Advanced Mathematics for Applications

Cover......Page 1
Half-title......Page 3
Title......Page 5
Dedication......Page 7
Contents......Page 9
Preface......Page 15
To the readers......Page 17
Tables......Page 19
Part 0: General Remarks and Basic Concepts......Page 21
1.1 Vector fields......Page 23
1.2.1 Boundary conditions......Page 26
1.2.2 Elliptic equations......Page 27
1.2.3 Hyperbolic equations......Page 28
1.2.4 Parabolic equations......Page 29
1.3 Diffusion......Page 30
1.4 Fluid mechanics......Page 32
1.5 Electromagnetism......Page 34
1.6 Linear elasticity......Page 36
1.7 Quantum mechanics......Page 38
1.8.1 Two dimensions......Page 39
1.8.2 N dimensions......Page 41
1.8.3 Functions and derivatives......Page 44
2.1 The method of eigenfunction expansions......Page 47
2.2 Variation of parameters......Page 50
2.3 The “δ-function”......Page 55
2.4 The idea of Green\'s functions......Page 59
2.5 Power series solution of ordinary differential equations......Page 62
Part I: Applications......Page 71
3.1 Summary of useful relations......Page 73
3.2.1 Homogeneous boundary conditions......Page 78
3.2.2 Other homogeneous boundary conditions......Page 81
3.2.3 Non-homogeneous boundary conditions......Page 82
3.2.4 A general case......Page 85
3.2.5 Mixed boundary conditions......Page 86
3.3 Waves on a string......Page 87
3.4 Poisson equation in a square......Page 88
3.5 Helmholtz equation in a semi-infinite strip......Page 92
3.6 Laplace equation in a disk......Page 93
3.7 The Poisson equation in a sector......Page 97
3.8 The quantum particle in a box: eigenvalues of the Laplacian......Page 98
3.9.1 Freely supported beam......Page 100
3.9.2 Cantilevered beam......Page 101
3.10 A comment on the method of separation of variables......Page 102
3.11 Other applications......Page 103
4.1 Useful formulae for the exponential transform......Page 104
4.1.3 Asymptotic behavior......Page 108
4.2 One-dimensional Helmholtz equation......Page 110
4.3 Schrödinger equation with a constant force......Page 113
4.4 Diffusion in an infinite medium......Page 115
4.4.1 One space dimension......Page 116
4.4.2 Three space dimensions......Page 118
4.4.3 Two space dimensions: Method of descent......Page 119
4.5.1 One space dimension......Page 120
4.5.2 Three space dimensions......Page 122
4.5.3 Two space dimensions......Page 124
4.5.4 The non-homogeneous problem......Page 125
4.6 Laplace equation in a strip and in a half-plane......Page 127
4.7 Example of an ill-posed problem......Page 128
4.8 The Hilbert transform and dispersion relations......Page 130
4.9 Fredholm integral equation of the convolution type......Page 132
4.10 Useful formulae for the sine and cosine transforms......Page 133
4.11 Diffusion in a semi-infinite medium......Page 136
4.12 Laplace equation in a quadrant......Page 138
4.13 Laplace equation in a semi-infinite strip......Page 140
4.14 One-sided transform......Page 141
4.15 Other applications......Page 142
5.1 Summary of useful relations......Page 143
5.1.1 Operational rules......Page 145
5.1.2 Small and large t......Page 147
5.2 Ordinary differential equations......Page 148
5.2.1 Homogeneous equation with constant coefficients......Page 149
5.2.2 Non-homogeneous equations with constant coefficients......Page 150
5.2.3 Non-constant coefficients......Page 152
5.3 Difference equations......Page 153
5.4 Differential-difference equations......Page 156
5.5.1 Infinite medium......Page 158
5.5.2 Finite interval......Page 159
5.5.3 Semi-in.nite medium, mixed boundary condition......Page 160
5.5.4 Duhamel’s theorem......Page 161
5.6 Integral equations......Page 164
