Efficient computation of oscillatory Bessel transforms with a singularity of Cauchy type

Abstract

In this article, we present and analyze efficient methods for computing the Cauchy principal value integral of the oscillatory Bessel function ∫abf(x)x−cJm(ωx)dx, where 0≤a<c<b. For the two cases of a>0 and a=0, we establish new steepest descent integration paths, and employ the complex line integration technique to obtain affordable quadrature rules. According to the relation formula between the Meijer G-function and the Bessel function, a crucial explicit expression of the required integral ∫0+∞xjx−cJm(ωx)dx can be derived in terms of the Meijer G-function. Furthermore, for the proposed quadrature formulae, we perform rigorous error analysis in inverse powers of ω. Particularly, compared with a combination method given in the work (Kang et al., 2022), the proposed method takes less time and produces more accurate results. We verify error analysis of the proposed method by experimental results. Numerical examples are presented to illustrate the accuracy and efficiency of the proposed numerical methods.