Efficient calculation and asymptotic expansions of many different oscillatory infinite integrals

Abstract

This paper introduces and analyzes quadrature rules and asymptotic expansions of a few highly oscillatory infinite integrals. We first derive a series of useful asymptotic expansions in inverse powers of the frequency parameter ω, which clarify the large ω behavior of these integrals. Then, based on the resulting asymptotic expansions, two different interpolatory quadrature rules are given. One is the so-called Filon-type methods based on standard Hermite interpolation of the non-oscillatory and non-singular part of the integrands at equidistant nodes. The other is the Filon–Clenshaw–Curtis-type method (FCC) by using special Hermite interpolation at N+1 Clenshaw–Curtis points and the fast computation of modified moments. The interpolation coefficients needed in the FCC method, can be computed by a numerically stable algorithm in O(Nlog N) operations based on fast Fourier transform (FFT). The required modified moments, can be accurately and efficiently calculated by some recurrence relation formulae. Moreover, for these quadrature rules, their error analyses in inverse powers of the frequency ω, are provided. The presented methods share the advantageous property that the accuracy improves greatly, for fixed N, as ω increases. Numerical examples show the accuracy and efficiency of the proposed methods.