Handbook of Computational Methods for Integration

During the past 20 years, there has been enormous productivity in theoretical as well as computational integration. Some attempts have been made to find an optimal or best numerical method and related computer code to put to rest the problem of numerical integration, but the research is continuously ongoing, as this problem is still very much open-ended. The importance of numerical integration in so many areas of science and technology has made a practical, up-to-date reference on this subject long overdue. The Handbook of Computational Methods for Integration discusses quadrature rules for finite and infinite range integrals and their applications in differential and integral equations, Fourier integrals and transforms, Hartley transforms, fast Fourier and Hartley transforms, Laplace transforms and wavelets. The practical, applied perspective of this book makes it unique among the many theoretical books on numerical integration and quadrature. It will be a welcomed addition to the libraries of applied mathematicians, scientists, and engineers in virtually every discipline.

Table of Contents:
Preface Notation Preliminaries Notation and Definitions Orthogonal Polynomials Finite and Divided Differences Interpolation Semi-Infinite Interval Convergence Accelerators Polynomial Splines Interpolatory Quadrature Riemann Integration Euler-Maclaurin Expansion Interpolatory Quadrature Rules Newton-Cotes Formulas Basic Quadrature Rules Repeated Quadrature Rules Romberg’s Scheme Gregory’s Correction Scheme Interpolatory Product Integration Iterative and Adaptive Schemes Test Integrals Gaussian Quadrature Gaussian Rules Extended Gaussian Rules Other Extended Rules Analytic Functions Bessel’s Rule Gaussian Rules for the Moments Finite Oscillatory Integrals Noninterpolatory Product Integration Test Integrals Improper Integrals Infinite Range Integrals Improper Integrals Slowly Convergent Integrals Oscillatory Integrals Product Integration Singular Integrals Quadrature Rules Product Integration Acceleration Methods Singular and Hypersingular Integrals Computer-Aided Derivations Fourier Integrals and Transforms Fourier Transforms Interpolatory Rules for Fourier Integrals Interpolatory Rules by Rational Functions Trigonometric Integrals Finite Fourier Transforms Discrete Fourier Transforms Hartley Transform Inversion of Laplace Transforms Use of Orthogonal Polynomials Interpolatory Methods Use of Gaussian Quadrature Rules Use of Fourier Series Use of Bromwich Contours Inversion by the Riemann Sum New Exact Laplace Inverse Transforms Wavelets Orthogonal Systems Trigonometric System Haar System Other Wavelet Systems Daubechies’ System Fast Daubechies Transforms Integral Equations Nyström System Integral Equations of the First Kind Integral Equations of the Second Kind Singular Integral Equations Weakly Singular Equations Cauchy Singular Equations of the First Kind Cauchy Singular Equations of the Second Kind Canonical Equation Finite-Part Singular Equations Integral Equations Over a Contour Appendix A: Quadrature Tables Cotesian Numbers, Tabulated for k£n/2, n=1(1)11 Weights for a Single Trapezoidal Rule and Repeated Simpson’s Rule Weights for Repeated Simpson’s Rule and a Single Trapezoidal Rule Weights for a Single 3/8-Rule and Repeated Simpson’s Rule Weights for Repeated Simpson’s Rule and a Single 3/8-Rule Gauss-Legendre Quadrature Gauss-Laguerre Quadrature Gauss-Hermite Quadrature Gauss-Radau Quadrature Gauss-Lobatto Quadrature Nodes of Equal-Weight Chebyshev Rule Gauss-Log Quadrature Gauss-Kronrod Quadrature Rule Patterson’s Quadrature Rule Filon’s Quadrature Formula Gauss-Cos Quadrature on [π/2, π/2] Gauss-Cos Quadrature on [0, π/2] Coefficients in (5.1.15) with w(x)=ln(1/x), 0