Mathematical Methods in Science and Engineering

852 Pages·2018·7.24 MB·English

by  Selçuk S. Bayin

MathematicalMethodsin
ScienceandEngineering
Mathematical Methods in Science
and Engineering
SelçukS¸.Bayın
InstituteofAppliedMathematics
MiddleEastTechnicalUniversity
AnkaraTurkey
SecondEdition
Thiseditionfirstpublished2018
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LibraryofCongressCataloguing-in-PublicationData:
Names:Bayın,S¸.Selçuk,1951-author.
Title:Mathematicalmethodsinscienceandengineering/bySelçukS¸.Bayın.
Description:Secondedition.|Hoboken,NJ:JohnWiley&Sons,2018.|
Includesbibliographicalreferencesandindex.|
Identifiers:LCCN2017042888(print)|LCCN2017048224(ebook)|ISBN
9781119425410(pdf)|ISBN9781119425458(epub)|ISBN9781119425397
(cloth)
Subjects:LCSH:Mathematicalphysics–Textbooks.|Engineering
mathematics–Textbooks.
Classification:LCCQC20(ebook)|LCCQC20.B352018(print)|DDC
530.15–dc23
LCrecordavailableathttps://lccn.loc.gov/2017042888
CoverDesign:Wiley
CoverImages:(Background)©Studio-Pro/Gettyimages;
(Imageinset)CourtesyofSelcukS.Bayin
Setin10/12ptWarnockProbySPiGlobal,Chennai,India
PrintedintheUnitedStatesofAmerica
10 9 8 7 6 5 4 3 2 1
v
Contents
Preface xix
1 LegendreEquationandPolynomials 1
1.1 Second-OrderDifferentialEquationsofPhysics 1
1.2 LegendreEquation 2
1.2.1 MethodofSeparationofVariables 4
1.2.2 SeriesSolutionoftheLegendreEquation 4
1.2.3 FrobeniusMethod–Review 7
1.3 LegendrePolynomials 8
1.3.1 RodriguezFormula 10
1.3.2 GeneratingFunction 10
1.3.3 RecursionRelations 12
1.3.4 SpecialValues 12
1.3.5 SpecialIntegrals 13
1.3.6 OrthogonalityandCompleteness 14
1.3.7 AsymptoticForms 17
1.4 AssociatedLegendreEquationandPolynomials 18
1.4.1 AssociatedLegendrePolynomialsPm(x) 20
l
1.4.2 Orthogonality 21
1.4.3 RecursionRelations 22
1.4.4 IntegralRepresentations 24
1.4.5 AssociatedLegendrePolynomialsform<0 26
1.5 SphericalHarmonics 27
1.5.1 AdditionTheoremofSphericalHarmonics 30
1.5.2 RealSphericalHarmonics 33
Bibliography 33
Problems 34
2 LaguerrePolynomials 39
2.1 CentralForceProblemsinQuantumMechanics 39
2.2 LaguerreEquationandPolynomials 41
vi Contents
2.2.1 GeneratingFunction 42
2.2.2 RodriguezFormula 43
2.2.3 Orthogonality 44
2.2.4 RecursionRelations 45
2.2.5 SpecialValues 46
2.3 AssociatedLaguerreEquationandPolynomials 46
2.3.1 GeneratingFunction 48
2.3.2 RodriguezFormulaandOrthogonality 49
2.3.3 RecursionRelations 49
Bibliography 49
Problems 50
3 HermitePolynomials 53
3.1 HarmonicOscillatorinQuantumMechanics 53
3.2 HermiteEquationandPolynomials 54
3.2.1 GeneratingFunction 56
3.2.2 RodriguezFormula 56
3.2.3 RecursionRelationsandOrthogonality 57
Bibliography 61
Problems 62
4 GegenbauerandChebyshevPolynomials 65
4.1 WaveEquationonaHypersphere 65
