Graduate Mathematical Physics with MATHEMATICA Supplements

481 Pages·2006·3.074 MB·English

by  J. Kelly

JamesJ.Kelly
GraduateMathematicalPhysics
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With MATHEMATICA supplements
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Preface
This textbook is intended to serve a course on mathematical methods of physics that is
oftentakenbygraduatestudentsintheirfirstsemesterorbyundergraduatesintheirsenior
year. I believe the most important topic for first-year graduate students in physics is the
theoryofanalyticfunctions.Somestudentsmayhavehadabriefexposuretothatsubject
asundergraduates,butfewareadequatelypreparedtoapplysuchmethodstophysicsprob-
lems.Therefore,Istartwiththetheoryofanalyticfunctionsandpracticallyallsubsequent
materialisbaseduponit.Theprimarytopicsinclude:theoryofanalyticfunctions,integral
transforms,generalizedfunctions,eigenfunctionexpansions,Greenfunctions,boundary-
valueproblems,andgrouptheory.Thiscourseisdesignedtopreparestudentsforadvanced
treatmentsofelectromagnetictheoryandquantummechanics,butthemethodsandappli-
cations are more general. Although this is a fairly standard course taught in most major
universities,Iwasnotsatisfiedwiththeavailabletextbooks.Somepopularbutencyclope-
dic books include a broader range of topics, much too broad to cover in one semester at
thedepththatIthoughtnecessaryforgraduatestudents.Otherswithamoremanageable
lengthappeartobetargetedprimarilyatundergraduatesandrelegatetoappendicessome
ofthetopicsthatIbelievetobemostimportant.Therefore,Isoonfoundthatpreparation
oflecturenotesfordistributiontostudentswasevolvingintoatextbook-writingproject.
I was not able to avoid producing too much material either. I usually chose to skip
most of the chapter on Legendre and Bessel functions, assuming that graduate students
already had some familiarity with them, and instead referred them to a summary of the
properties that are useful for the chapter on boundary-value problems. Other instructors
might choose to omit the chapter on dispersion theory instead because most of it will
probablybecoveredinthesubsequentcourseonelectromagnetism,butIfindthatsubject
more interesting and more fun to discuss than special functions. The chapter on group
theorywaspreparedattherequestofreviewers;althoughIneverreachedthattopicinone
semester, I hope that it will be useful for those teaching a two-semester course or as a
resourcethatstudentswilluselateron.Itmayalsobeusefulforone-semestercoursesat
institutionswheretheaveragestudentalreadyhasasufficientlystrongmasteryofanalytic
functions that the first couple of chapters can be abbreviated or omitted. I believe that
it should be possible to cover most of the remaining material well in a single semester
at any mid-level university. I assume that the calculus of variations will be covered in a
concurrent course on classical mechanics and that the students are already comfortable
with linear algebra, differential equations, and vector calculus. Probability theory, tensor
analysis,anddifferentialgeometryareomitted.
A CD containing detailed solutions to all of the problems is available to instructors.
These solutions often employ (cid:1)(cid:2) to perform some of the routine but tedious
manipulations and to prepare figures. Some of these solutions may also be presented as
additionalexamplesofthetechniquescoveredinthiscourse.
