Preface ...................................................... xiii
To the reader .................................................. xv
List of tables ............................................... xvii
Part 0 General Remarks and Basic Concepts ..................... l
1 The Classical Field Equations ................................ 3
1.1 Vector fields ........................................... 3
1.2 The fundamental equations ............................... 6
1.3 Diffusion .............................................. 10
1.4 Fluid mechanics ........................................ 12
1.5 Electromagnetism ....................................... 14
1.6 Linear elasticity ...................................... 16
1.7 Quantum mechanics ...................................... 18
1.8 Eigenfunction expansions: introduction ................. 19
2 Some Simple Preliminaries ................................... 27
2.1 The method of eigenfunction expansions ................. 27
2.2 Variation of parameters ................................ 30
2.3 The "S-function" ....................................... 35
2.4 The idea of Green's functions .......................... 39
2.5 Power series solution .................................. 42
Part I Applications ........................................... 51
3 Fourier Series: Applications ................................ 53
3.1 Summary of useful relations ............................ 53
3.2 The diffusion equation ................................. 58
3.3 Waves on a string ...................................... 67
3.4 Poisson equation in a square ........................... 68
3.5 Helmholtz equation in a semi-infinite strip ............ 72
3.6 Laplace equation in a disk ............................. 73
3.7 The Poisson equation in a sector ....................... 77
3.8 The quantum particle in a box: eigenvalues of the
Laplacian .............................................. 78
3.9 Elastic vibrations ..................................... 80
3.10 A comment on the method of separation of variables ..... 82
3.11 Other applications ..................................... 83
4 Fourier Transform: Applications ............................. 84
4.1 Useful formulae for the exponential transform .......... 84
4.2 One-dimensional Helmholtz equation ..................... 90
4.3 Schrцdinger equation with a constant force ............. 93
4.4 Diffusion in an infinite medium ........................ 95
4.5 Wave equation ......................................... 100
4.6 Laplace equation in a strip and in a half-plane ....... 107
4.7 Example of an ill-posed problem ....................... 108
4.8 The Hilbert transform and dispersion relations ........ 110
4.9 Fredholm integral equation of the convolution type .... 112
4.10 Useful formulae for the sine and cosine transforms .... 113
4.11 Diffusion in a semi-infinite medium ................... 116
4.12 Laplace equation in a quadrant ........................ 118
4.13 Laplace equation in a semi-infinite strip ............. 120
4.14 One-sided transform ................................... 121
4.15 Other applications .................................... 122
5 Laplace Transform: Applications ............................ 123
5.1 Summary of useful relations ........................... 123
5.2 Ordinary differential equations ....................... 128
5.3 Difference equations .................................. 133
5.4 Differential-difference equations ..................... 136
5.5 Diffusion equation .................................... 138
5.6 Integral equations .................................... 144
5.7 Other applications .................................... 145
6 Cylindrical Systems ........................................ 146
6.1 Cylindrical coordinates ............................... 146
6.2 Summary of useful relations ........................... 147
6.3 Laplace equation in a cylinder ........................ 154
6.4 Fundamental solution of the Poisson equation .......... 157
6.5 Transient diffusion in a cylinder ..................... 159
6.6 Formulae for the Hankel transform ..................... 162
6.7 Laplace equation in a half-space ...................... 163
6.8 Axisymmetric waves on a liquid surface ................ 166
6.9 Dual integral equations ............................... 168
7 Spherical Systems .......................................... 170
7.1 Spherical polar coordinates ........................... 171
7.2 Spherical harmonics: useful relations ................. 173
7.3 A sphere in a uniform field ........................... 178
7.4 Half-sphere on a plane ................................ 179
7.5 General solution of the Laplace and Poisson
equations ............................................. 181
7.6 The Poisson equation in free space .................... 182
7.7 General solution of the biharmonic equation ........... 183
7.8 Exterior Poisson formula for the Dirichlet problem .... 184
7.9 Point source near a sphere ............................ 185
7.10 A domain perturbation problem ......................... 187
7.11 Conical boundaries .................................... 191
7.12 Spherical Bessel functions: useful relations .......... 194
7.13 Fundamental solution of the Helmholtz equation ........ 196
7.14 Scattering of scalar waves ............................ 200
7.15 Expansion of a plane vector wave ...................... 203
7.16 Scattering of electromagnetic waves ................... 205
7.17 The elastic sphere .................................... 207
7.18 Toroidal-poloidal decomposition and viscous flow ...... 208
Part II Essential Tools ...................................... 213
8 Sequences and Series ....................................... 215
8.1 Numerical sequences and series ........................ 215
8.2 Sequences of functions ................................ 219
8.3 Series of functions ................................... 224
8.4 Power series .......................................... 228
8.5 Other definitions of the sum of a series .............. 230
8.6 Double series ......................................... 232
8.7 Practical summation methods ........................... 234
9 Fourier Series: Theory ..................................... 242
9.1 The Fourier exponential basis functions ............... 242
9.2 Fourier series in exponential form .................... 244
9.3 Point-wise convergence of the Fourier series .......... 245
9.4 Uniform and absolute convergence ...................... 248
9.5 Behavior of the coefficients .......................... 249
9.6 Trigonometric form .................................... 252
9.7 Sine and cosine series ................................ 253
9.8 Term-by-term integration and differentiation .......... 256
9.9 Change of scale ....................................... 258
9.10 The Gibbs phenomenon .................................. 261
9.11 Other modes of convergence ............................ 263
9.12 The conjugate series .................................. 265
10 The Fourier and Hankel Transforms .......................... 266
10.1 Heuristic motivation .................................. 266
10.2 The exponential Fourier transform ..................... 268
10.3 Operational formulae .................................. 270
10.4 Uncertainty relation .................................. 271