4.2 GegenbauerEquationandPolynomials 68
4.2.1 OrthogonalityandtheGeneratingFunction 68
4.2.2 AnotherRepresentationoftheSolution 69
4.2.3 TheSecondSolution 70
4.2.4 ConnectionwiththeGegenbauerPolynomials 71
4.2.5 EvaluationoftheNormalizationConstant 72
4.3 ChebyshevEquationandPolynomials 72
4.3.1 ChebyshevPolynomialsoftheFirstKind 72
4.3.2 ChebyshevandGegenbauerPolynomials 73
4.3.3 ChebyshevPolynomialsoftheSecondKind 73
4.3.4 OrthogonalityandGeneratingFunction 74
4.3.5 AnotherDefinition 75
Bibliography 76
Problems 76
5 BesselFunctions 81
5.1 Bessel’sEquation 83
5.2 BesselFunctions 83
5.2.1 AsymptoticForms 84
5.3 ModifiedBesselFunctions 86
Contents vii
5.4 SphericalBesselFunctions 87
5.5 PropertiesofBesselFunctions 88
5.5.1 GeneratingFunction 88
5.5.2 IntegralDefinitions 89
5.5.3 RecursionRelationsoftheBesselFunctions 89
5.5.4 OrthogonalityandRootsofBesselFunctions 90
5.5.5 BoundaryConditionsfortheBesselFunctions 91
5.5.6 WronskianofPairsofSolutions 94
5.6 TransformationsofBesselFunctions 95
5.6.1 CriticalLengthofaRod 96
Bibliography 98
Problems 99
6 HypergeometricFunctions 103
6.1 HypergeometricSeries 103
6.2 HypergeometricRepresentationsofSpecialFunctions 107
6.3 ConfluentHypergeometricEquation 108
6.4 PochhammerSymbolandHypergeometricFunctions 109
6.5 ReductionofParameters 113
Bibliography 115
Problems 115
7 Sturm–LiouvilleTheory 119
7.1 Self-AdjointDifferentialOperators 119
7.2 Sturm–LiouvilleSystems 120
7.3 HermitianOperators 121
7.4 PropertiesofHermitianOperators 122
7.4.1 RealEigenvalues 122
7.4.2 OrthogonalityofEigenfunctions 123
7.4.3 CompletenessandtheExpansionTheorem 123
7.5 GeneralizedFourierSeries 125
7.6 TrigonometricFourierSeries 126
7.7 HermitianOperatorsinQuantumMechanics 127
Bibliography 129
Problems 130
8 FactorizationMethod 133
8.1 AnotherFormfortheSturm–LiouvilleEquation 133
8.2 MethodofFactorization 135
8.3 TheoryofFactorizationandtheLadderOperators 136
8.4 SolutionsviatheFactorizationMethod 141
8.4.1 CaseI(m>0and𝜇(m)isanincreasingfunction) 141
8.4.2 CaseII(m>0and𝜇(m)isadecreasingfunction) 142
viii Contents
8.5 TechniqueandtheCategoriesofFactorization 143
8.5.1 PossibleFormsfork(z,m) 143
8.5.1.1 Positivepowersofm 143
8.5.1.2 Negativepowersofm 146
8.6 AssociatedLegendreEquation(TypeA) 148
8.6.1 DeterminingtheEigenvalues,𝜆 149
l
8.6.2 ConstructionoftheEigenfunctions 150
8.6.3 LadderOperatorsform 151
8.6.4 InterpretationoftheL andL Operators 153
+ −
8.6.5 LadderOperatorsforl 155
8.6.6 CompleteSetofLadderOperators 159
8.7 SchrödingerEquationandSingle-ElectronAtom(TypeF) 160
8.8 GegenbauerFunctions(TypeA) 162
8.9 SymmetricTop(TypeA) 163
8.10 BesselFunctions(TypeC) 164
8.11 HarmonicOscillator(TypeD) 165
8.12 DifferentialEquationfortheRotationMatrix 166
8.12.1 Step-Up/DownOperatorsform 166