5.7 Other applications......Page 165
6.1 Cylindrical coordinates......Page 166
6.2 Summary of useful relations......Page 167
6.2.1 Bessel functions......Page 169
6.2.2 Inhomogeneous Bessel equation......Page 172
6.2.3 Fourier–Bessel series......Page 173
6.3 Laplace equation in a cylinder......Page 174
6.4 Fundamental solution of the Poisson equation......Page 177
6.5 Transient diffusion in a cylinder......Page 179
6.6 Formulae for the Hankel transform......Page 182
6.7 Laplace equation in a half-space......Page 183
6.8 Axisymmetric waves on a liquid surface......Page 186
6.9 Dual integral equations......Page 188
7 Spherical Systems......Page 190
7.1 Spherical polar coordinates......Page 191
7.2 Spherical harmonics: useful relations......Page 193
7.2.1 The axisymmetric case......Page 196
7.3 General solution of the Laplace and Poisson equations......Page 198
7.4 A sphere in a uniform field......Page 199
7.5 Half-sphere on a plane......Page 201
7.6 The Poisson equation in free space......Page 202
7.7 General solution of the biharmonic equation......Page 203
7.8 Exterior Poisson formula for the Dirichlet problem......Page 204
7.9 Point source near a sphere......Page 205
7.10 A domain perturbation problem......Page 207
7.11 Conical boundaries......Page 211
7.12 Spherical Bessel functions: useful relations......Page 214
7.13 Fundamental solution of the Helmholtz equation......Page 216
7.14 Scattering of scalar waves......Page 220
7.15 Expansion of a plane vector wave in vector harmonics......Page 223
7.16 Scattering of electromagnetic waves......Page 225
7.17 The elastic sphere......Page 227
7.18 Toroidal--poloidal decomposition and viscous flow......Page 228
7.18.1 Sphere in a time-dependent viscous flow......Page 230
Part III: Essential Tools......Page 233
8.1 Numerical sequences and series......Page 235
8.1.1 Numerical series......Page 236
8.1.2 Absolute convergence......Page 237
8.2 Sequences of functions......Page 239
8.2.1 Uniform convergence......Page 240
8.2.2 Convergence almost everywhere......Page 242
8.2.3 Other modes of convergence......Page 243
8.3 Series of functions......Page 244
8.4 Power series......Page 248
8.5 Other definitions of the sum of a series......Page 250
8.6 Double series......Page 252
8.7 Practical summation methods......Page 254
9.1 The Fourier exponential basis functions......Page 262
9.2 Fourier series in exponential form......Page 264
9.3 Point-wise convergence of the Fourier series......Page 265
9.4 Uniform and absolute convergence......Page 268
9.5 Behavior of the coefficients......Page 269
9.6 Trigonometric form......Page 272
9.7 Sine and cosine series......Page 273
9.8 Term-by-term integration and differentiation......Page 276
9.9 Change of scale......Page 278
9.10 The Gibbs phenomenon......Page 281
9.11 Other modes of convergence......Page 283
9.12 The conjugate series......Page 285
10.1 Heuristic motivation......Page 286
10.2 The exponential Fourier transform......Page 288
10.3 Operational formulae......Page 290
10.4 Uncertainty relation......Page 291
10.5 Sine and cosine transforms......Page 292
10.6 One-sided and complex Fourier transform......Page 294
10.7 Integral asymptotics......Page 296
10.8 Asymptotic behavior of the sine and cosine transforms......Page 301
10.9 The Hankel transform......Page 302
11.1 Direct and inverse Laplace transform......Page 305
11.2 Operational formulae......Page 307
11.3 Inversion of the Laplace transform......Page 311
11.4.1 Small t......Page 315
11.4.2 Large t......Page 317
11.4.3 Tauberian theorems......Page 320
12.1 Introduction......Page 322
12.2 The Bessel functions......Page 324
