Asymptotics of some integrals involving modified Bessel and hyper-Bessel functions

TL;DR

This paper studies the asymptotic behavior of integrals involving modified Bessel and hyper-Bessel functions for large parameters, extending previous work on classical Bessel integrals and providing numerical validation.

Contribution

It derives the leading asymptotic expansions of integrals with modified Bessel and hyper-Bessel functions, including new results for hyper-Bessel functions over finite intervals.

Findings

Asymptotic formulas for integrals with modified Bessel functions for large n

Leading behavior of hyper-Bessel functions at infinity

Numerical examples confirming the accuracy of asymptotic expansions

Abstract

We investigate the asymptotic expansion of integrals analogous to Ball's integral \[\int_0^\infty \left(\frac{\Gamma(1+\nu)|J_\nu(x)|}{(x/2)^\nu}\right)^{\!n}dx\] for large $n$ in which the Bessel function $J_\nu(x)$ is replaced by the modified Bessel functions $I_\nu(x)$ and $K_\nu(x)$ together with appropriate exponential factors $e^{\mp x}$, respectively. The above integral with $J_\nu(x)$ replaced by a hyper-Bessel function of the type recently discussed in Aktas {\it et al.} [The Ramanujan J., 2019] and taken over a finite interval determined by the first positive zero of the function is also considered for $n\to\infty$. We give the leading asymptotic behaviour of the hyper-Bessel function for $x\to+\infty$ in an appendix. Numerical examples are given to illustrate the accuracy of the various expansions obtained.