Numerical approximations to integrals with a highly oscillatory Bessel kernel

In this paper, we consider a new numerical method for computing highly oscillatory Bessel transforms. We begin our analysis by using the integral form of Bessel function and its analytic continuation. Then we transform the integrals into the forms on [0,+∞) that the integrand does not oscillate and decays exponentially fast, which can be efficiently computed by using Gauss–Laguerre quadrature rule. Moreover, we derive corresponding error bounds in terms of the frequency r and the point number n. Numerical examples based on theoretical results are presented to demonstrate the efficiency and accuracy of the proposed method.