Analytic methods in physics

Title......Page 1
Preface......Page 3
Contents......Page 5
1.1 Introduction......Page 11
1.2 The Cartesian Coordinate System......Page 14
1.3 Differentiation of Vector Functions......Page 24
1.4 Integration of Vector Functions......Page 29
1.5 Orthogonal Curvilinear Coordinates......Page 41
1.6 Problems......Page 47
1.7 Appendix I: Systbme International (SI) Units......Page 50
1.8 Appendix 11: Properties of Determinants......Page 51
1.9 Summary of Some Properties of Determinants......Page 54
2.1 Introduction......Page 56
2.2 Matrix Analysis......Page 57
2.3 Essentials of Vector Spaces......Page 68
2.4 Essential Algebraic Structures......Page 77
2.5 Problems......Page 86
3.2 Complex Variables and Their Representations......Page 91
3.3 The de Moivre Theorem......Page 94
3.4 Analytic Functions of a Complex Variable......Page 95
3.5 Contour Integrals......Page 98
3.6 The Taylor Series and Zeros of f (z)......Page 102
3.7 The Laurent Expansion......Page 104
3.8 Problems......Page 109
3.9 Appendix: Series......Page 111
4.1 Isolated Singular Points......Page 115
4.2 Evaluation of Residues......Page 117
4.3 The Cauchy Residue Theorem......Page 121
4.4 The Cauchy Principal Value......Page 122
4.5 Evaluation of Definite Integrals......Page 123
4.6 Dispersion Relations......Page 128
4.7 Conformal Transformations......Page 130
4.8 Multi-valued Functions......Page 133
4.9 Problems......Page 137
5.1 Introduction......Page 139
5.3 Change of Interval......Page 140
5.4 Complex Form of the Fourier Series......Page 141
5.5 Generalized Fourier Series and the Dirac Delta Function......Page 145
5.6 Summation of the Fourier Series......Page 147
5.7 The Gibbs Phenomenon......Page 149
5.8 Summary of Some Properties of Fourier Series......Page 150
5.9 Problems......Page 151
6.1 Introduction......Page 153
6.2 Cosine and Sine Transforms......Page 155
6.3 The Transforms of Derivatives......Page 158
6.4 The Convolution Theorem......Page 160
6.5 Parseval\'s Relation......Page 161
6.6 Problems......Page 162
7.1 Introduction......Page 163
7.2 First-Order Linear Differential Equations......Page 164
7.3 The Bernoulli Differential Equation......Page 169
7.4 Second-Order Linear Differential Equations......Page 170
7.5 Some Numerical Methods......Page 182
7.6 Problems......Page 185
8.1 Introduction......Page 191
8.2 The Method of Separation of Variables......Page 193
8.3 Green\'s Functions in Potential Theory......Page 202
8.4 Some Numerical Methods......Page 204
8.5 Problems......Page 206
9.1 Introduction......Page 210
9.2 The Sturm-Liouville Theory......Page 211
9.3 The Hermite Polynomials......Page 218
9.4 The Helmholtz Differential Equation in Spherical Coordinates......Page 220
9.5 The Helmholtz Differential Equation in Cylindrical Coordinates......Page 228
9.6 The Hypergeometric Function......Page 231
9.7 The Confluent Hypergeometric Function......Page 234
9.8 Other Special Functions used in Physics......Page 235
9.9 Problems......Page 237
10.1 Introduction......Page 259
10.2 Integral Equations with Separable Kernels......Page 261
10.4 The Neurnann Series Method......Page 263
10.5 The Abel Problem......Page 264
10.6 Problems......Page 266
11.1 Introduction......Page 268
11.2 Stationary Values of Certain Functions and Functionals......Page 269
11.3 Hamilton\'s Variational Principle in Mechanics......Page 275
11.4 Formulation of Harniltonian Mechanics......Page 278
11.6 Transitions to Quantum Mechanics......Page 281
11.7 Problems......Page 284
12.1 Introduction......Page 286
12.2 Transformation of Coordinates in Linear Spaces......Page 287
12.3 Contravariant and Covariant Tensors......Page 289
12.4 Tensor Algebra......Page 292
12.5 The Line Element......Page 294
12.6 Tensor Calculus......Page 295
12.7 The Equation of the Geodesic Line......Page 299
12.8 Special Equations Involving the Metric Tensor......Page 300
12.9 Exterior Differential Forms......Page 303
12.10 Problems......Page 308
Bibliography......Page 310
Index......Page 313