Efficient Numerical Quadrature for Highly Oscillatory Integrals with Bessel Function Kernels

In this paper, we investigate efficient numerical methods for highly oscillatory integrals with Bessel function kernels over finite and infinite domains. Initially, we decompose the two types of integrals into the sum of two integrals. For one of these integrals, we reformulate the Bessel function $J_{\nu }\left(z\right)$ as a linear combination of the modified Bessel function of the second kind $K_{\nu }\left(z\right)$, subsequently transforming it into a line integral over an infinite interval on the complex plane. This transformation allows for efficient approximation using the Cauchy residue theorem and appropriate Gaussian quadrature rules. For the other integral, we achieve efficient computation by integrating special functions with Gaussian quadrature rules. Furthermore, we conduct an error analysis of the proposed methods and validate their effectiveness through numerical experiments. The proposed methods are applicable for any real number $\nu$ and require only the first $\lfloor \nu \rfloor$ derivatives of f at 0, rendering them more efficient than existing methods that typically necessitate higher-order derivatives.