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

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.