We consider the asymptotic behavior of the incomplete gamma functions $gamma (-a,-z)$ and $Gamma (-a,-z)$ as $atoinfty$. Uniform expansions are needed to describe the transition area $z sim a$, in which case error functions are used as main approximants. We use integral representations of the incomplete gamma functions and derive a uniform expansion by applying techniques used for the existing uniform expansions for $gamma (a,z)$ and $Gamma (a,z)$. The result is compared with Olver's uniform expansion for the generalized exponential integral. A numerical verification of the expansion is given in a final section.

Incomplete beta and gamma functions (error functions, probability integral, Fresnel integrals) (msc 33B20), Asymptotic approximations, asymptotic expansions (steepest descent, etc.) (msc 41A60)
CWI
Department of Analysis, Algebra and Geometry [AM]
Computational Dynamics

Temme, N.M. (1996). Uniform asymptotics for the incomplete gamma functions starting from negative values of the parameters. Department of Analysis, Algebra and Geometry [AM]. CWI.