A note on a modified Bessel function integral

TL;DR

This paper derives closed-form expressions for a class of integrals involving the modified Bessel function, specifically for cases where parameters are non-negative integers of different parity, expanding analytical tools for such integrals.

Contribution

It provides new closed-form solutions for integrals of hyperbolic cosine powers multiplied by modified Bessel functions with integer parameters of different parity.

Findings

Closed-form expressions involving exponential and polynomial terms.

Results depend on the parity difference of the parameters.

Enhances analytical methods for Bessel-related integrals.

Abstract

We investigate the integral \[\int_0^\infty \cosh^\mu\!t\,K_\nu(z\cosh t)\,dt \qquad \Re(z)>0,\] where $K$ denotes the modified Bessel function, for non-negative integer values of the parameters $\mu$ and $\nu$. When the integers are of different parity, closed-form expressions are obtained in terms of $z^{-1}e^{-z}$ multiplied by a polynomial in $z^{-1}$ of degree dependent on the sign of $\mu-\nu$.