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This page intentionally left blank MATHEMATICAL METHODS FOR PHYSICISTS SIXTH EDITION This page intentionally left blank MATHEMATICAL METHODS FOR PHYSICISTS SIXTH EDITION George B. Arfken MiamiUniversity Oxford,OH Hans J. Weber UniversityofVirginia Charlottesville,VA Amsterdam Boston Heidelberg London NewYork Oxford Paris SanDiego SanFrancisco Singapore Sydney Tokyo AcquisitionsEditor TomSinger ProjectManager SimonCrump MarketingManager LindaBeattie CoverDesign EricDeCicco Composition VTEXTypesettingServices CoverPrinter PhoenixColor InteriorPrinter TheMaple–VailBookManufacturingGroup ElsevierAcademicPress 30CorporateDrive,Suite400,Burlington,MA01803,USA 525BStreet,Suite1900,SanDiego,California92101-4495,USA 84Theobald’sRoad,LondonWC1X8RR,UK Thisbookisprintedonacid-freepaper.(cid:1)∞ Copyright©2005,ElsevierInc.Allrightsreserved. Nopartofthispublicationmaybereproducedortransmittedinanyformorbyanymeans,electronicorme- chanical,includingphotocopy,recording,oranyinformationstorageandretrievalsystem,withoutpermissionin writingfromthepublisher. PermissionsmaybesoughtdirectlyfromElsevier’sScience&TechnologyRightsDepartmentinOxford,UK: phone:(+44)1865843830,fax:(+44)1865853333,e-mail:[email protected] yourrequeston-lineviatheElsevierhomepage(http://elsevier.com),byselecting“CustomerSupport”andthen “ObtainingPermissions.” LibraryofCongressCataloging-in-PublicationData Appicationsubmitted BritishLibraryCataloguinginPublicationData AcataloguerecordforthisbookisavailablefromtheBritishLibrary ISBN:0-12-059876-0Casebound ISBN:0-12-088584-0InternationalStudentsEdition ForallinformationonallElsevierAcademicPressPublications visitourWebsiteatwww.books.elsevier.com PrintedintheUnitedStatesofAmerica 05 06 07 08 09 10 9 8 7 6 5 4 3 2 1 C ONTENTS Preface xi 1 VectorAnalysis 1 1.1 Definitions,ElementaryApproach . . . . . . . . . . . . . . . . . . . . . 1 1.2 RotationoftheCoordinateAxes . . . . . . . . . . . . . . . . . . . . . . 7 1.3 ScalarorDotProduct . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4 VectororCrossProduct . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.5 TripleScalarProduct,TripleVectorProduct . . . . . . . . . . . . . . . 25 1.6 Gradient,∇ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1.7 Divergence,∇ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 1.8 Curl,∇× . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 1.9 SuccessiveApplicationsof∇ . . . . . . . . . . . . . . . . . . . . . . . 49 1.10 VectorIntegration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 1.11 Gauss’Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 1.12 Stokes’Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 1.13 PotentialTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 1.14 Gauss’Law,Poisson’sEquation . . . . . . . . . . . . . . . . . . . . . . 79 1.15 DiracDeltaFunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 1.16 Helmholtz’sTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 AdditionalReadings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 2 VectorAnalysisinCurvedCoordinatesandTensors 103 2.1 OrthogonalCoordinatesinR3 . . . . . . . . . . . . . . . . . . . . . . . 103 2.2 DifferentialVectorOperators . . . . . . . . . . . . . . . . . . . . . . . 110 2.3 SpecialCoordinateSystems:Introduction . . . . . . . . . . . . . . . . 114 2.4 CircularCylinderCoordinates. . . . . . . . . . . . . . . . . . . . . . . 115 2.5 SphericalPolarCoordinates . . . . . . . . . . . . . . . . . . . . . . . . 123 v vi Contents 2.6 TensorAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 2.7 Contraction,DirectProduct . . . . . . . . . . . . . . . . . . . . . . . . 139 2.8 QuotientRule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 2.9 Pseudotensors,DualTensors . . . . . . . . . . . . . . . . . . . . . . . 142 2.10 GeneralTensors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 2.11 TensorDerivativeOperators . . . . . . . . . . . . . . . . . . . . . . . . 160 AdditionalReadings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 3 DeterminantsandMatrices 165 3.1 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 3.2 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 3.3 OrthogonalMatrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 3.4 HermitianMatrices,UnitaryMatrices . . . . . . . . . . . . . . . . . . 208 3.5 DiagonalizationofMatrices . . . . . . . . . . . . . . . . . . . . . . . . 215 3.6 NormalMatrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 AdditionalReadings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 4 GroupTheory 241 4.1 IntroductiontoGroupTheory . . . . . . . . . . . . . . . . . . . . . . . 241 4.2 GeneratorsofContinuousGroups . . . . . . . . . . . . . . . . . . . . . 246 4.3 OrbitalAngularMomentum . . . . . . . . . . . . . . . . . . . . . . . . 261 4.4 AngularMomentumCoupling . . . . . . . . . . . . . . . . . . . . . . . 266 4.5 HomogeneousLorentzGroup . . . . . . . . . . . . . . . . . . . . . . . 278 4.6 LorentzCovarianceofMaxwell’sEquations . . . . . . . . . . . . . . . 