8.12.2 Step-Up/DownOperatorsform′ 167
8.12.3 NormalizedFunctionswithm=m′ =l 168
8.12.4 FullMatrixforl=2 168
8.12.5 Step-Up/DownOperatorsforl 170
Bibliography 171
Problems 171
9 CoordinatesandTensors 175
9.1 CartesianCoordinates 175
9.1.1 AlgebraofVectors 176
9.1.2 DifferentiationofVectors 177
9.2 OrthogonalTransformations 178
9.2.1 RotationsAboutCartesianAxes 182
9.2.2 FormalPropertiesoftheRotationMatrix 183
9.2.3 EulerAnglesandArbitraryRotations 183
9.2.4 ActiveandPassiveInterpretationsofRotations 185
9.2.5 InfinitesimalTransformations 186
9.2.6 InfinitesimalTransformationsCommute 188
9.3 CartesianTensors 189
9.3.1 OperationswithCartesianTensors 190
9.3.2 TensorDensitiesorPseudotensors 191
9.4 CartesianTensorsandtheTheoryofElasticity 192
9.4.1 StrainTensor 192
9.4.2 StressTensor 193
9.4.3 ThermodynamicsandDeformations 194
Contents ix
9.4.4 ConnectionbetweenShearandStrain 196
9.4.5 Hook’sLaw 200
9.5 GeneralizedCoordinatesandGeneralTensors 201
9.5.1 ContravariantandCovariantComponents 202
9.5.2 MetricTensorandtheLineElement 203
9.5.3 GeometricInterpretationofComponents 206
9.5.4 InterpretationoftheMetricTensor 207
9.6 OperationswithGeneralTensors 214
9.6.1 EinsteinSummationConvention 214
9.6.2 ContractionofIndices 214
9.6.3 MultiplicationofTensors 214
9.6.4 TheQuotientTheorem 214
9.6.5 EqualityofTensors 215
9.6.6 TensorDensities 215
9.6.7 DifferentiationofTensors 216
9.6.8 SomeCovariantDerivatives 219
9.6.9 RiemannCurvatureTensor 220
9.7 Curvature 221
9.7.1 ParallelTransport 222
9.7.2 RoundTripsviaParallelTransport 223
9.7.3 AlgebraicPropertiesoftheCurvatureTensor 225
9.7.4 ContractionsoftheCurvatureTensor 226
9.7.5 CurvatureinnDimensions 227
9.7.6 Geodesics 229
9.7.7 InvarianceVersusCovariance 229
9.8 SpacetimeandFour-Tensors 230
9.8.1 MinkowskiSpacetime 230
9.8.2 LorentzTransformationsandSpecialRelativity 231
9.8.3 TimeDilationandLengthContraction 233
9.8.4 AdditionofVelocities 233
9.8.5 Four-TensorsinMinkowskiSpacetime 234
9.8.6 Four-Velocity 237
9.8.7 Four-MomentumandConservationLaws 238
9.8.8 MassofaMovingParticle 240
9.8.9 WaveFour-Vector 240
9.8.10 DerivativeOperatorsinSpacetime 241
9.8.11 RelativeOrientationofAxesinK andK Frames 241
9.9 Maxwell’sEquationsinMinkowskiSpacetime 243
9.9.1 TransformationofElectromagneticFields 246
9.9.2 Maxwell’sEquationsinTermsofPotentials 246
9.9.3 CovarianceofNewton’sDynamicTheory 247
Bibliography 248
Problems 249
x Contents
10 ContinuousGroupsandRepresentations 257
10.1 DefinitionofaGroup 258
10.1.1 Nomenclature 258
10.2 InfinitesimalRingorLieAlgebra 259
10.2.1 PropertiesofrG 260
10.3 LieAlgebraoftheRotationGroupR(3) 260
10.3.1 AnotherApproachtorR(3) 262
10.4 GroupInvariants 264
10.4.1 LorentzTransformations 266
