Uniform Asymptotics for the Incomplete Gamma Functions Starting from NegativeValues of the Parameters

摘要

We consider the asymptotic behavior of the incomplete gamma function gamma (-a,-z) and gamma (-a,-z) as approaches infinity. Uniform expansions are needed to describe the transition area z a, in which case error functions are used as mained 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) 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.