Definite integrals involving Bessel functions expressed as a series of special functions

TL;DR

This paper develops a contour integration method to evaluate integrals involving products of Bessel functions, expressing the results as series of hypergeometric and special functions, extending previous work by Landau et al.

Contribution

It introduces a contour integration approach to derive series representations of Bessel function integrals, expanding the analytical tools available for such evaluations.

Findings

Derived new series representations for Bessel function integrals

Extended previous results by Landau et al.

Provided analytical expressions involving hypergeometric functions

Abstract

Series involving hypergeometric functions are used to derive, extend and evaluate integrals involving the product of two Bessel functions of the first kind $J_{u}(a z)$ $J_{v}(b z)$ with order $u,v$, studied by Landau et al. The method used in this work is contour integration.