Uniform asymptotics for the incomplete gamma functions starting from negative values of the parameters
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)|
|Department of Analysis, Algebra and Geometry [AM]|
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.