Chapter 1．THE INITIAL VALUE PROBLEM FOR ORDINARY DIFFERENTIAL EQUATIONS 1
1．Successive Approximations 1
1．Existence and uniqueness of the solution of the ordinary differential equation of the first order 1
2．Remark on approximate solutions 6
3．Integration constants 8
4．Solution by power series expansion 10
5．Differential equations containing parameters．Perturbation theory 14
6．Existence and uniqueness of the solution of a system of differential equations 17
2．Linear Differential Equations of the nth Order 21
7．Singular points for linear differential equations 21
8．Fundamental system of solutions 23
9．Wronskian．Liouville＇s formula 27
10．Lagrange＇s method of variation of constants and D＇Alembert＇s method of reduction of order 29
11．Linear differential equations with constant coefficients 31
3．Second Order Differential Equations of the Fuchs Type 37
12．Regular singular points．Fuchs＇theorem 37
13．Gauss differential equations 45
14．Legendre differential equations 48
15．Bessel differential equations 51
Chapter 2．THE BOUNDARY VALUE PROBLEM FOR LINEAR DIFFERENTIAL EQUATIONS OF SECOND ORDER 61
1．Boundary Value Problem 61
16．Boundary value problem of the Sturm-Liouville type 61
17．Green＇s function．Reduction to integral equations 64
18．Periodic solutions．Generalized Green＇s function 68
2．Hilbert-Schmidt Theory of Integral Equations with Symmetric Kernels 76
19．The Ascoli-Arzelà theorem 76
20．Existence proof for the eigenvalues 80
21．The Bessel inequality．The Hilbert-Schmidt expansion theorem 83
22．Approximations of eigenvalues．Rayleigh＇s principle and the Kryloff-Weinstein theorem 91
23．Inhomogeneous integral equations 96
24．Hermite，Laguerre and Legendre polynomials 100
3．Asymptotic Expression of Eigenvalues and Eigenfunctions，Liouville＇s Method 110
25．The Liouville transformation 110
26．Asymptotic expressions of eigenvalues and eigenfunctions 112
Chapter 3．FREDHOLM INTEGRAL EQUATIONS 115
1．Fredholm Alternative Theorem 115
27．The case when ∫ba∫ba|K(s，t)|2 ds dt<1 115
28．The general case 118
29．Fredholm＇s alternative theorem 125
2．The Schmidt Expansion Theorem and the Mercer Expansion Theorem 127
30．Operator-theoretical notations 127
31．The Schmidt expansion theorem 128
32．Application to Fredholm integral equation of the first kind 131
33．Positive definite kernels．Mercer＇s expansion theorem 132
3．Singular Integral Equations 139
34．Discontinuous kernels 140
35．Examples．Band spectrum 141
Chapter 4．VOLTERRA INTEGRAL EQUATIONS 145
1．Volterra Integral Equations of the Second Kind 145
36．Existence and uniqueness of solutions 145
37．Resolvent kernels 147
38．Application to linear differential equations 149
39．The singular kernel P(s，t)/(s，t)a 151
2．Volterra Integral Equations of the First Kind 153
40．Reduction to integral equations of the second kind 153
41．Abel integral equations 154
Chapter 5．THE GENERAL EXPANSION THEOREM(WEYL-STONE-TITCHMARSH-KODAIRA＇S THEOREM) 159
1．Classification of Singular Boundary Points 160
42．Green＇s formula 160
43．Limit point case and limit circle case 162
44．Definition of m1(λ) and m2(λ) 170
2．The General Expansion Theorem 173
45．Application of the Hilbert-Schmidt expansion theorem 173
46．Helly＇s theorem and Poisson＇s integral formula 177
47．The Weyl-Stone-Titchmarsh-Kodaira theorem 183
48．Density matrix 190
3．Examples 192
49．The Fourier series expansion 192
50．The Fourier integral theorem 194
51．The Hermite function expansion 196
52．The Hankel integral theorem 199
53．The Fourier-Bessel series expansion 202
54．The Laguerre function expansion 205
Chapter 6．NON-LINEAR INTEGRAL EQUATIONS 209
55．Non-linear Volterra integral equations 209
56．Non-linear Fredholm integral equations 210
Appendix．FROM THE THEORY OF FUNCTIONS OF A COMPLEX VARIABLE 213
A theorem on normal family of regular functions(Part 44) 213
Hurwitz＇s theorem(Part 47) 213
The Poisson integral formula(Part 46) 214
BIBLIOGRAPHY 217
INDEX 219