Chapter I GENERAL INTRODUCTION 1 §1 Fredholm and Volterra equations 1 1.1 Fredholm equations 1 1.2 Equations with a weak singularity 6 1.3 Volterra equations 6 §2 Other classes of integral equations 8 / 2.1 Equations with convolution kernels 8 * 2.2 The Wiener-Hopf equations 9 2.3 Dual equations 9 2.4 Integral transforms 10 * 2.5 Singular integral equations 11 2.6 Non-linear integral equations 12 §3 Some inversion formulas 14 3.1 The inversion of integral transforms 15 3.2 Inversion formulas for equations with convolution kernels 17 * 3.3 Volterra's equations with one independent variable and a convolution kernel 20 3.4 The Abel equation 20 3.5 Integral equations with kernels denned by hyper-geometric functions 22 Chapter II THE FREDHOLM THEORY 26 §1 Basic concepts and the Fredholm theorems 26 1.1 Basic concepts 26 1.2 The basic theorems 31 §2 The solution of Fredholm equations: The method of successive approximation 33 Contents 2.1 The construction of approximations: The Neumann series 33 2.2 The resolvent kernel 35 §3 The solution of Fredholm equations: Degenerate equations and the general case 37 3.1 Equations with degenerate kernels 37 3.2 The general case 40 §4 The Fredholm resolvent 43 4.1 The Fredholm resolvent 43 4.2 Properties of the resolvent 44 §5 The solution of Fredholm equations: The Fredholm series 44 5.1 The Fredholm series. Fredholm determinants and minors 45 5.2 The representation of the eigenfunctions of a kernel in terms of the minors of Fredholm 47 §6 Equations with a weak singularity 48 6.1 Boundedness of the integral operator with a weak singularity 48 6.2 Iteration of a kernel with a weak singularity 49 6.3 The method of successive approximation 49 §7 Systems of integral equations 50 7.1 The vector form for systems of integral equations 50 7.2 Methods of solution for Fredholm kernels 50 7.3 Methods of solution for kernels with a weak singularity 51 §8 The structure of the resolvent in the neighbourhood of a characteristic value 51 8.1 Orthogonal kernels 51 8.2 The principal kernels 52 8.3 The canonical kernels 53 §9 The rate of growth of eigenvalues 54 Chapter III SYMMETRIC EQUATIONS 57 §1 Basic properties 57 1.1 Symmetric kernels 57 1.2 Basic theorems connected with symmetric kernels 58 1.3 Systems of characteristic values and eigenfunctions 58 1.4 Orthogonalization 58 §2 The Hilbert-Schmidt series and its properties 59 2.1 Hilbert-Schmidt theorem 59 2.2 The solution of symmetric integral equations 60 vi Contents …