10.5 Sine and cosine transforms ............................ 272
10.6 One-sided and complex Fourier transform ............... 274
10.7 Integral asymptotics .................................. 276
10.8 Asymptotic behavior ................................... 281
10.9 The Hankel transform .................................. 282
11 The Laplace Transform ...................................... 285
11.1 Direct and inverse Laplace transform .................. 285
11.2 Operational formulae .................................. 287
11.3 Inversion of the Laplace transform .................... 291
11.4 Behavior for small and large .......................... 295
12 The Bessel Equation ........................................ 302
12.1 Introduction .......................................... 302
12.2 The Bessel functions .................................. 304
12.3 Spherical Bessel functions ............................ 309
12.4 Modified Bessel functions ............................. 310
12.5 The Fourier-Bessel and Dini series .................... 312
12.6 Other series expansions ............................... 315
13 The Legendre Equation ...................................... 317
13.1 Introduction .......................................... 317
13.2 The Legendre equation ................................. 318
13.3 Legendre polynomials .................................. 319
13.4 Expansion in series of Legendre polynomials ........... 324
13.5 Legendre functions .................................... 326
13.6 Associated Legendre functions ......................... 328
13.7 Conical boundaries .................................... 330
13.8 Extensions ............................................ 332
13.9 Orthogonal polynomials ................................ 334
14 Spherical Harmonics ........................................ 337
14.1 Introduction .......................................... 337
14.2 Spherical harmonics ................................... 338
14.3 Expansion in series of spherical harmonics ............ 343
14.4 Vector harmonics ...................................... 345
15 Green's Functions: Ordinary Differential Equations ......... 348
15.1 Two-point boundary value problems ..................... 348
15.2 The regular eigenvalue Sturm-Liouville problem ........ 359
15.3 The eigenfunction expansion ........................... 366
15.4 Singular Sturm-Liouville problems ..................... 375
15.5 Initial-value problem ................................. 388
15.6 A broader perspective on Green's functions ............ 392
15.7 Modified Green's function ............................. 396
16 Green's Functions: Partial Differential Equations .......... 400
16.1 Poisson equation ...................................... 400
16.2 The diffusion equation ................................ 410
16.3 Wave equation ......................................... 415
17 Analytic Functions ......................................... 418
17.1 Complex algebra ....................................... 418
17.2 Analytic functions .................................... 421
17.3 Integral of an analytic function ...................... 423
17.4 The Taylor and Laurent series ......................... 432
17.5 Analytic continuation I ............................... 436
17.6 Multi-valued functions ................................ 446
17.7 Riemann surfaces ...................................... 453
17.8 Analytic continuation II .............................. 455
17.9 Residues and applications ............................. 459
17.10 Conformal mapping .................................... 471
18 Matrices and Finite-Dimensional Linear Spaces .............. 491
18.1 Basic definitions and properties ...................... 491
18.2 Determinants .......................................... 496
18.3 Matrices and linear operators ......................... 499
18.4 Change of basis ....................................... 503
18.5 Scalar product ........................................ 505
18.6 Eigenvalues and eigenvectors .......................... 509
18.7 Simple, normal and Hermitian matrices ................. 511
18.8 Spectrum and singular values .......................... 515
18.9 Projections ........................................... 519
18.10 Defective matrices ................................... 522
18.11 Functions of matrices ................................ 528
18.12 Systems of ordinary differential equations ........... 530
Part III Some Advanced Tools ................................. 535
19 Infinite-Dimensional Spaces ................................ 537
19.1 Linear vector spaces .................................. 537
19.2 Normed spaces ......................................... 541
19.3 Hilbert spaces ........................................ 548
19.4 Orthogonality ......................................... 554
19.5 Sobolev spaces ........................................ 560
19.6 Linear functional ..................................... 563
20 Theory of Distributions .................................... 568
20.1 Introduction .......................................... 568
20.2 Test functions and distributions ...................... 569
20.3 Examples .............................................. 572
20.4 Operations on distributions ........................... 575
20.5 The distributional derivative ......................... 580
20.6 Sequences of distributions ............................ 584
20.7 Multi-dimensional distributions ....................... 588
20.8 Distributions and analytic functions .................. 591
20.9 Convolution ........................................... 593
20.10 The Fourier transform of distributions ............... 595
20.11 The Fourier series of distributions .................. 600
20.12 Laplace transform of distributions ................... 605
20.13 Miscellaneous applications ........................... 608
20.14 The Hilbert transform ................................ 616
21 Linear Operators in Infinite-Dimensional Spaces ............ 620
21.1 Linear operators ...................................... 621
21.2 Bounded operators ..................................... 623
21.3 Compact operators ..................................... 636
21.4 Unitary operators ..................................... 644
21.5 The inverse of an operator ............................ 647
21.6 Closed operators and their adjoint .................... 649
21.7 Solvability conditions and the Fredholm alternative ... 657
21.8 Invariant subspaces and reduction ..................... 660
21.9 Resolvent and spectrum ................................ 661
21.10 Analytic properties of the resolvent ................. 669
21.11 Spectral theorems .................................... 671
Appendix ...................................................... 678
A.l Sets .................................................. 678
A.2 Measure ............................................... 681
A.3 Functions ............................................. 682
A.4 Integration ........................................... 686
A.5 Curves ................................................ 696
A.6 Bounds and limits ..................................... 697
References .................................................... 699
Index ......................................................... 707
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