The resurgence properties of the large order asymptotics of the Anger--Weber function I

TL;DR

This paper develops new asymptotic representations for the Anger--Weber function using advanced steepest descent methods, providing detailed properties, error bounds, and insights into Stokes phenomena for large order expansions.

Contribution

It introduces novel asymptotic representations and detailed analysis of the large order behavior of the Anger--Weber function, including error bounds and Stokes transition insights.

Findings

Explicit error bounds for asymptotic expansions

Asymptotics for late coefficients of the series

Exponentially improved asymptotic expansions and Stokes transition analysis

Abstract

The aim of this paper is to derive new representations for the Anger--Weber function, exploiting the reformulation of the method of steepest descents by C. J. Howls (Howls, Proc. R. Soc. Lond. A 439 (1992) 373--396). Using these representations, we obtain a number of properties of the large order asymptotic expansions of the Anger--Weber function, including explicit and realistic error bounds, asymptotics for the late coefficients, exponentially improved asymptotic expansions, and the smooth transition of the Stokes discontinuities.