Mathematical Methods for Physics

Preface 3
Contents 5
1 Functions of a Complex Variable 9
11 A Brief Review of Analytic Functions 9
12 Cauchy Residue Theorem and Its Applications 23
13 Poisson's Integral and Mittag-Le_er's Expansion 53
14 Evaluations of Inverse Laplace Transform 58
Exercise 64

2 Conformal Mapping 69
21 Examples of Conformal Mappings 69
22 Transformation of Harmonic Functions 78
23 Applications to Steady Temperatures 81
24 Applications to Electrostatic Potential 90
25 Schwarz-Christo_el Transformation 101
26 Applications to Fluid Flow 113
Exercise 124

3 Elliptic Functions 129
31 Introduction 129
32 Elliptic Integrals 135
33 Parametric Equation of the Ellipse 145
34 Reduction to the Standard Form 157
35 Complex Argument 169
36 _Conformal Mapping 174
37 _Applications 181
Exercise 196

4 Tensor Calculus 201
41 Tensor Algebra 201
42 Fundamental Tensor (Metric) 207
43 Parallel Displacement 213
44 Christo_el Symbols 214
45 Covariant Di_erentiation 221
46 Geodesics 228
47 Frenet-Serret Formulas 236
48 Riemann-Christo_el Tensors 241
49 Gravity as a Metric Phenomenon 255
Exercise 269

5 Sturm-Liouville Theory 271
51 Adjoint and Hermitian Operators 272
52 Properties of the Hermitian Operators 279
53 Bessel Inequality and Schwarz Inequality 284
54 Green Function 290
55 Gram-Schmidt Orthogonalization 317
Exercise 321

6 Gamma Function 323
61 De_nition and Properties of Gamma Functions (z) 323
62 Integral Expression of (z) 328
63 Cauchy and Saalschutz Extension of (z) with Re(z) < 0 332
64 Digamma Functions And Polygamma Functions 333
65 Bernoulli Numbers And Bernoulli Functions 339
66 Euler-Maclaurin Integration Formula 342
67 Beta Function and Incomplete Functions 346
68 Error Functions 353
69 Dirichlet Integral 357 Exercise 360

7 Bessel Functions 365
71 Generating Function 365
72 Recurrence Relations 368
73 Integral Expressions of Bessel Function Jn(x) 370
74 Bessel Functions J_(x) with Noninteger _ 371
75 Contour Expression of Bessel Functions 380
76 Orthogonality of Bessel Functions 383
77 The Second Kind Bessel Functions N_(x) 391
78 Hankel Functions H(1;2)
_ (x) 394
79 Saddle-Point Method (Steepest Descent) 396
710 Wronskian Formulas 401
711 Modi_ed Bessel Functions 403
712 Spherical Bessel Functions 410
713 Modi_ed Spherical Bessel Functions 419
Exercise 421

8 Legendre Functions 425
81 Generating Function 425
82 Recurrence Relations 429
83 Orthogonality 433
84 Rodrigues Formula of Legendre Functions 439
85 Legendre Functions of the Second Kind 444
86 Laplace Integral Representation of Legendre Function 450
87 Associated Legendre Functions 452
88 Spherical Harmonic Functions 462
89 Angular Momentum 468
810 Addition Theorem 474
811 _Integrals of the Product of Three Spherical Harmonic Functions 479
Exercise 481

9 Other Special Functions 485
91 Hermite Functions 485
92 Laguerre Functions 502
93 Associated Laguerre Functions 506
94 Chebyshev Polynomials 514
95 Hypergeometric Functions 526
96 Conuent Hypergeometric Functions 535
Exercise 543

10 Fourier Series and Fourier Transform 547
101 Fourier Series 547
102 Complex Fourier Series 561
103 Applications to Solving Di_erential Equations 563
104 Fourier Integral 569
105 Properties of Fourier Transform 583
106 Dirac _-Function 601
Exercise 610

11 Laplace Transform 615
111 De_nition of Laplace Transform 615
112 Properties of Laplace Transform 618
113 Applications to Special Functions and Di_erential Equations 630
114 Inverse Laplace Transform 647
115 Operator Calculus 656
116 Useful Integrals 662
Exercise 669

12 Mellin and Hankel Transform 673
121 De_nition of Integral Transform 673
122 Mellin Transform 678
123 Properties of Mellin Transform 687
124 Hankel Transform 691
125 Properties of Hankel Transform 702
126 Relation Between Hankel and Fourier Transforms 707
127 _Dual Integral Equations 714
128 Finite Hankel Transform 722 Exercise 738

