Preface xi
1 Vectors and linear operators 1
1.1 The linearity of physical phenomena 1
1.2 Vector spaces 2
1.3 Inner products and orthogonality 10
1.4 Operators and matrices 16
1.5 Eigenvectors and their role in representing operators 36
1.6 Hilbert space: Infinite-dimensional vector space 43
Exercises 47
2 Sturm-Liouville theory 51
2.1 Second-order differential equations 52
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
3.2 Separation of variables and the Helmholtz equation 76
3.3 The paraxial approximation 83
3.4 The three types of linear PDEs 84
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
5.2 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
6.2 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
7.2 Properties of the Bessel functions, Jn(x) 175
7.3 Orthogonality 180
7.4 Bessel series 182
7.5 The Fourier-Bessel transform 185
7.6 Spherical Bessel functions 186
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
8.3 Contour integrals 202
8.4 Integrating analytic functions 206
8.5 Cauchy integral formulas 210
8.6 Taylor and Laurent series 213
8.7 Singularities and residues 217
8.8 Definite integrals 221
8.9 Meromorphic functions 228
8.10 Approximation of integrals 230
8.11 The analytic signal 236
8.12 The Laplace transform 242
Summary 245
Exercises 245
9 Inhomogeneous differential equations 251
9.1 The method of Green functions 251
9.2 Poisson equation 260
9.3 Helmholtz equation 266
9.4 Diffusion equation 272
9.5 Wave equation 279
9.6 The Kirchhoff integral theorem 283
Summary 284
Exercises 284
10 Integral equations 287
10.1 Introduction 287
10.2 Classification of integral equations 290
10.3 Neumann series 291
10.4 Integral transform methods 293
10.5 Separable kernels 295
10.6 Self-adjoint kernels 297
10.7 Numerical approaches 302
Summary 314
Exercises 315
11 Tensor analysis 319
11.1 Once over lightly: A quick intro to tensors 319
11.2 Transformation properties 327
11.3 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
