A uniform asymptotic expansion for the incomplete gamma functions revisited

TL;DR

This paper revisits a uniform asymptotic expansion for the incomplete gamma function, correcting previous errors and providing detailed analysis near the transition point where z equals a.

Contribution

It corrects a misprint in the coefficients of a previously published expansion and offers a detailed discussion of the behavior at the transition point z=a.

Findings

Corrected coefficients for the asymptotic expansion.

Enhanced understanding of the transition behavior at z=a.

Clarified the case when z equals a.

Abstract

A new uniform asymptotic expansion for the incomplete gamma function $\Gamma(a,z)$ valid for large values of $z$ was given by the author in {\it J. Comput. Appl. Math.} {\bf 148} (2002) 323--339. This expansion contains a complementary error function of an argument measuring transition across the point $z=a$, with easily computable coefficients that do not involve a removable singularity in the neighbourhood of this point. In this note we correct a misprint in the listing of certain coefficients in this expansion and discuss in more detail the situation corresponding to $\Gamma(a,a)$.