Preface xi
1 Vectors and linear operators 1
1.1 The linearity of physical phenomena 1
1.2 Vector spaces 2
1.2.1 A word on notation 4
1.2.2 Linear independence, bases, and dimensionality 5
1.2.3 Subspaces 7
1.2.4 Isomorphism of N-dimensional spaces 8
1.2.5 Dual spaces 8
1.3 Inner products and orthogonality 10
1.3.1 Inner products 10
1.3.2 The Schwarz inequality 11
1.3.3 Vector norms 12
1.3.4 Orthonormal bases and the Gram-Schmidt process 12
1.3.5 Complete sets of orthonormal vectors 15
1.4 Operators and matrices 16
1.4.1 Linear operators 17
1.4.2 Representing operators with matrices 18
1.4.3 Matrix algebra 20
1.4.4 Rank and nullity 22
1.4.5 Bounded operators 23
1.4.6 Inverses 24
1.4.7 Change of basis and the similarity transformation 25
1.4.8 Adjoints and Hermitian operators 27
1.4.9 Determinants and the matrix inverse 29
1.4.10 Unitary operators 33
1.4.11 The trace of a matrix 35
1.5 Eigenvectors and their role in representing operators 36
1.5.1 Eigenvectors and eigenvalues 36
1.5.2 The eigenproblem for Hermitian and unitary operators 39
1.5.3 Diagonalizing matrices 40
1.6 Hilbert space: Infinite-dimensional vector space 43
Exercises 47
2 Sturm-Liouville theory 51
2.1 Second-order differential equations 52
2.1.1 Uniqueness and linear independence 52
2.1.2 The adjoint operator 55
2.1.3 Self-adjoint operator 56
2.2 Sturm-Liouville systems 57
2.3 The Sturm-Liouville eigenproblem 60
2.4 The Dirac delta function 64
2.5 Completeness 66
2.6 Recap 68
Summary 68
Exercises 69
3 Partial differential equations 71
3.1 A survey of partial differential equations 71
3.1.1 The continuity equation 71
3.1.2 The diffusion equation 72
3.1.3 The free-particle Schrödinger equation 73
3.1.4 The heat equation 73
3.1.5 The inhomogeneous diffusion equation 74
3.1.6 Schrödinger equation for a particle in a potential field 74
3.1.7 The Poisson equation 74
3.1.8 The Laplace equation 75
3.1.9 The wave equation 75
3.1.10 Inhomogeneous wave equation 76
3.1.11 Summary of PDEs 76
3.2 Separation of variables and the Helmholtz equation 76
3.2.1 Rectangular coordinates 78
3.2.2 Cylindrical coordinates 80
3.2.3 Spherical coordinates 82
3.3 The paraxial approximation 83
3.4 The three types of linear PDEs 84
3.4.1 Hyperbolic PDEs 85
3.4.2 Parabolic PDEs 87
3.4.3 Elliptic PDEs 87
3.5 Outlook 88
Summary 88
Exercises 89
4 Fourier analysis 91
4.1 Fourier series 91
4.2 The exponential form of Fourier series 96
4.3 General intervals 98
4.4 Parseval's theorem 103
4.5 Back to the delta function 105
4.6 Fourier transform 107
4.7 Convolution integral 111
Summary 115
Exercises 116
5 Series solutions of ordinary differential equations 121
5.1 The Frobenius method 122
5.1.1 Power series 122
5.1.2 Introductory example 123
5.1.3 Ordinary points 125
5.1.4 Regular singular points 130
5.2 Wronskian method for obtaining a second solution 137
5.3 Bessel and Neumann functions 137
5.4 Legendre polynomials 142
Summary 144
Exercises 145
6 Spherical harmonics 147
6.1 Properties of the Legendre polynomials, Pl(x) 148
6.1.1 Rodrigues formula 148
6.1.2 Orthogonality 150
6.1.3 Completeness 151
6.1.4 Generating function 152
6.1.5 Recursion relations 155
6.2 Associated Legendre functions, Pm l (x) 157
6.3 Spherical harmonic functions, Yml (¿, ¿) 158
6.4 Addition theorem for Ym l (¿, ¿) 160
6.5 Laplace equation in spherical coordinates 166
Summary 167
Exercises 168
7 Bessel functions 173
7.1 Small-argument and asymptotic forms 173
7.1.1 Limiting forms for small argument 173
7.1.2 Asymptotic forms for large argument 174
7.1.3 Hankel functions 174
7.2 Properties of the Bessel functions, Jn(x) 175
7.2.1 Series associated with the generating function 175
7.2.2 Recursion relations 177
7.2.3 Integral representation 178
7.3 Orthogonality 180
7.4 Bessel series 182
7.5 The Fourier-Bessel transform 185
7.6 Spherical Bessel functions 186
7.6.1 Reduction to elementary functions 186
7.6.2 Small-argument forms 188
7.6.3 Asymptotic forms 188
7.6.4 Orthogonality and completeness 189
7.7 Expansion of plane waves in spherical harmonics 190
Summary 192
Exercises 192
8 Complex analysis 195
8.1 Complex functions 195
8.2 Analytic functions: differentiable in a region 197
8.2.1 Continuity, differentiability, and analyticity 197
