Efficient Quadrature Rules for Cauchy Principal Value Integrals with Highly Oscillatory Bessel Kernel

Abstract

In this paper, we address the fast computation of highly oscillatory Bessel transforms involving algebraic-type singularities and Cauchy principal value integrals. By modifying the numerical steepest-descent method, we propose a new efficient quadrature rule. We first divide the considered integrals into two parts Iω1[f,s] and Iω2[f,s], which can be transformed into the infinite integrals on [0,∞) by the Cauchy residue theorem. Based on the asymptotic properties of the kernel function, we construct suitable Gaussian quadrature rules to enable efficient evaluation of the resulting infinite integrals. We next develop two new quadrature rules for this transform, applicable regardless of whether the endpoint a is zero or nonzero. Error bounds are established for both methods, and their effectiveness is validated through several numerical experiments.