An asymptotic for the K-Bessel function using the saddle-point method

TL;DR

This paper derives an asymptotic expression for the K-Bessel function with large complex order and argument using the saddle-point method, providing an elementary proof and a new integral representation involving a single saddle point.

Contribution

It offers an elementary saddle-point method proof of known asymptotics for the K-Bessel function and introduces a novel integral representation with a single saddle point.

Findings

Asymptotic formula for $K_{r+it}(y)$ as $y o \infty$

New integral representation for $K_{r+it}(y)$ with $t= y ext{cosh} \mu$ involving one saddle point

Elementary proof elucidating the origin of the asymptotics

Abstract

Using the saddle-point method, we compute an asymptotic, as $y \rightarrow \infty$, for the $K$-Bessel function $K_{r + i t}(y)$ with positive, real argument $y$ and of large complex order $r+it$ where $r$ is bounded and $t = y \sin \theta$ for a fixed parameter $0\leq \theta\leq \pi/2$ or $t= y \cosh \mu$ for a fixed parameter $\mu>0$. Our method gives an illustrative proof, using elementary tools, of this known result and explains how these asymptotics come about. As part of our proof, we prove a new result, namely a novel integral representation for $K_{r + i t}(y)$ in the case $t= y \cosh \mu$. This integral representation involves only one saddle point.