12.2.2 Recurrence relations......Page 327
12.2.4 Asymptotic behavior......Page 328
12.3 Spherical Bessel functions......Page 329
12.4 Modified Bessel functions......Page 330
12.5 The Fourier–Bessel and Dini series......Page 332
12.5.2 Eigenfunction expansion......Page 333
12.6 Other series expansions......Page 335
13.1 Introduction......Page 337
13.2 The Legendre equation......Page 338
13.3 Legendre polynomials......Page 339
13.3.1 Integral representations......Page 340
13.3.2 Generating function......Page 341
13.3.3 Recurrence relations......Page 342
13.3.4 Orthogonality......Page 343
13.4 Expansion in series of Legendre polynomials......Page 344
13.5 Legendre functions......Page 346
13.6 Associated Legendre functions......Page 348
13.7 Conical boundaries......Page 350
13.8 Extensions......Page 352
13.9 Orthogonal polynomials......Page 354
14.1 Introduction......Page 357
14.2 Spherical harmonics......Page 358
14.2.1 Solid harmonics......Page 361
14.2.2 Addition theorem......Page 362
14.3 Expansion in series of spherical harmonics......Page 363
14.4 Vector harmonics......Page 365
14.4.1 Solenoidal and irrotational vector fields......Page 366
15.1 Two-point boundary value problems......Page 368
15.1.1 Solution for the Green’s function......Page 370
15.1.2 Separated boundary conditions......Page 372
15.1.3 Properties of the Green’s function for separated boundary conditions......Page 375
15.1.4 Mixed boundary conditions......Page 377
15.2 The regular eigenvalue Sturm–Liouville problem......Page 379
15.2.1 Properties of Sturm–Liouville eigenvalues and eigenfunctions......Page 381
15.2.2 Mixed boundary conditions......Page 384
15.3.1 The Hilbert–Schmidt theorem......Page 386
15.3.2 Direct proof of the expansion theorem......Page 387
15.3.3 Remarks......Page 392
15.3.4 Mixed boundary conditions......Page 394
15.4 Singular Sturm–Liouville problems......Page 395
15.4.1 Limit-point case......Page 399
15.4.2 Limit-circle case......Page 402
15.4.3 “Eigenfunction expansion” with a continuous spectrum......Page 406
15.5 Initial-value problem for ordinary differential equations......Page 408
15.6 A broader perspective on Green\'s functions......Page 412
15.6.1 Symmetry properties of Green’s functions......Page 414
15.7 Modified Green\'s function......Page 416
16.1 Poisson equation......Page 420
16.1.1 Dirichlet problem......Page 423
16.1.2 Neumann problem......Page 425
16.1.3 The boundary integral method......Page 427
16.1.4 Eigenvalue problem......Page 429
16.2 The diffusion equation......Page 430
16.2.1 Dirichlet problem......Page 432
16.2.2 Neumann problem......Page 434
16.3 Wave equation......Page 435
17.1 Complex algebra......Page 438
17.2 Analytic functions......Page 441
17.3 Integral of an analytic function......Page 443
17.3.1 Cauchy’s theorem......Page 445
17.3.2 Cauchy’s integral formula......Page 448
17.3.3 Further consequences......Page 451
17.4 The Taylor and Laurent series......Page 452
17.4.1 The Laurent series and singular points......Page 454
17.5 Analytic continuation I......Page 456
17.5.1 The Gamma function......Page 462
17.6 Multi-valued functions......Page 466
17.6.1 Branch cuts......Page 467
17.7 Riemann surfaces......Page 473
17.8 Analytic continuation II......Page 475
17.8.1 Functional relations in the complex domain......Page 476
17.8.2 Continuation across a boundary......Page 478
17.9 Residues and applications......Page 479
17.9.1 Jordan’s lemma. Fourier and Laplace integrals......Page 481
17.9.3 Multivalued functions......Page 485
17.9.4 Summation of series......Page 487
17.9.5 The logarithmic residue......Page 489
17.10 Conformal mapping......Page 491
