On the evaluation of infinite integrals involving Bessel functions

Abstract

In this paper we consider the numerical method for computing the infinite highly oscillatory Bessel integrals of the form ∫a∞f(x)Cv(ωx)dx, where Cv(ωx) denotes Bessel function Jv(ωx) of the first kind, Yv(ωx) of the second kind, Hv(1)(ωx) and Hv(2)(ωx) of the third kind, f is a smooth function on [a,∞),limx→∞f(k)(x)=0(k=0,1,2,…),ω is large and a⩾1ωk with k≤1. We construct the method based on approximating f by a combination of the shifted Chebyshev polynomial so that the generalized moments can be evaluated efficiently by the truncated formula of Whittaker W function. The method is very efficient in obtaining very high precision approximations if ω is sufficiently large. Furthermore, we give the error which depends on the endpoint “a”. Numerical examples are provided to confirm our results.