The resurgence properties of the incomplete gamma function I

TL;DR

This paper develops new representations for the incomplete gamma function using steepest descents, leading to improved asymptotic expansions, explicit error bounds, and insights into Stokes phenomena for large arguments.

Contribution

It introduces novel representations and detailed asymptotic properties of the incomplete gamma function, enhancing understanding of its behavior for large arguments.

Findings

Explicit error bounds for asymptotic expansions

Exponentially improved asymptotic expansions

Analysis of Stokes discontinuities and smooth transitions

Abstract

In this paper we derive new representations for the incomplete gamma 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 asymptotic expansions of the incomplete gamma function with large arguments, including explicit and realistic error bounds, asymptotics for the late coefficients, exponentially improved asymptotic expansions, and the smooth transition of the Stokes discontinuities.