Contents
Editors' Foreword, iii
Preface, v
Introduction, 1
1. Dirichlet's Problem And Poisson's Theorem, 6
1-1. The Equation Of Steady State Heat Conduction, 7
1-2. The Solution Of The Differential Equation By Products, 10
1-3. The Fourier Coefficients, 13
1-4. Poisson's Kernel, 14
1-5. Assumption Of Boundary Values: Poisson's Theorem, 16
1-6. Two Simple Consequences Of Poisson's Theorem, 20
1-7. Uniqueness Of The Solution Of The Heat Problem, 21
1-8. The Pointwise Convergence Of Fourier Series, 24
2. The Method Of Separation Of Variables, 29
2-1. Sine And Cosine Series, 29
2-2. The Vibrating String, 33
2-3. Generalities On The Method Of Separation, 38
3. Some Applications Of Poisson's Theorem, 42
3-1. Uniform Approximation, 42
3-2. Least Squares Approximation, 45
3-3. Inner Products And Schwarz's Inequality, 48
4. Fourier Transforms, 57
4-1. Improper Integrals, 58
4-2. The Dirichlet Problem In A Half Plane, 65
4-3. Poisson's Kernel For A Half Plane, 68
4-4. The Maximum Principle For Harmonic Functions, And The Question Of Uniqueness, 72
4-5. Fourier Inversion And The Plancherel Formula, 77
4-6. Fourier Transforms And Derivatives, 84
4-7. Convolution, 87
4-8. The Time-Dependent Heat Equation, 91
Bibliography, 97
Answers To Exercises, 99
Index, 103