The asymptotic expansion of Kratzel's integral and an integral related to an extension of the Whittaker function

TL;DR

This paper derives asymptotic expansions for Krätzels integral and an extended Whittaker function-related integral using steepest descents and Mellin-Barnes methods, including special cases and numerical validation.

Contribution

It provides new asymptotic formulas for Krätzels integral and an extended Whittaker function, with alternative derivations and analysis of special cases.

Findings

Asymptotic expansion for Krätzels integral as |x|→∞

Alternative Mellin-Barnes derivation of the expansion

Numerical validation of the asymptotic formulas

Abstract

We consider the asymptotic expansion of Kr\"atzel's integral \[F_{p,\nu}(x)=\int_0^\infty t^{\nu-1} e^{-t^p-x/t}\,dt\qquad (|\arg\,x|<\pi/2),\] for $p>0$ as $|x|\to \infty$ in the sector $|\arg\,x|<\pi/2$ employing the method of steepest descents. An alternative derivation of this expansion is given using a Mellin-Barnes integral approach. The cases $p<0$, $\Re (\nu)<0$ and when $x$ and $\nu$ ($p>0$) are both large are also considered. A second section discusses the asymptotic expansion of an integral involving a modified Bessel function that has recently been introduced as an extension of the Whittaker function $M_{\kappa,\mu}(z)$. Numerical examples are provided to illustrate the accuracy of the various expansions obtained.