Preface ........................................................ xv
1 Vector algebra ............................................... 1
1.1 Preliminaries ........................................... 1
1.2 Coordinate system invariance ............................ 4
1.3 Vector multiplication ................................... 9
1.4 Useful products of vectors ............................. 12
1.5 Linear vector spaces ................................... 13
1.6 Focus: periodic media and reciprocal lattice vectors ... 17
1.7 Additional reading ..................................... 24
1.8 Exercises .............................................. 24
2 Vector calculus ............................................. 28
2.1 Introduction ........................................... 28
2.2 Vector integration ..................................... 29
2.3 The gradient,  ........................................ 35
2.4 Divergence, .......................................... 37
2.5 The curl, x ........................................... 41
2.6 Further applications of .............................. 43
2.7 Gauss' theorem (divergence theorem) .................... 45
2.8 Stokes' theorem ........................................ 47
2.9 Potential theory ....................................... 48
2.10 Focus: Maxwell's equations in integral and
differential form ...................................... 51
2.11 Focus: gauge freedom in Maxwell's equations ............ 57
2.12 Additional reading ..................................... 60
2.13 Exercises .............................................. 60
3 Vector calculus in curvilinear coordinate systems ........... 64
3.1 Introduction: systems with different symmetries ........ 64
3.2 General orthogonal coordinate systems .................. 65
3.3 Vector operators in curvilinear coordinates ............ 69
3.4 Cylindrical coordinates ................................ 73
3.5 Spherical coordinates .................................. 76
3.6 Exercises .............................................. 79
4 Matrices and linear algebra ................................. 83
4.1 Introduction: Polarization and Jones vectors ........... 83
4.2 Matrix algebra ......................................... 88
4.3 Systems of equations, determinants, and inverses ....... 93
4.4 Orthogonal matrices ................................... 102
4.5 Hermitian matrices and unitary matrices ............... 105
4.6 Diagonalization of matrices, eigenvectors, and
eigenvalues ........................................... 107
4.7 Gram-Schmidt orthonormalization ....................... 115
4.8 Orthonormal vectors and basis vectors ................. 118
4.9 Functions of matrices ................................. 120
4.10 Focus: matrix methods for geometrical optics .......... 120
4.11 Additional reading .................................... 133
4.12 Exercises ............................................. 133
5 Advanced matrix techniques and tensors ..................... 139
5.1 Introduction: Foldy-Lax scattering theory ............. 139
5.2 Advanced matrix terminology ........................... 142
5.3 Left-right eigenvalues and biorthogonality ............ 143
5.4 Singular value decomposition .......................... 146
5.5 Other matrix manipulations ............................ 153
5.6 Tensors ............................................... 159
5.7 Additional reading .................................... 174
5.8 Exercises ............................................. 174
6 Distributions .............................................. 177
6.1 Introduction: Gauss'law and the Poisson equation ...... 177
6.2 Introduction to delta functions ....................... 181
6.3 Calculus of delta functions ........................... 184
6.4 Other representations of the delta function ........... 185
6.5 Heaviside step function ............................... 187
6.6 Delta functions of more than one variable ............. 188
6.7 Additional reading .................................... 192
6.8 Exercises ............................................. 192
7 Infinite series ............................................ 195
7.1 Introduction: the Fabry-Perot interferometer .......... 195
7.2 Sequences and series .................................. 198
7.3 Series convergence .................................... 201
7.4 Series of functions ................................... 210
7.5 Taylor series ......................................... 213
7.6 Taylor series in more than one variable ............... 218
7.7 Power series .......................................... 220
7.8 Focus: convergence of the Born series ................. 221
7.9 Additional reading .................................... 226
7.10 Exercises ............................................. 226
8 Fourier series ............................................. 230
8.1 Introduction: diffraction gratings .................... 230
8.2 Real-valued Fourier series ............................ 233
8.3 Examples .............................................. 236
8.4 Integration range of the Fourier series ............... 239
8.5 Complex-valued Fourier series ......................... 239
8.6 Properties of Fourier series .......................... 240
8.7 Gibbs phenomenon and convergence in the mean .......... 243
8.8 Focus: X-ray diffraction from crystals ................ 246
8.9 Additional reading .................................... 249
8.10 Exercises ............................................. 249
9 Complex analysis ........................................... 252
9.1 Introduction: electric potential in an infinite
cylinder .............................................. 252
9.2 Complex algebra ....................................... 254
9.3 Functions of a complex variable ....................... 258
