The resurgence properties of the Incomplete gamma function II

TL;DR

This paper introduces a new representation for the incomplete gamma function, providing bounds and asymptotic expansions for large parameters, and rigorously proves an exponentially improved asymptotic series.

Contribution

It develops a novel representation based on steepest descents, enabling precise bounds and asymptotic analysis of the incomplete gamma function for large arguments.

Findings

Derived a numerically computable bound for the remainder term

Obtained an asymptotic expansion for late coefficients

Provided a rigorous proof of Dingle's exponential improvement

Abstract

In this paper we derive a new representation 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 this representation, we obtain numerically computable bounds for the remainder term of the asymptotic expansion of the incomplete gamma function $\Gamma \left( { - a,\lambda a} \right)$ with large $a$ and fixed positive $\lambda$, and an asymptotic expansion for its late coefficients. We also give a rigorous proof of Dingle's formal result regarding the exponentially improved version of the asymptotic series of $\Gamma \left( { - a,\lambda a} \right)$.