1 Fourier series
Piecewise Continuous Functions
Fourier Cosine Series 4
Examples 6
Fourier Sine Series 9
Examples 10
Fourier Series 14
3 Partial Differential Equations of Physics
Linear Boundary Value Problems 60
One-Dimensional Heat Equation 62
Related Equations 65
Laplacian in Cylindrical and Spherical Coordinates 67
Derivations 70
Boundary Conditions 73
Duhamel's Principle 75
A Vibrating String 80
Vibrations of Bars and Membranes 83
General Solution of a Wave Equation 87
Types of Equations and Boundary Conditions 92
4 The Fourier method
Linear Operators
Principle of Superposition 97
Examples 99
Eigenvalues and Eigenfunctions 102
A Temperature Problem 104
A Vibrating String Problem 107
Historical Development 111
5 Boundary value Problems
A Slab with Faces at Prescribed Temperatures 114
Related Temperature Problems 118
Temperatures in a Sphere 122
A Slab with Internally Generated Heat 125
Steady Temperatures in Rectangular Coordinates 131
Steady Temperatures in Cylindrical Coordinates 136
A String with Prescribed Initial Conditions 140
Resonance 146
An elastic Bar 149
Double fourier series 152
Periodic Boundary Conditions 155
Examples 16
Adaptations to Other Intervals 20
2 Convergence of Fourier Series
One-Sided Derivatives 25
A Property of Fourier Coefficients 28
Two Lemmas 31
A Fourier Theorem 35
A Related Fourier Theorem 38
Examples 39
Convergence on Other Intervals 43
A Lemma 47
Absolute and Uniform Convergence of Fourier Series 49
The Gibbs phenomenon 51
Differentiation of Fourier Series 54
Integration of Fourier Series 55
6 Fourier Integrals and Applications
The Fourier Integral Formula 161
Dirichlet's Integral 163
Two Lemmas 165
A Fourier Integral Theorem 169
The Cosine and Sine Integrals 173
Some eigenvalue problems on Unbounded Intervals 174
More on Superposition of Solutions 177
Steady Temperatures in a Semi-Infinite Strip 179
Temperatures in a semi-Infinite Solid 182
Temperatures in an Unlimited Medium 187
7 Orthonormal sets
Inner Products and Orthonormal sets 189
Examples 191
Generalized Fourier Series 195
Examples 196
Best Approximation in the Mean 200
Bessel's Inequality and Parseval,s Equation 203
Applications to Fourier Series 205
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems 210
Modifications 212
Orthogonal Eigenfunctions and Real Eigenvalues 213
Real-Valued Eigenfunctions 218
Nonnegative Eigenvalues 220
Methods of solution 221
Examples of Eigenfunction Expansions 227
A Temperature Problem in Rectangular Coordinates 234
Steady Temperatures 238
Other Coordinates 242
A Modification of the Method 245
Another modification 249
A Vertically Hung Elastic Bar 252
9 Bessel Functions and Applications
The Gamma Function 261
Bessel Functions Jv(x) 263
Solutions when v=0.1.2
Recurrence Relations 273
Bessel's Integral Form 277
Some Consequences of the Integral Forms 279
The Zeros of Jn(x) 283
Zeros of related Functions 286
Orthogonal Sets of Bessel Functions 288
Proof of the Theorems 290
Two Lemmas 295
Fourier-Bessel Series 298
Examples 300
Temperatures in a Long Cylinder 305
A Temperature Problem in Shrunken Fittings 312
Internally Generated Heat 314
Temperatures in a Long Cylindrical Wedge 319
Vibration of a Circular Membrane 321
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation 326
Legendre Polynomials 328
Rodrigues' Formula 333
Laplace's Integral Form 336
Some Consequences of the Integral Form 338
Orthogonality of Legendre Polynomials 341
Normalized Legendre Polynomials 343
Legendre Series 345
The Eigenfunctions Pn(cos 8) 350
Dirichlet Problems in Spherical Regions 352
Steady Temperatures in a Hemisphere 356
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence 362
Verification of Solution of Temperature Problem 365
Uniqueness of Solutions of the Heat Equation 368
Verification of Solution of Vibrating String Problem 372
Uniqueness of Solutions of the Wave Equation 374
Appendixes
Bibliography 377
Some Fourier Series Expansions 381
Solutions of Some Regular Sturm-Liouville Problems 383
Some Fourier-Bessel Series Expansions 387
Index
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