13 Integral Equations 741
131 Linear Di_erential Equations And Integral Equations 742
132 Sturm-Liouville Equation into Integral Equation 747
133 Integral Transforms 759
134 Iteration Method 767
135 Separable Kernels 769
136 Eigenvalues and Eigenfunctions 772
137 Variation-Iteration Method 777
138 Two-Dimensional Green Function 782
139 Three-Dimensional Green Function 789
1310Applications to Heat, Wave, and Schrodinger Equations 796
Exercise 807

14 Calculus of Variations 813
141 Variational Calculus 814
142 Hamiltonian Principle 821
143 One Dependence, Several Independent Variables 824
144 Several Dependent, Several Independent Variables 827
145 Lagrangian Multipliers 830
146 Variation Subject to Constraints 835
147 Rayleigh-Ritz Method 842
148 Variational Formulation of Eigenfunction Problems 844
149 Eigenfunction Problems by the Ratio Method 850
Exercise 855
Bibliography 859
Index 861

Preface 3
Contents 5
1 Functions of a Complex Variable 9
11 A Brief Review of Analytic Functions 9
12 Cauchy Residue Theorem and Its Applications 23
13 Poisson's Integral and Mittag-Le_er's Expansion 53
14 Evaluations of Inverse Laplace Transform 58
Exercise 64

2 Conformal Mapping 69
21 Examples of Conformal Mappings 69
22 Transformation of Harmonic Functions 78
23 Applications to Steady Temperatures 81
24 Applications to Electrostatic Potential 90
25 Schwarz-Christo_el Transformation 101
26 Applications to Fluid Flow 113
Exercise 124

3 Elliptic Functions 129
31 Introduction 129
32 Elliptic Integrals 135
33 Parametric Equation of the Ellipse 145
34 Reduction to the Standard Form 157
35 Complex Argument 169
36 _Conformal Mapping 174
37 _Applications 181
Exercise 196

4 Tensor Calculus 201
41 Tensor Algebra 201
42 Fundamental Tensor (Metric) 207
43 Parallel Displacement 213
44 Christo_el Symbols 214
45 Covariant Di_erentiation 221
46 Geodesics 228
47 Frenet-Serret Formulas 236
48 Riemann-Christo_el Tensors 241
49 Gravity as a Metric Phenomenon 255
Exercise 269

5 Sturm-Liouville Theory 271
51 Adjoint and Hermitian Operators 272
52 Properties of the Hermitian Operators 279
53 Bessel Inequality and Schwarz Inequality 284
54 Green Function 290
55 Gram-Schmidt Orthogonalization 317
Exercise 321

6 Gamma Function 323
61 De_nition and Properties of Gamma Functions

9 Other Special Functions 485
91 Hermite Functions 485
92 Laguerre Functions 502
93 Associated Laguerre Functions 506
94 Chebyshev Polynomials 514
95 Hypergeometric Functions 526
96 Conuent Hypergeometric Functions 535
Exercise 543

10 Fourier Series and Fourier Transform 547
101 Fourier Series 547
102 Complex Fourier Series 561
103 Applications to Solving Di_erential Equations 563
104 Fourier Integral 569
105 Properties of Fourier Transform 583
106 Dirac _-Function 601
Exercise 610

11 Laplace Transform 615
111 De_nition of Laplace Transform 615
112 Properties of Laplace Transform 618
113 Applications to Special Functions and Di_erential Equations 630
114 Inverse Laplace Transform 647
115 Operator Calculus 656
116 Useful Integrals 662
Exercise 669

12 Mellin and Hankel Transform 673
121 De_nition of Integral Transform 673
122 Mellin Transform 678
123 Properties of Mellin Transform 687
124 Hankel Transform 691
125 Properties of Hankel Transform 702
126 Relation Between Hankel and Fourier Transforms 707
127 _Dual Integral Equations 714
128 Finite Hankel Transform 722 Exercise 738

13 Integral Equations 741
131 Linear Di_erential Equations And Integral Equations 742
132 Sturm-Liouville Equation into Integral Equation 747
133 Integral Transforms 759
134 Iteration Method 767
135 Separable Kernels 769
136 Eigenvalues and Eigenfunctions 772
137 Variation-Iteration Method 777
138 Two-Dimensional Green Function 782
139 Three-Dimensional Green Function 789
1310Applications to Heat, Wave, and Schrodinger Equations 796
Exercise 807

14 Calculus of Variations 813
141 Variational Calculus 814
142 Hamiltonian Principle 821
143 One Dependence, Several Independent Variables 824
144 Several Dependent, Several Independent Variables 827
145 Lagrangian Multipliers 830
146 Variation Subject to Constraints 835
147 Rayleigh-Ritz Method 842
148 Variational Formulation of Eigenfunction Problems 844
149 Eigenfunction Problems by the Ratio Method 850
Exercise 855
Bibliography 859
Index 861