The resurgence properties of the Hankel and Bessel functions of nearly equal order and argument

TL;DR

This paper develops new asymptotic representations for Hankel and Bessel functions of nearly equal order and argument, providing explicit error bounds, improved expansions, and insights into Stokes phenomena.

Contribution

It introduces novel asymptotic representations based on Berry and Howls' method, enhancing understanding of these functions' behaviors and transition properties.

Findings

Derived explicit error bounds for asymptotic expansions

Established exponentially improved asymptotic formulas

Analyzed the smooth transition of Stokes discontinuities

Abstract

The aim of this paper is to derive new representations for the Hankel functions, the Bessel functions and their derivatives, 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 asymptotic expansions of the Hankel and Bessel functions and their derivatives of nearly equal order and argument, including explicit and numerically computable error bounds, asymptotics for the late coefficients, exponentially improved asymptotic expansions, and the smooth transition of the Stokes discontinuities.