMenofScienceforYearsInfluenced TheCareersofManyPeople,IncludingMyself. JWB CONTENTS Preface x 1 Complex Numbers 1 Sumsand Products 1 BasicAlgebraicProperties 3 FurtherProperties 5 Vectorsand Moduli 9 ComplexConjugates 13 ExponentialForm 16 Productsand PowersinExponentialForm 18 Argumentsof Productsand Quotients 20 RootsofComplexNumbers 24 Examples 27 RegionsintheComplexPlane 31 2 Analytic Functions 35 Functionsof aComplexVariable 35 Mappings 38 MappingsbytheExponentialFunction 42 Limits 45 Theoremson Limits 48 v vi contents LimitsInvolvingthePointat Infinity 50 Continuity 53 Derivatives 56 DifferentiationFormulas 60 Cauchy–RiemannEquations 63 Sufficient ConditionsforDifferentiability 66 Polar Coordinates 68 AnalyticFunctions 73 Examples 75 HarmonicFunctions 78 UniquelyDetermined AnalyticFunctions 83 Reflection Principle 85 3 Elementary Functions 89 The ExponentialFunction 89 The LogarithmicFunction 93 Branchesand DerivativesofLogarithms 95 Some IdentitiesInvolvingLogarithms 98 ComplexExponents 101 TrigonometricFunctions 104 HyperbolicFunctions 109 InverseTrigonometricandHyperbolicFunctions 112 4 Integrals 117 DerivativesofFunctionsw(t) 117 DefiniteIntegralsof Functionsw(t) 119 Contours 122 ContourIntegrals 127 Some Examples 129 Examples withBranchCuts 133 Upper BoundsforModuliofContourIntegrals 137 Antiderivatives 142 Proofof theTheorem 146 Cauchy–GoursatTheorem 150 Proofof theTheorem 152 contents vii SimplyConnectedDomains 156 MultiplyConnectedDomains 158 CauchyIntegralFormula 164 AnExtensionoftheCauchyIntegralFormula 165 SomeConsequencesoftheExtension 168 Liouville’sTheoremand theFundamentalTheorem ofAlgebra 172 MaximumModulusPrinciple 175 5 Series 181 ConvergenceofSequences 181 ConvergenceofSeries 184 TaylorSeries 189 ProofofTaylor’sTheorem 190 Examples 192 Laurent Series 197 ProofofLaurent’sTheorem 199 Examples 202 Absoluteand UniformConvergenceofPower Series 208 Continuityof SumsofPower Series 211 IntegrationandDifferentiationof PowerSeries 213 UniquenessofSeriesRepresentations 217 Multiplicationand Divisionof PowerSeries 222 6 Residues and Poles 229 IsolatedSingularPoints 229 Residues 231 Cauchy’sResidueTheorem 234 Residueat Infinity 237 TheThree TypesofIsolatedSingularPoints 240 Residuesat Poles 244 Examples 245 ZerosofAnalyticFunctions 249 Zerosand Poles 252 BehaviorofFunctionsNear IsolatedSingularPoints 257 viii contents 7 Applications of Residues 261 EvaluationofImproperIntegrals 261 Example 264 ImproperIntegralsfromFourierAnalysis 269 Jordan’sLemma 272 IndentedPaths 277 An IndentationAroundaBranchPoint 280 IntegrationAlonga BranchCut 283 DefiniteIntegralsInvolvingSinesand Cosines 288 ArgumentPrinciple 291 Rouche´’sTheorem 294 InverseLaplace Transforms 298 Examples 301 8 Mapping by Elementary Functions 311 Linear Transformations 311 The Transformationw=1/z 313 Mappingsby1/z 315 Linear FractionalTransformations 319 An ImplicitForm 322 MappingsoftheUpperHalfPlane 325 The Transformationw=sinz 330 Mappingsbyz2 and Branchesofz1/2 336 SquareRootsofPolynomials 341 Riemann Surfaces 347 Surfaces forRelatedFunctions 351 9 Conformal Mapping 355 Preservationof Angles 355 Scale Factors 358 Local Inverses 360 HarmonicConjugates 363 TransformationsofHarmonicFunctions 365 TransformationsofBoundaryConditions 367 contents ix 10 Applications of Conformal Mapping 373 Steady Temperatures 373 Steady TemperaturesinaHalfPlane 375 A RelatedProblem 377 TemperaturesinaQuadrant 379 ElectrostaticPotential 385 Potentialina CylindricalSpace 386 Two-DimensionalFluidFlow 391 TheStreamFunction 393 FlowsAroundaCornerandArounda Cylinder 395 11 The Schwarz–Christoffel Transformation 403 MappingtheReal AxisOntoa Polygon 403 Schwarz–ChristoffelTransformation 405 Trianglesand Rectangles 408 DegeneratePolygons 413 FluidFlow inaChannelThroughaSlit 417 FlowinaChannelWithan Offset 420 ElectrostaticPotentialAboutanEdgeof aConductingPlate 422 12 Integral Formulas of the Poisson Type 429 PoissonIntegralFormula 429 DirichletProblemfora Disk 432 RelatedBoundaryValueProblems 437 SchwarzIntegralFormula 440 DirichletProblemfora HalfPlane 441 Neumann Problems 445 Appendixes 449 Bibliography 449 TableofTransformationsof Regions 452 Index 461