8.2.2 Cauchy-Riemann conditions 198
8.2.3 Analytic functions are functions only of z = x + iy 201
8.2.4 Useful definitions 201
8.3 Contour integrals 202
8.4 Integrating analytic functions 206
8.5 Cauchy integral formulas 210
8.5.1 Derivatives of analytic functions 211
8.5.2 Consequences of the Cauchy formulas 212
8.6 Taylor and Laurent series 213
8.6.1 Taylor series 213
8.6.2 The zeros of analytic functions are isolated 215
8.6.3 Laurent series 215
8.7 Singularities and residues 217
8.7.1 Isolated singularities, residue theorem 217
8.7.2 Multivalued functions, branch points, and branch cuts 220
8.8 Definite integrals 221
8.8.1 Integrands containing cos ¿ and sin ¿ 222
8.8.2 Infinite integrals 223
8.8.3 Poles on the contour of integration 226
8.9 Meromorphic functions 228
8.10 Approximation of integrals 230
8.10.1 The method of steepest descent 233
8.10.2 The method of stationary phase 235
8.11 The analytic signal 236
8.11.1 The Hilbert transform 237
8.11.2 Paley-Wiener and Titchmarsh theorems 239
8.11.3 Is the analytic signal, analytic? 241
8.12 The Laplace transform 242
Summary 245
Exercises 245
9 Inhomogeneous differential equations 251
9.1 The method of Green functions 251
9.1.1 Boundary conditions 252
9.1.2 Reciprocity relation: G(x, x') = G(x', x) 253
9.1.3 Matching conditions 254
9.1.4 Direct construction of G(x, x') 255
9.1.5 Eigenfunction expansions 257
9.2 Poisson equation 260
9.2.1 Boundary conditions and reciprocity relations 261
9.2.2 So, what's the Green function? 263
9.3 Helmholtz equation 266
9.3.1 Green function for two-dimensional problems 267
9.3.2 Free-space Green function for three dimensions 270
9.3.3 Expansion in spherical harmonics 270
9.4 Diffusion equation 272
9.4.1 Boundary conditions, causality, and reciprocity 272
9.4.2 Solution to the diffusion equation 274
9.4.3 Free-space Green function 275
9.5 Wave equation 279
9.6 The Kirchhoff integral theorem 283
Summary 284
Exercises 284
10 Integral equations 287
10.1 Introduction 287
10.1.1 Equivalence of integral and differential equations 287
10.1.2 Role of coordinate systems in capturing boundary data 288
10.2 Classification of integral equations 290
10.3 Neumann series 291
10.4 Integral transform methods 293
10.4.1 Difference kernels 293
10.4.2 Fourier kernels 294
10.5 Separable kernels 295
10.6 Self-adjoint kernels 297
10.7 Numerical approaches 302
10.7.1 Matrix form 302
10.7.2 Measurement space 303
10.7.3 The generalized inverse 306
Summary 314
Exercises 315
11 Tensor analysis 319
11.1 Once over lightly: A quick intro to tensors 319
11.2 Transformation properties 327
11.2.1 The two types of vector: Contravariant and covariant 327
11.2.2 Coordinate transformations 328
11.2.3 Contravariant vectors and tensors 332
11.2.4 Covariant vectors and tensors 336
11.2.5 Mixed tensors 339
11.2.6 Covariant equations 339
11.3 Contraction and the quotient theorem 340
11.4 The metric tensor 342
11.5 Raising and lowering indices 344
11.6 Geometric properties of covariant vectors 347
11.7 Relative tensors 350
11.8 Tensors as operators 353
11.9 Symmetric and antisymmetric tensors 356
11.10 The Levi-Civita tensor 357
11.11 Pseudotensors 360
11.12 Covariant differentiation of tensors 363
Summary 373
Exercises 374
A Vector calculus 377
A.1 Scalar fields 377
A.1.1 The directional derivative 377
A.1.2 The gradient 378
A.2 Vector fields 379
A.2.1 Divergence 379
A.2.2 Curl 380
A.2.3 The Laplacian 380
A.2.4 Vector operator formulae 381
A.3 Integration 382
A.3.1 Line integrals 382
A.3.2 Surface integrals 383
A.4 Important integral theorems in vector calculus 384
A.4.1 Green's theorem in the plane 384
A.4.2 The divergence theorem 386
A.4.3 Stokes' theorem 386
A.4.4 Conservative fields 387
A.4.5 The Helmholtz theorem 389
A.5 Coordinate systems 390
A.5.1 Orthogonal curvilinear coordinates 390
A.5.2 Unit vectors 391
A.5.3 Differential displacement 392
A.5.4 Differential surface and volume elements 393
A.5.5 Transformation of vector components 393
A.5.6 Cylindrical coordinates 394
B Power series 401
C The gamma function, ¿(x) 403
Recursion relation 403
Limit formula 404
Reflection formula 405
Digamma function 405
D Boundary conditions for Partial Differential Equations 409
Summary 417
References 419
Index 421
