Fourier analysis and its applications

Contents......Page 7
1 Overture......Page 9
1.1 Some equations of mathematical physics......Page 10
1.2 Linear differential operators......Page 16
1.3 Separation of variables......Page 20
2.1 The Fourier series of a periodic function......Page 26
2.2 A convergence theorem......Page 39
2.3 Derivatives, integrals, and uniform convergence......Page 46
2.4 Fourier series on intervals......Page 51
2.5 Some applications......Page 56
2.6 Further remarks on Fourier series......Page 65
3.1 Vectors and inner products......Page 70
3.2 Functions and inner products......Page 76
3.3 Convergence and completeness......Page 80
3.4 More about L2 spaces; the dominated convergence theorem......Page 89
3.5 Regular Sturm-Liouville problems......Page 94
3.6 Singular Sturm-Liouville problems......Page 103
4 Some Boundary Value Problems......Page 105
4.1 Some useful techniques......Page 106
4.2 One-dimensional heat flow......Page 109
4.3 One-dimensional wave motion......Page 116
4.4 The Dirichlet problem......Page 122
4.5 Multiple Fourier series and applications......Page 129
5 Bessel Functions......Page 135
5.1 Solutions of Bessel\'s equation......Page 136
5.2 Bessel function identities......Page 141
5.3 Asymptotics and zeros of Bessel functions......Page 146
5.4 Orthogonal sets of Bessel functions......Page 151
5.5 Applications of Bessel functions......Page 157
5.6 Variants of Bessel functions......Page 166
6.1 Introduction......Page 172
6.2 Legendre polynomials......Page 174
6.3 Spherical coordinates and Legendre functions......Page 182
6.4 Hermite polynomials......Page 192
6.5 Laguerre polynomials......Page 198
6.6 Other orthogonal bases......Page 204
7 The Fourier Transform......Page 212
7.1 Convolutions......Page 214
7.2 The Fourier transform......Page 221
7.3 Some applications......Page 233
7.4 Fourier transforms and Sturm-Liouville problems......Page 244
7.5 Multivariable convolutions and Fourier transforms......Page 249
7.6 Transforms related to the Fourier transform......Page 257
8.1 The Laplace transform......Page 264
8.2 The inversion formula......Page 274
8.3 Applications: Ordinary differential equations......Page 281
8.4 Applications: Partial differential equations......Page 287
8.5 Applications: Integral equations......Page 294
8.6 Asymptotics of Laplace transforms......Page 300
9 Generalized Functions......Page 311
9.1 Distributions......Page 312
9.2 Convergence, convolution, and approximation......Page 322
9.3 More examples: Periodic distributions and finite parts......Page 328
9.4 Tempered distributions and Fourier transforms......Page 338
9.5 Weak solutions of differential equations......Page 349
10 Green\'s Functions......Page 357
10.1 Green\'s functions for ordinary differential operators......Page 358
10.2 Green\'s functions for partial differential operators......Page 366
10.3 Green\'s functions and regular Sturm-Liouville problems......Page 377
10.4 Green\'s functions and singular Sturm-Liouville problems......Page 387
1 Some physical derivations......Page 394
2 Summary of complex variable theory......Page 400
3 The gamma function......Page 407
4 Calculations in polar coordinates......Page 412
5 The fundamental theorem of ordinary differential equations......Page 417
Answers to the Exercises......Page 421
References......Page 434
Index of Symbols......Page 437
Index......Page 438