A method for efficient computation of integrals with oscillatory and singular integrand

Abstract

A method based on modification of numerical steepest descent method to efficiently compute highly oscillatory integrals having endpoint singularities of algebraic and logarithmic type is proposed in this paper. The three-term recursion coefficients for orthogonal polynomials with respect to Gautschi’s weight function wG(t;s)=ts(t−1−logt)e−t\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$w^{G}(t;s)=t^{s}(t-1-\\log t){\\mathrm {e}}^{-t}$\\end{document} (s > −\u20091) on (0,∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(0,\\infty )$\\end{document}, as well as the corresponding quadrature formulas of Gaussian type, are used in this method. Finally, in order to illustrate the efficiency of the presented method a few numerical examples are included. The obtained results show that the proposed method is very efficient and economical in terms of computation time.