A method to generate generalized quadrature rule for oscillatory integrals

Abstract In a recent paper Evans and Webster produced a quadrature rule for oscillatory integrals with a trigonometric kernel. Their method is a modification of an earlier method due to Levin, and involves making a quadrature rule exact for a set of functions for which the modified moments have a simple closed form. In this paper we extend their method to deal with any oscillatory kernel, provided only that it satisfies a (known) linear ordinary differential equation. Some specific examples involving Bessel functions and Bessel–Trig combinations are given to illustrate the practical use of the method.