Uniform asymptotics for the incomplete gamma functions starting from negative values of the parameters
Abstract
We consider the asymptotic behavior of the incomplete gamma functions gamma(-a,-z) and Gamma(-a,-z) as a goes to infinity. Uniform expansions are needed to describe the transition area z~a in which case error functions are used as main approximants. We use integral representations of the incomplete gamma functions and derive a uniform equation 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.
- Publication:
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arXiv Mathematics e-prints
- Pub Date:
- March 1996
- DOI:
- 10.48550/arXiv.math/9603218
- arXiv:
- arXiv:math/9603218
- Bibcode:
- 1996math......3218T
- Keywords:
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- Mathematics - Classical Analysis and ODEs