GraduateMathematicalPhysics.JamesJ.Kelly
Copyright©2006WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim
ISBN:3-527-40637-9
Contents
Preface V
NotetotheReader XV
1 AnalyticFunctions 1
1.1 ComplexNumbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 MotivationandDefinitions . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 TriangleInequalities . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.3 PolarRepresentation . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.4 ArgumentFunction . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 TakeCarewithMultivaluedFunctions . . . . . . . . . . . . . . . . . . . 8
1.3 FunctionsasMappings . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.1 Mapping:w(cid:1)(cid:2)z . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3.2 Mapping:w(cid:1)Sin(cid:3)z(cid:4) . . . . . . . . . . . . . . . . . . . . . . . . 16
1.4 ElementaryFunctionsandTheirInverses. . . . . . . . . . . . . . . . . . 17
1.4.1 ExponentialandLogarithm. . . . . . . . . . . . . . . . . . . . . 17
1.4.2 Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.4.3 TrigonometricandHyperbolicFunctions . . . . . . . . . . . . . 19
1.4.4 StandardBranchCuts . . . . . . . . . . . . . . . . . . . . . . . 20
1.5 Sets,Curves,RegionsandDomains . . . . . . . . . . . . . . . . . . . . 21
1.6 LimitsandContinuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.7 Differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.7.1 Cauchy–RiemannEquations . . . . . . . . . . . . . . . . . . . . 23
1.7.2 DifferentiationRules . . . . . . . . . . . . . . . . . . . . . . . . 25
1.8 PropertiesofAnalyticFunctions . . . . . . . . . . . . . . . . . . . . . . 26
1.9 Cauchy–GoursatTheorem . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.9.1 SimplyConnectedRegions . . . . . . . . . . . . . . . . . . . . . 28
1.9.2 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.9.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.10 CauchyIntegralFormula . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.10.1 IntegrationAroundNonanalyticRegions . . . . . . . . . . . . . 32
1.10.2 CauchyIntegralFormula . . . . . . . . . . . . . . . . . . . . . . 34
1.10.3 Example:YukawaField . . . . . . . . . . . . . . . . . . . . . . 34
1.10.4 DerivativesofAnalyticFunctions . . . . . . . . . . . . . . . . . 36
1.10.5 Morera’sTheorem . . . . . . . . . . . . . . . . . . . . . . . . . 37
1.11 ComplexSequencesandSeries . . . . . . . . . . . . . . . . . . . . . . . 37
1.11.1 ConvergenceTests . . . . . . . . . . . . . . . . . . . . . . . . . 37
1.11.2 UniformConvergence . . . . . . . . . . . . . . . . . . . . . . . 40
GraduateMathematicalPhysics.JamesJ.Kelly
Copyright©2006WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim
ISBN:3-527-40637-9
VIII Contents
1.12 DerivativesandTaylorSeriesforAnalyticFunctions . . . . . . . . . . . 41
1.12.1 TaylorSeries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
1.12.2 CauchyInequality . . . . . . . . . . . . . . . . . . . . . . . . . 44
1.12.3 Liouville’sTheorem . . . . . . . . . . . . . . . . . . . . . . . . 44
1.12.4 FundamentalTheoremofAlgebra . . . . . . . . . . . . . . . . . 44
1.12.5 ZerosofAnalyticFunctions . . . . . . . . . . . . . . . . . . . . 45
1.13 LaurentSeries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
1.13.1 Derivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
1.13.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
1.13.3 ClassificationofSingularities . . . . . . . . . . . . . . . . . . . 49
1.13.4 PolesandResidues . . . . . . . . . . . . . . . . . . . . . . . . . 50
1.14 MeromorphicFunctions. . . . . . . . . . . . . . . . . . . . . . . . . . . 51
1.14.1 PoleExpansion . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
1.14.2 Example:Tan(cid:3)z(cid:4) . . . . . . . . . . . . . . . . . . . . . . . . . . 53
1.14.3 ProductExpansion . . . . . . . . . . . . . . . . . . . . . . . . . 54