10.5 UnitaryGroupinTwoDimensionsU(2) 267
10.5.1 SpecialUnitaryGroupSU(2) 269
10.5.2 LieAlgebraofSU(2) 270
10.5.3 AnotherApproachtorSU(2) 272
10.6 LorentzGroupandItsLieAlgebra 274
10.7 GroupRepresentations 279
10.7.1 Schur’sLemma 279
10.7.2 GroupCharacter 280
10.7.3 UnitaryRepresentation 280
10.8 RepresentationsofR(3) 281
10.8.1 SphericalHarmonicsandRepresentationsofR(3) 281
10.8.2 AngularMomentuminQuantumMechanics 281
10.8.3 RotationofthePhysicalSystem 282
10.8.4 RotationOperatorinTermsoftheEulerAngles 282
10.8.5 RotationOperatorintheOriginalCoordinates 283
10.8.6 EigenvalueEquationsforL ,L ,andL2 287
z ±
10.8.7 FourierExpansioninSphericalHarmonics 287
10.8.8 MatrixElementsofL ,L ,andL 289
x y z
10.8.9 RotationMatricesoftheSphericalHarmonics 290
10.8.10 Evaluationofthedl (𝛽)Matrices 292
m′m
10.8.11 Inverseofthedl (𝛽)Matrices 292
m′m
10.8.12 DifferentialEquationfordl (𝛽) 293
m′m
10.8.13 AdditionTheoremforSphericalHarmonics 296
10.8.14 DeterminationofI intheAdditionTheorem 298
l
10.8.15 ConnectionofDl (𝛽)withSphericalHarmonics 300
mm′
10.9 IrreducibleRepresentationsofSU(2) 302
10.10 RelationofSU(2)andR(3) 303
10.11 GroupSpaces 306
10.11.1 RealVectorSpace 306
10.11.2 InnerProductSpace 307
10.11.3 Four-VectorSpace 307
10.11.4 ComplexVectorSpace 308
10.11.5 FunctionSpaceandHilbertSpace 308
10.11.6 Completeness 309
Contents xi
10.12 HilbertSpaceandQuantumMechanics 310
10.13 ContinuousGroupsandSymmetries 311
10.13.1 PointGroupsandTheirGenerators 311
10.13.2 TransformationofGeneratorsandNormalForms 312
10.13.3 TheCaseofMultipleParameters 314
10.13.4 ActionofGeneratorsonFunctions 315
10.13.5 ExtensionorProlongationofGenerators 316
10.13.6 SymmetriesofDifferentialEquations 318
Bibliography 321
Problems 322
11 ComplexVariablesandFunctions 327
11.1 ComplexAlgebra 327
11.2 ComplexFunctions 329
11.3 ComplexDerivativesandCauchy–RiemannConditions 330
11.3.1 AnalyticFunctions 330
11.3.2 HarmonicFunctions 332
11.4 Mappings 334
11.4.1 ConformalMappings 348
11.4.2 ElectrostaticsandConformalMappings 349
11.4.3 FluidMechanicsandConformalMappings 352
11.4.4 Schwarz–ChristoffelTransformations 358
Bibliography 368
Problems 368
12 ComplexIntegralsandSeries 373
12.1 ComplexIntegralTheorems 373
12.1.1 Cauchy–GoursatTheorem 373
12.1.2 CauchyIntegralTheorem 374
12.1.3 CauchyTheorem 376
12.2 TaylorSeries 378
12.3 LaurentSeries 379
12.4 ClassificationofSingularPoints 385
12.5 ResidueTheorem 386
12.6 AnalyticContinuation 389
12.7 ComplexTechniquesinTakingSomeDefiniteIntegrals 392
12.8 GammaandBetaFunctions 399
12.8.1 GammaFunction 399
12.8.2 BetaFunction 401
12.8.3 UsefulRelationsoftheGammaFunctions 403
12.8.4 IncompleteGammaandBetaFunctions 403
12.8.5 AnalyticContinuationoftheGammaFunction 404
12.9 CauchyPrincipalValueIntegral 406