17.10.1 The Schwarz–Christoffel transformation......Page 496
17.10.2 Multiply connected domains......Page 502
17.10.3 The Laplace equation......Page 503
17.10.4 Orthogonal coordinates......Page 507
18.1 Basic definitions and properties......Page 511
18.2 Determinants......Page 516
18.3 Matrices and linear operators......Page 519
18.4 Change of basis......Page 523
18.5 Scalar product......Page 525
18.6 Eigenvalues and eigenvectors......Page 529
18.7 Simple, normal and Hermitian matrices......Page 531
18.8 Spectrum and singular values......Page 535
18.9 Projections......Page 539
18.10 Defective matrices......Page 542
18.11 Functions of matrices......Page 548
18.12 Systems of ordinary differential equations......Page 550
Part IV: Some Advanced Tools......Page 555
19.1 Linear vector spaces......Page 557
19.2 Normed spaces......Page 561
19.2.1 Convergence in a normed space......Page 564
19.3 Hilbert spaces......Page 568
19.3.1 Weak convergence......Page 572
19.4 Orthogonality......Page 574
19.4.1 Gram–Schmidt orthogonalization procedure......Page 575
19.4.2 Best Approximation......Page 576
19.4.3 Orthonormal basis......Page 578
19.5 Sobolev spaces......Page 580
19.6 Linear functionals......Page 583
20.1 Introduction......Page 588
20.2 Test functions and distributions......Page 589
20.3 Examples......Page 592
20.4 Operations on distributions......Page 595
20.5 The distributional derivative......Page 600
20.6 Sequences of distributions......Page 604
20.7 Multi-dimensional distributions......Page 608
20.8 Distributions and analytic functions......Page 611
20.9 Convolution......Page 613
20.10 The Fourier transform of distributions......Page 615
20.11 The Fourier series of distributions......Page 620
20.12 Laplace transform of distributions......Page 625
20.13.1 Approximate evaluation of integrals......Page 628
20.13.2 Algebraic equations......Page 629
20.13.3 Generalized solutions of differential equations......Page 632
20.13.4 Convolution and fundamental solutions......Page 633
20.14 The Hilbert transform......Page 636
21 Linear Operators in Infinite-Dimensional Spaces......Page 640
21.1 Linear operators......Page 641
21.2 Bounded operators......Page 643
21.2.1 The adjoint of a bounded operator......Page 648
21.2.2 Self-adjoint bounded operators......Page 650
21.2.3 Contraction mapping......Page 652
21.2.4 The Neumann series......Page 653
21.3 Compact operators......Page 656
21.3.1 Self-adjoint operators with a compact resolvent......Page 662
21.3.2 Non-self-adjoint compact operators......Page 663
21.4 Unitary operators......Page 664
21.4.1 Bochner’s theorem......Page 666
21.5 The inverse of an operator......Page 667
21.6 Closed operators and their adjoint......Page 669
21.6.1 Adjoint of a closed operator......Page 672
21.7 Solvability conditions and the Fredholm alternative......Page 677
21.8 Invariant subspaces and reduction......Page 680
21.9 Resolvent and spectrum......Page 681
21.9.1 Bounded operators......Page 686
21.9.2 Unbounded closed operators......Page 688
21.10 Analytic properties of the resolvent......Page 689
21.11 Spectral theorems......Page 691
21.11.1 Continuous spectrum......Page 693
A.1 Sets......Page 698
A.2 Measure......Page 701
A.3 Functions......Page 702
A.4 Integration......Page 706
A.4.1 The Riemann integral......Page 707
A.4.2 The Riemann–Stieltjes integral......Page 708
A.4.3 The Lebesgue integral......Page 709
A.4.5 Differentiation under the integral sign......Page 711
A.4.6 Sequences......Page 713
A.4.7 Series......Page 714
A.5 Curves......Page 716
A.6 Bounds and limits......Page 717
References......Page 719
Index......Page 727