283 4.7 DiscreteGroups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 4.8 DifferentialForms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 AdditionalReadings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 5 InfiniteSeries 321 5.1 FundamentalConcepts . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 5.2 ConvergenceTests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 5.3 AlternatingSeries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 5.4 AlgebraofSeries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 5.5 SeriesofFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 5.6 Taylor’sExpansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 5.7 PowerSeries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 5.8 EllipticIntegrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370 5.9 BernoulliNumbers,Euler–MaclaurinFormula . . . . . . . . . . . . . . 376 5.10 AsymptoticSeries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 5.11 InfiniteProducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396 AdditionalReadings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 6 FunctionsofaComplexVariableIAnalyticProperties,Mapping 403 6.1 ComplexAlgebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404 6.2 Cauchy–RiemannConditions . . . . . . . . . . . . . . . . . . . . . . . 413 6.3 Cauchy’sIntegralTheorem. . . . . . . . . . . . . . . . . . . . . . . . . 418 Contents vii 6.4 Cauchy’sIntegralFormula . . . . . . . . . . . . . . . . . . . . . . . . . 425 6.5 LaurentExpansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430 6.6 Singularities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438 6.7 Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 6.8 ConformalMapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451 AdditionalReadings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 7 FunctionsofaComplexVariableII 455 7.1 CalculusofResidues . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455 7.2 DispersionRelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482 7.3 MethodofSteepestDescents . . . . . . . . . . . . . . . . . . . . . . . . 489 AdditionalReadings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497 8 TheGammaFunction(FactorialFunction) 499 8.1 Definitions,SimpleProperties . . . . . . . . . . . . . . . . . . . . . . . 499 8.2 DigammaandPolygammaFunctions . . . . . . . . . . . . . . . . . . . 510 8.3 Stirling’sSeries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516 8.4 TheBetaFunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520 8.5 IncompleteGammaFunction . . . . . . . . . . . . . . . . . . . . . . . 527 AdditionalReadings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533 9 DifferentialEquations 535 9.1 PartialDifferentialEquations . . . . . . . . . . . . . . . . . . . . . . . 535 9.2 First-OrderDifferentialEquations . . . . . . . . . . . . . . . . . . . . 543 9.3 SeparationofVariables. . . . . . . . . . . . . . . . . . . . . . . . . . . 554 9.4 SingularPoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562 9.5 SeriesSolutions—Frobenius’Method . . . . . . . . . . . . . . . . . . . 565 9.6 ASecondSolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578 9.7 NonhomogeneousEquation—Green’sFunction . . . . . . . . . . . . . 592 9.8 HeatFlow,orDiffusion,PDE . . . . . . . . . . . . . . . . . . . . . . . 611 AdditionalReadings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618 10 Sturm–LiouvilleTheory—OrthogonalFunctions 621 10.1 Self-AdjointODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622 10.2 HermitianOperators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634 10.3 Gram–SchmidtOrthogonalization . . . . . . . . . . . . . . . . . . . . . 642 10.4 CompletenessofEigenfunctions . . . . . . . . . . . . . . . . . . . . . . 649 10.5 Green’sFunction—EigenfunctionExpansion . . . . . . . . . . . . . . . 662 AdditionalReadings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674 11 BesselFunctions 675 11.1 BesselFunctionsoftheFirstKind,J (x) . . . . . . . . . . . . . . . . . 675 ν 11.2 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694 11.3 NeumannFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 699 11.4 HankelFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707 11.5 ModifiedBesselFunctions,I (x)andK (x) . . . . . . . . . . . . . . . 713 ν ν viii Contents 11.6 AsymptoticExpansions . . . . . . . . . . . . . . . . . . . . . . . . . . . 719 11.7 SphericalBesselFunctions. . . . . . . . . . . . . . . . . . . . . . . . . 725 AdditionalReadings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 739 12 LegendreFunctions 741 12.1 GeneratingFunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 741 12.2 RecurrenceRelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 749 12.3 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 756 12.4 AlternateDefinitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 767 12.5 AssociatedLegendreFunctions . . . . . . . . . . . . . . . . . . . . . . 771 12.6 SphericalHarmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786 12.7 OrbitalAngularMomentumOperators . . . . . . . . . . . . . . . . . . 793 12.8 AdditionTheoremforSphericalHarmonics . . . . . . . . . . . . . . . 