1.14.4 Example:Sin(cid:3)z(cid:4). . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2 Integration 65
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.2 GoodTricks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.2.1 ParametricDifferentiation . . . . . . . . . . . . . . . . . . . . . 65
2.2.2 ConvergenceFactors . . . . . . . . . . . . . . . . . . . . . . . . 66
2.3 ContourIntegration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
2.3.1 ResidueTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . 66
2.3.2 DefiniteIntegralsoftheForm(cid:1)2Π f(cid:3)sinΘ,cosΘ(cid:4)(cid:7)Θ . . . . . . . . 67
0(cid:8)
2.3.3 DefiniteIntegralsoftheForm(cid:1) f(cid:3)x(cid:4)(cid:7)x . . . . . . . . . . . . . 69
(cid:9)(cid:8)
2.3.4 FourierIntegrals . . . . . . . . . . . . . . . . . . . . . . . . . . 70
2.3.5 CustomContours . . . . . . . . . . . . . . . . . . . . . . . . . . 72
2.4 IsolatedSingularitiesontheContour . . . . . . . . . . . . . . . . . . . . 73
2.4.1 RemovableSingularity . . . . . . . . . . . . . . . . . . . . . . . 73
2.4.2 CauchyPrincipalValue . . . . . . . . . . . . . . . . . . . . . . . 75
2.5 IntegrationAroundaBranchPoint . . . . . . . . . . . . . . . . . . . . . 77
2.6 ReductiontoTabulatedIntegrals . . . . . . . . . . . . . . . . . . . . . . 79
2.6.1 Example:(cid:1)(cid:8) (cid:2)(cid:9)x4(cid:7)x . . . . . . . . . . . . . . . . . . . . . . . . 80
(cid:9)(cid:8)
2.6.2 Example:TheBetaFunction . . . . . . . . . . . . . . . . . . . . 81
2.6.3 Example:(cid:1)(cid:8) Ωn (cid:7)Ω . . . . . . . . . . . . . . . . . . . . . . . 81
0 (cid:2)ΒΩ(cid:9)1
2.7 IntegralRepresentationsforAnalyticFunctions . . . . . . . . . . . . . . 82
2.8 Using(cid:1)(cid:2)toEvaluateIntegrals . . . . . . . . . . . . . . . . . 86
2.8.1 SymbolicIntegration . . . . . . . . . . . . . . . . . . . . . . . . 86
2.8.2 NumericalIntegration . . . . . . . . . . . . . . . . . . . . . . . 88
2.8.3 FurtherInformation. . . . . . . . . . . . . . . . . . . . . . . . . 89
3 AsymptoticSeries 95
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
Contents IX
3.2 MethodofSteepestDescent . . . . . . . . . . . . . . . . . . . . . . . . 96
3.2.1 Example:GammaFunction . . . . . . . . . . . . . . . . . . . . 99
3.3 PartialIntegration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
3.3.1 Example:ComplementaryErrorFunction . . . . . . . . . . . . . 102
3.4 ExpansionofanIntegrand . . . . . . . . . . . . . . . . . . . . . . . . . 104
3.4.1 Example:ModifiedBesselFunction . . . . . . . . . . . . . . . . 105
4 GeneralizedFunctions 111
4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.2 PropertiesoftheDiracDeltaFunction . . . . . . . . . . . . . . . . . . . 113
4.3 OtherUsefulGeneralizedFunctions . . . . . . . . . . . . . . . . . . . . 115
4.3.1 HeavisideStepFunction . . . . . . . . . . . . . . . . . . . . . . 115
4.3.2 DerivativesoftheDiracDeltaFunction . . . . . . . . . . . . . . 116
4.4 GreenFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
4.5 MultidimensionalDeltaFunctions . . . . . . . . . . . . . . . . . . . . . 120
5 IntegralTransforms 125
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.2 FourierTransform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.2.2 DefinitionandInversion . . . . . . . . . . . . . . . . . . . . . . 128
5.2.3 BasicProperties . . . . . . . . . . . . . . . . . . . . . . . . . . 130
5.2.4 Parseval’sTheorem . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.2.5 ConvolutionTheorem . . . . . . . . . . . . . . . . . . . . . . . 132
5.2.6 CorrelationTheorem . . . . . . . . . . . . . . . . . . . . . . . . 133
5.2.7 UsefulFourierTransforms . . . . . . . . . . . . . . . . . . . . . 134
5.2.8 FourierTransformofDerivatives. . . . . . . . . . . . . . . . . . 138
5.2.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.3 GreenFunctionsviaFourierTransform . . . . . . . . . . . . . . . . . . 139
5.3.1 Example:GreenFunctionforOne-DimensionalDiffusion . . . . 139
5.3.2 Example:Three-DimensionalGreenFunctionforDiffusion
