The resurgence properties of the large order asymptotics of the Hankel and Bessel functions

TL;DR

This paper develops new representations for Hankel and Bessel functions to analyze their large order asymptotics, providing explicit error bounds, improved expansions, and insights into Stokes phenomena.

Contribution

It introduces novel representations based on Berry and Howls' method, enhancing understanding of large order asymptotics and Stokes transitions for these functions.

Findings

Explicit error bounds for asymptotic expansions

Exponentially improved asymptotic formulas

Analysis of Stokes discontinuities and transitions

Abstract

The aim of this paper is to derive new representations for the Hankel and Bessel functions, exploiting the reformulation of the method of steepest descents by M. V. Berry and C. J. Howls (Berry and Howls, Proc. R. Soc. Lond. A 434 (1991) 657--675). Using these representations, we obtain a number of properties of the large order asymptotic expansions of the Hankel and Bessel functions due to Debye, including explicit and realistic error bounds, asymptotics for the late coefficients, exponentially improved asymptotic expansions, and the smooth transition of the Stokes discontinuities.