797 12.9 IntegralsofThreeY’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803 12.10 LegendreFunctionsoftheSecondKind . . . . . . . . . . . . . . . . . . 806 12.11 VectorSphericalHarmonics . . . . . . . . . . . . . . . . . . . . . . . . 813 AdditionalReadings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816 13 MoreSpecialFunctions 817 13.1 HermiteFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 817 13.2 LaguerreFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 837 13.3 ChebyshevPolynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 848 13.4 HypergeometricFunctions . . . . . . . . . . . . . . . . . . . . . . . . . 859 13.5 ConfluentHypergeometricFunctions . . . . . . . . . . . . . . . . . . . 863 13.6 MathieuFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 869 AdditionalReadings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 879 14 FourierSeries 881 14.1 GeneralProperties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 881 14.2 Advantages,UsesofFourierSeries . . . . . . . . . . . . . . . . . . . . 888 14.3 ApplicationsofFourierSeries . . . . . . . . . . . . . . . . . . . . . . . 892 14.4 PropertiesofFourierSeries . . . . . . . . . . . . . . . . . . . . . . . . 903 14.5 GibbsPhenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 910 14.6 DiscreteFourierTransform . . . . . . . . . . . . . . . . . . . . . . . . 914 14.7 FourierExpansionsofMathieuFunctions . . . . . . . . . . . . . . . . 919 AdditionalReadings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 929 15 IntegralTransforms 931 15.1 IntegralTransforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 931 15.2 DevelopmentoftheFourierIntegral. . . . . . . . . . . . . . . . . . . . 936 15.3 FourierTransforms—InversionTheorem . . . . . . . . . . . . . . . . . 938 15.4 FourierTransformofDerivatives . . . . . . . . . . . . . . . . . . . . . 946 15.5 ConvolutionTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 951 15.6 MomentumRepresentation . . . . . . . . . . . . . . . . . . . . . . . . . 955 15.7 TransferFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 961 15.8 LaplaceTransforms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965 Contents ix 15.9 LaplaceTransformofDerivatives . . . . . . . . . . . . . . . . . . . . . 971 15.10 OtherProperties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 979 15.11 Convolution(Faltungs)Theorem . . . . . . . . . . . . . . . . . . . . . 990 15.12 InverseLaplaceTransform . . . . . . . . . . . . . . . . . . . . . . . . . 994 AdditionalReadings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1003 16 IntegralEquations 1005 16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1005 16.2 IntegralTransforms,GeneratingFunctions . . . . . . . . . . . . . . . . 1012 16.3 NeumannSeries,Separable(Degenerate)Kernels . . . . . . . . . . . . 1018 16.4 Hilbert–SchmidtTheory . . . . . . . . . . . . . . . . . . . . . . . . . . 1029 AdditionalReadings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1036 17 CalculusofVariations 1037 17.1 ADependentandanIndependentVariable . . . . . . . . . . . . . . . . 1038 17.2 ApplicationsoftheEulerEquation . . . . . . . . . . . . . . . . . . . . 1044 17.3 SeveralDependentVariables. . . . . . . . . . . . . . . . . . . . . . . . 1052 17.4 SeveralIndependentVariables . . . . . . . . . . . . . . . . . . . . . . . 1056 17.5 SeveralDependentandIndependentVariables . . . . . . . . . . . . . . 1058 17.6 LagrangianMultipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . 1060 17.7 VariationwithConstraints . . . . . . . . . . . . . . . . . . . . . . . . . 1065 17.8 Rayleigh–RitzVariationalTechnique . . . . . . . . . . . . . . . . . . . 1072 AdditionalReadings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076 18 NonlinearMethodsandChaos 1079 18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1079 18.2 TheLogisticMap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1080 18.3 SensitivitytoInitialConditionsandParameters . . . . . . . . . . . . . 1085 18.4 NonlinearDifferentialEquations . . . . . . . . . . . . . . . . . . . . . 1088 AdditionalReadings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1107 19 Probability 1109 19.1 Definitions,SimpleProperties . . . . . . . . . . . . . . . . . . . . . . . 1109 19.2 RandomVariables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1116 19.3 BinomialDistribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 1128 19.4 PoissonDistribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1130 19.5 Gauss’NormalDistribution . . . . . . . . . . . . . . . . . . . . . . . . 1134 19.6 Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1138 AdditionalReadings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1150 GeneralReferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1150 Index 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Jan 4, 2015 No part of this publication may be reproduced or transmitted in any 05 06 07 08 09 10 9 8 7 6 5 4 3 2 1 Special Coordinate Systems: Introduction . The book also incorporates a deliberate focus on problem-solving skills by parallelogram addition, the equilibrium condition of zero re

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