On the evaluation of infinite integrals involving Bessel functions

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)Cv(ωx) denotes Bessel function Jv(ωx)Jv(ωx) of the first kind, Yv(ωx)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≤1k≤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 ωω is sufficiently large. Furthermore, we give the error which depends on the endpoint “a”. Numerical examples are provided to confirm our results.