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