Integrals of products of four modified Bessel functions

TL;DR

This paper derives general formulas for integrals involving four modified Bessel functions multiplied by a power function, expressed through advanced special functions, and explores special cases with simpler or elementary solutions, including new Airy function integrals.

Contribution

It introduces comprehensive formulas for complex integrals of four Bessel functions using Meijer G and hypergeometric functions, and identifies special cases with elementary solutions.

Findings

Formulas in terms of Meijer G and hypergeometric functions

Special cases with elementary solutions

New integrals involving four Airy functions

Abstract

We evaluate definite integrals involving the product of four modified Bessel functions of the first and second kind and a power function. We provide general formulas expressed in terms of the Meijer $G$-function and generalized hypergeometric and Lauricella $F_C$ functions, and study a number of special cases in which the integrals can be evaluated in terms of simpler special functions or indeed take an elementary form. As a consequence, we deduce some new formulas for definite integrals of products of four Airy functions.