1 Introduction -- 1 1 Notation and Definitions -- 1.2 Classification -- 1.3 Function Spaces -- 1.4 Convergence -- 1.5 Inverse Operator -- 1.6 Nyström System -- 1.7 Other Types of Kernels -- 1.8 Neumann Series -- 1.9 Resolvent Operator -- 1.10 Fredholm Alternative -- 2 Eigenvalue Problems -- 2.1 Linear Symmetric Equations -- 2.2 Residual Methods -- 2.3 Degenerate Kernels -- 2.4 Replacement by a Degenerate Kernel -- 2.5 Baterman's Method -- 2.6 Generallized Eigenvalue Problem -- 2.7 Applications -- 3 Equations of the Second Kind -- 3.1 Fredholm Equations -- 3.2 Volterra Equations -- 4 Classical Methods for FK2 -- 4.1 Expansion Method -- 4.2 Product-Integration Method -- 4.3 Quadrature Method -- 4.4 Deferred Correction Methods -- 4.5 A Modified Quadrature Method -- 4.6 Collocation Methods -- 4.7 Elliott's Modification -- 5 Variational Methods -- 5.1 Galerkin Method -- 5.2 Ritz-Galerkin Methods -- 5.3 Special Cases -- 5.4 Fredholm-Nyström System -- 6 Iteration Methods -- 6.1 Simple Iterations -- 6.2 Quadrature Formulas -- 6.3 Error Analysis -- 6.4 Iterative Scheme -- 6.5 Krylov-Bogoliubov Method -- 7 Singular Equations -- 7.1 Singularities in Linear Equations -- 7.2 Fredholm Theorems -- 7.3 Modified Quadrature Rule -- 7.4 Convolution-Type Kernels -- 7.5 Volterra-Type Singular Equations -- 7.6 Convolution Methods -- 7.7 Asymptotic Methods for Log-Singular Equations -- 7.8 Iteration Methods -- 7.9 Singular Equations with the Hilbert Kernel -- 7.10 Finite-Part Singular Equations -- 8 Weakly Singular Equations -- 8.1 Weakly Singular Kernel -- 8.2 Taylor's Series Method -- 8.3 Lp-Approximation Method -- 8.4 Product-Integration Method -- 8.5 Splines Method -- 8.6 Weakly Singular Volterra Equations -- 9 Cauchy Singular Equations -- 9.1 Cauchy Singular Equations of the First Kind -- 9.2 Approximation by Trigonometric Polynomials -- 9.3 Cauchy Singular Equations of the Second Kind -- 9.4 From CSK2 to FK2 -- 9.5 Gauss-Jacobi Quadrature -- 9.6 Collocation Method for CSK1 -- 10 Sinc-Galerkin Methods -- 10.1 Sine Function Approximations -- 10.2 Conformal Maps and Interpolation -- 10.3 Approximation Theory -- 10.4 Convergence -- 10.5 Sinc-Galerkin Scheme -- 10.6 Computation Guidelines -- 10.7 Sine-Collocation Method -- 10.8 Single-Layer Potential -- 10.9 Double-Layer Problem -- 11 Equations of the First Kind -- 11.1 Inherent Ill-Posedness -- 11.2 Separable Kernels -- 11.3 Some Theorems -- 11.4 Numerical Methods -- 11.5 Volterra Equations of the First Kind -- 11.6 Abel's Equation -- 11.7 Iterative Schemes -- 12 Inversion of Laplace Transforms -- 12.1 Laplace Transforms -- 12.2 General Interpolating Scheme -- 12.3 Inversion by Fourier Series -- 12.4 Inversion by the Riemann Sum -- 12.5 Approximate Formulas -- A Quadrature Rules -- A. 1 Newton-Cotes Quadratures -- A.2 Gaussian Quadratures -- A.3 Integration of Products -- A.4 Singular Integrals -- A.5 Infinite-Range Integrals -- A. 6 Linear Transformation of Quadratures -- A.7 Trigonometric Polynomials -- A.8 Condition Number -- A.7 Quadrature Tables -- B Orthogonal Polynomials -- B.l Zeros of Some Orthogonal Polynomials -- C Whittaker's Cardinal Function -- C. 1 Basic Results -- C.2 Approximation of an Integral -- D Singular Integrals -- D.l Cauchy's Principal-Value Integrals -- D.2 P.V. of a Singular Integral on a Contour -- D.3 Hadamard's Finite-Part Integrals -- D.4 Two-Sided Finite-Part Integrals -- D.5 One-Sided Finite-Part Integrals -- D.6 Examples of Cauchy P.V. Integrals -- D.7 Examples of Hadamard's Finite-Part Integrals.