9.4 Complex derivatives and analyticity ................... 261
9.5 Complex integration and Cauchy's integral theorem ..... 265
9.6 Cauchy's integral formula ............................. 269
9.7 Taylor series ......................................... 271
9.8 Laurent series ........................................ 273
9.9 Classification of isolated singularities .............. 276
9.10 Branch points and Riemann surfaces .................... 278
9.11 Residue theorem ....................................... 285
9.12 Evaluation of definite integrals ...................... 288
9.13 Cauchy principal value ................................ 297
9.14 Focus: Kramers-Rronig relations ....................... 299
9.15 Focus: optical vortices ............................... 302
9.16 Additional reading .................................... 308
9.17 Exercises ............................................. 308
10 Advanced complex analysis .................................. 312
10.1 Introduction .......................................... 312
10.2 Analytic continuation ................................. 312
10.3 Stereographic projection .............................. 316
10.4 Conformal mapping ..................................... 325
10.5 Significant theorems in complex analysis .............. 332
10.6 Focus: analytic properties of wavefields .............. 340
10.7 Focus: optical cloaking and transformation optics ..... 345
10.8 Exercises ............................................. 348
11 Fourier transforms ......................................... 350
11.1 Introduction: Fraunhofer diffraction .................. 350
11.2 The Fourier transform and its inverse ................. 352
11.3 Examples of Fourier transforms ........................ 354
11.4 Mathematical properties of the Fourier transform ...... 358
11.5 Physical properties of the Fourier transform .......... 365
11.6 Eigenfunctions of the Fourier operator ................ 372
11.7 Higher-dimensional transforms ......................... 373
11.8 Focus: spatial filtering .............................. 375
11.9 Focus: angular spectrum representation ................ 377
11.10 Additional reading ................................... 382
11.11 Exercises ............................................ 383
12 Other integral transforms .................................. 386
12.1 Introduction: the Fresnel transform ................... 386
12.2 Linear canonical transforms ........................... 391
12.3 The Laplace transform ................................. 395
12.4 Fractional Fourier transform .......................... 400
12.5 Mixed domain transforms ............................... 402
12.6 The wavelet transform ................................. 406
12.7 The Wigner transform .................................. 409
12.8 Focus: the Radon transform and computed axial
tomography (CAT) ...................................... 410
12.9 Additional reading .................................... 416
12.10 Exercises ............................................ 416
13 Discrete transforms ........................................ 419
13.1 Introduction: the sampling theorem .................... 419
13.2 Sampling and the Poisson sum formula .................. 423
13.3 The discrete Fourier transform ........................ 427
13.4 Properties of the DFT ................................. 430
13.5 Convolution ........................................... 432
13.6 Fast Fourier transform ................................ 433
13.7 The z-transform ....................................... 437
13.8 Focus: z-transforms in the numerical solution of
Maxwell's equations ................................... 445
13.9 Focus: the Talbot effect .............................. 449
13.10 Exercises ............................................ 456
14 Ordinary differential equations ............................ 458
14.1 Introduction: the classic ODEs ........................ 458
14.2 Classification of ODEs ................................ 459
14.3 Ordinary differential equations and phase space ....... 460
14.4 First-order ODEs ...................................... 469
14.5 Second-order ODEs with constant coefficients .......... 474
14.6 The Wronskian and associated strategies ............... 476
14.7 Variation of parameters ............................... 478
14.8 Series solutions ...................................... 480
14.9 Singularities, complex analysis, and general
Frobenius solutions ................................... 481
14.10 Integral transform solutions ......................... 485
14.11 Systems of differential equations .................... 486
14.12 Numerical analysis of differential equations ......... 488
14.13 Additional reading ................................... 501
14.14 Exercises ............................................ 501
15 Partial differential equations ............................. 505
15.1 Introduction: propagation in a rectangular waveguide .. 505
15.2 Classification of second-order linear PDEs ............ 508
15.3 Separation of variables ............................... 517
15.4 Hyperbolic equations .................................. 519
15.5 Elliptic equations .................................... 525
15.6 Parabolic equations ................................... 530
15.7 Solutions by integral transforms ...................... 534
15.8 Inhomogeneous problems and eigenfunction solutions .... 538
15.9 Infinite domains; the d'Alembert solution ............. 539
15.10 Method of images ..................................... 544
15.11 Additional reading ................................... 545
15.12 Exercises ............................................ 545
16 Bessel functions ........................................... 550
16.1 Introduction: propagation in a circular waveguide ..... 550
