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.