Gaussian Quadrature of int_0^1 f(x) log^m(x) dx and int_(-1)^1 f(x) cos(pi*x/2) dx

TL;DR

This paper provides tabulated abscissae and weights for Gaussian quadrature methods tailored to integrals with specific singular and symmetric weight functions, facilitating accurate numerical integration.

Contribution

It introduces explicit Gaussian quadrature rules for integrals with singular and symmetric weights, including up to 128 nodes, enhancing computational techniques for these integrals.

Findings

Tabulated abscissae and weights for integrals with (-log x)^m weights.

Tabulated abscissae and weights for integrals with cos(pi*x/2) weight.

Explicit quadrature rules up to 128 nodes for practical use.

Abstract

We tabulate the abscissae and associated weights for numerical integration of integrals with either the singular weight function (-log x)^m for exponents m=1, 2 or 3, or the symmetric weight function cos(pi*x/2). Standard brute force arithmetics generates explicit pairs of these values for up to 128 nodes.