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.3.3 Example:GreenFunctionforDampedOscillator . . . . . . . . . 143
5.3.4 OperatorMethod . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.4 CosineorSineTransformsforEvenorOddFunctions. . . . . . . . . . . 147
5.5 DiscreteFourierTransform . . . . . . . . . . . . . . . . . . . . . . . . . 148
5.5.1 Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
5.5.2 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
5.5.3 TemporalCorrelation . . . . . . . . . . . . . . . . . . . . . . . . 156
5.5.4 PowerSpectrumEstimation . . . . . . . . . . . . . . . . . . . . 160
5.6 LaplaceTransform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
5.6.1 DefinitionandInversion . . . . . . . . . . . . . . . . . . . . . . 165
5.6.2 LaplaceTransformsforElementaryFunctions. . . . . . . . . . . 167
5.6.3 LaplaceTransformofDerivatives . . . . . . . . . . . . . . . . . 170
X Contents
5.6.4 ConvolutionTheorem . . . . . . . . . . . . . . . . . . . . . . . 171
5.6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
5.7 GreenFunctionsviaLaplaceTransform . . . . . . . . . . . . . . . . . . 173
5.7.1 Example:SeriesRCCircuit . . . . . . . . . . . . . . . . . . . . 174
5.7.2 Example:DampedOscillator . . . . . . . . . . . . . . . . . . . . 175
5.7.3 Example:DiffusionwithConstantBoundaryValue . . . . . . . . 176
6 AnalyticContinuationandDispersionRelations 191
6.1 AnalyticContinuation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
6.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
6.1.2 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
6.1.3 ReflectionPrinciple. . . . . . . . . . . . . . . . . . . . . . . . . 194
6.1.4 PermanenceofAlgebraicForm . . . . . . . . . . . . . . . . . . 195
6.1.5 Example:GammaFunction . . . . . . . . . . . . . . . . . . . . 195
6.2 DispersionRelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
6.2.1 Causality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
6.2.2 OscillatorModel . . . . . . . . . . . . . . . . . . . . . . . . . . 200
6.2.3 Kramers–KronigRelations . . . . . . . . . . . . . . . . . . . . . 203
6.2.4 SumRules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
6.3 HilbertTransform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
6.4 SpreadingofaWavePacket. . . . . . . . . . . . . . . . . . . . . . . . . 209
6.5 Solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
7 Sturm–LiouvilleTheory 223
7.1 Introduction:TheGeneralStringEquation . . . . . . . . . . . . . . . . . 223
7.2 HilbertSpaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
7.2.1 SchwartzInequality . . . . . . . . . . . . . . . . . . . . . . . . 229
7.2.2 Gram–SchmidtOrthogonalization . . . . . . . . . . . . . . . . . 230
7.3 PropertiesofSturm–LiouvilleSystems . . . . . . . . . . . . . . . . . . . 232
7.3.1 Self-Adjointness . . . . . . . . . . . . . . . . . . . . . . . . . . 232
7.3.2 RealityofEigenvaluesandOrthogonalityofEigenfunctions . . . 233
7.3.3 DiscretenessofEigenvalues . . . . . . . . . . . . . . . . . . . . 235
7.3.4 CompletenessofEigenfunctions . . . . . . . . . . . . . . . . . . 235
7.3.5 Parseval’sTheorem . . . . . . . . . . . . . . . . . . . . . . . . . 237
7.3.6 RealityofEigenfunctions . . . . . . . . . . . . . . . . . . . . . 238
7.3.7 InterleavingofZeros . . . . . . . . . . . . . . . . . . . . . . . . 238
7.3.8 ComparisonTheorems . . . . . . . . . . . . . . . . . . . . . . . 240
7.4 GreenFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
7.4.1 InterfaceMatching . . . . . . . . . . . . . . . . . . . . . . . . . 242
7.4.2 EigenfunctionExpansionofGreenFunction. . . . . . . . . . . . 246
7.4.3 Example:VibratingString . . . . . . . . . . . . . . . . . . . . . 252
7.5 PerturbationTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
7.5.1 Example:BeadatCenterofaString . . . . . . . . . . . . . . . . 255
7.6 VariationalMethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256