Extended Hadamard expansions for the Airy functions

TL;DR

This paper introduces a new convergent series expansion for the Airy function based on a refined steepest descent method, enabling high-precision computations with fewer terms and providing insights into the Stokes phenomenon.

Contribution

It presents a novel finite-path splitting approach for steepest descent, converting asymptotic series into convergent series, and relates the Stokes phenomenon to path transitions.

Findings

The new series converges for all complex values of z.

Fewer terms are needed for high-precision calculations.

The approach clarifies the Stokes phenomenon in the Airy function.

Abstract

A new series expansion for the the Airy function is presented here that stems from the method of steepest descents and can be related to the Hadamard expansions as presented in prevous works cited in the manuscript, and which is convergent for all values of the complex variable. Hadamard expansions were introduced as an extension of the method of steepest descents and are defined in terms of a large number of non-systematic integration path subdivisions. Unlike them, the expansions in the present work originate in the splitting of the steepest descent in a number of segments that is not only finite but very small, and which are defined on the basis of the location of the branch points. One of the segments reaches to infinity and this gives rise to the presence of upper incomplete Gamma functions. This is one of the most important differences with the Hadamard series as defined in the aforementioned references, where all the incomplete Gamma functions are of the lower type. The theoretical interest of the new series expansion is twofold. First of all, it shows how to convert an asymptotic series into a convergent one with a finite splitting of the steepest descent path. Secondly, the inverse of the phase function that is part of the Laplace-type equation is Taylor-expanded around branch points to produce Puiseux series when necessary. In addition to this, the proposed analysis shows again how the Stokes phenomenon for the Airy function is related to the transition of the steepest descent paths at $\arg z = \pm 2 \pi/3$ from one to two. In regard to its computational application, these series expansions require a relatively small number of terms for each of them to reach a very high precision.