CHAPTER Ⅰ INTRODUCTION 1
1.The Two Related Problems 1
2.Linear Differential Equations 2
3.Infinite Series of Solutions 5
4.Boundary Value Problems 6
CHAPTER Ⅱ PARTIAL DIFFERENTIAL EQUATIONS OF PHYSICS 10
5.Gravitational Potential 10
6.Laplace’s Equation 12
7.Cylindrical and Spherical Coordinates 13
8.The Flux of Heat 15
9.The Heat Equation 17
10.Other Cases of the Heat Equation 19
11.The Equation of the Vibrating String 21
12.Other Equations.Types 23
13.A Problem in Vibrations of a String 24
14.Example.The Plucked String 28
15.The Fourier Sine Series 29
16.Imaginary Exponential Functions 31
CHAPTER Ⅲ ORTHOGONAL SETS OF FUNCTIONS 34
17.Inner Product of Two Vectors.Orthogonality 34
18.Orthonormal Sets of Vectors 35
19.Functions as Vectors.Orthogonality 37
20.Generalized Fourier Series 39
21.Approximation in the Mean 40
22.Closed and Complete Systems 42
23.Other Types of Orthogonality 44
24.Orthogonal Functions Generated by Differential Equations 46
25.Orthogonality of the Characteristic Functions 49
CHAPTER Ⅳ FOURIER SERIES 53
26.Definition 53
27.Periodicity of the Function.Example 55
28.Fourier Sine Series.Cosine Series 57
29.Illustration 59
30.Other Forms of Fourier Series 61
31.Sectionally Continuous Functions 64
32.Preliminary Theory 67
33.A Fourier Theorem 70
34.Diacussion of the Theorem 72
35.The Orthonormal Trigonometric Functions 74
CHAPTER Ⅴ FURTHER PROPERTIES OF FOURIER SERIES;FOURIER INTEGRALS 78
36.Differentiation of Fourier Series 78
37.Integration of Fourier Series 80
38.Uniform Convergence 82
39.Concerning More General Conditions 85
40.The Fourier Integral 88
41.Other Forms of the Fourier Integral 91
CHAPTER Ⅵ SOLUTION OF BOUNDARY VALUE PROBLEMS BY THE USE OF FOURIER SERIES AND INTEGRALS 94
42.Formal and Rigorous Solutions 94
43.The Vibrating String 95
44.Variations of the Problem 98
45.Temperatures in a Slab with Faces at Temperature Zero 102
46.The Above Solution Established.Uniqueness 105
47.Variations of the Problem of Temperatures in a Slab 108
48.Temperatures in a Sphere 112
49.Steady Temperatures in a Rectangular Plate 114
50.Displacements in a Membrane.Fourier Series in Two Variables 116
51.Temperatures in an Infinite Bar.Application of Fourier Integrals 120
52.Temperatures in a Semi-infinite Bar 122
53.Further Applications of the Series and Integrals 123
CHAPTER Ⅶ UNIQUENESS OF SOLUTIONS 127
54.Introduction 127
55.Abel’s Test for Uniform Convergence of Series 127
56.Uniqueness Theorems for Temperature Problems 130
57.Example 133
58.Uniqueaess of the Potential Function 134
59.An Application 137
CHAPTER Ⅷ BESSEL FUNCTIONS AND APPLICATIONS 143
60.Derivation of the Functions Jn(x) 143
61.The Functions of Integral Orders 145
62.Differentiation and Recursion Formulas 148
63.Integral Forms of Jn(x) 149
64.The Zeros of Jn(x) 153
65.The Orthogonality of Bessel Functions 157
66.The Ortbonormal Functions 161
67.Fourier-Bessel Expansions of Functions 162
68.Temperatures in an Infinite Cylinder 165
69.Radiation at the Surface of the Cylinder 168
70.The Vibration of a Circular Membrane 170
CHAPTER Ⅸ LEGENDRE POLYNOMIALS AND APPLICATIONS 175
71.Derivation of the Legendre Polynomials 175
72.Other Legendre Functions 177
73.Generating Functions for Pn(x) 179
74.The Legendre Coefficients 181
75.The Orthogonality of Pn(x).Norms 183
76.The Functions Pn(x) as a Complete Orthogonal Set 185
77.The Expansion of xm 187
78.Derivatives of the Polynomials 189
79.An Expansion Theorem 191
80.The Potential about a Spherical Surface 193
81.The Gravitational Potential Due to a Circular Plate 198
INDEX 203