16.2 Bessel's equation and series solutions ................ 552
16.3 The generating function ............................... 555
16.4 Recurrence relations .................................. 557
16.5 Integral representations .............................. 560
16.6 Hankel functions ...................................... 564
16.7 Modified Bessel functions ............................. 565
16.8 Asymptotic behavior of Bessel functions ............... 566
16.9 Zeros of Bessel functions ............................. 567
16.10 Orthogonality relations .............................. 569
16.11 Bessel functions of fractional order ................. 572
16.12 Addition theorems, sum theorems, and product
relations ............................................ 576
16.13 Focus: nondiffracting beams .......................... 579
16.14 Additional reading ................................... 582
16.15 Exercises ............................................ 582
17 Legendre functions and spherical harmonics ................. 585
17.1 Introduction: Laplace's equation in spherical
coordinates ........................................... 585
17.2 Series solution of the Legendre equation .............. 587
17.3 Generating function ................................... 589
17.4 Recurrence relations .................................. 590
17.5 Integral formulas ..................................... 592
17.6 Orthogonality ......................................... 594
17.7 Associated Legendre functions ......................... 597
17.8 Spherical harmonics ................................... 602
17.9 Spherical harmonic addition theorem ................... 605
17.10 Solution of PDEs in spherical coordinates ............ 608
17.11 Gegenbauer polynomial ................................ 610
17.12 Focus: multipole expansion for static electric
fields ............................................... 611
17.13 Focus: vector spherical harmonics and radiation
fields ............................................... 614
17.14 Exercises ............................................ 618
18 Orthogonal functions ....................................... 622
18.1 Introduction: Sturm-Liouville equations ............... 622
18.2 Hermite polynomials ................................... 627
18.3 Laguerre functions .................................... 641
18.4 Chebyshev polynomials ................................. 650
18.5 Jacobi polynomials .................................... 654
18.6 Focus: Zernike polynomials ............................ 655
18.7 Additional reading .................................... 662
18.8 Exercises ............................................. 662
19 Green's functions .......................................... 665
19.1 Introduction: the Huygens-Fresnel integral ............ 665
19.2 Inhomogeneous Sturm-Liouville equations ............... 669
19.3 Properties of Green's functions ....................... 674
19.4 Green's functions of second-order PDEs ................ 676
19.5 Method of images ...................................... 685
19.6 Modal expansion of Green's functions .................. 689
19.7 Integral equations .................................... 693
19.8 Focus: Rayleigh-Sommerfeld diffraction ................ 701
19.9 Focus: dyadic Green's function for Maxwell's
equations ............................................. 704
19.10 Focus: scattering theory and the Born series ......... 709
19.11 Exercises ............................................ 712
20 The calculus of variations ................................. 715
20.1 Introduction: principle of Fermat ..................... 715
20.2 Extrema of functions and functionals .................. 718
20.3 Euler's equation ...................................... 721
20.4 Second form of Euler's equation ....................... 727
20.5 Calculus of variations with several dependent
variables ............................................. 730
20.6 Calculus of variations with several independent
variables ............................................. 732
20.7 Euler's equation with auxiliary conditions: Lagrange
multipliers ........................................... 734
20.8 Hamiltonian dynamics .................................. 739
20.9 Focus: aperture apodization ........................... 742
20.10 Additional reading ................................... 745
20.11 Exercises ............................................ 745
21 Asymptotic techniques ...................................... 748
21.1 Introduction: foundations of geometrical optics ....... 748
21.2 Definition of an asymptotic series .................... 753
21.3 Asymptotic behavior of integrals ...................... 756
21.4 Method of stationary phase ............................ 763
21.5 Method of steepest descents ........................... 766
21.6 Method of stationary phase for double integrals ....... 771
21.7 Additional reading .................................... 772
21.8 Exercises ............................................. 773
Appendix A The gamma function ................................. 775
A.l Definition ............................................ 775
A.2 Basic properties ...................................... 776
A.3 Stirling's formula .................................... 778
A.4 Beta function ......................................... 779
A.5 Useful integrals ...................................... 780
Appendix В Hypergeometric functions ........................... 783
B.1 Hypergeometric function ............................... 784
B.2 Confluent hypergeometric function ..................... 785
B.3 Integral representations .............................. 785
References .................................................... 787
Index ......................................................... 793
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