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Zbl 1076.33001
Ferreira, Chelo; López, José L.; Sinus{\'\i}a, Ester Pérez
Incomplete gamma functions for large values of their variables. (English)
[J] Adv. Appl. Math. 34, No. 3, 467-485 (2005). ISSN 0196-8858

The authors derive simple asymptotic expansions of the incomplete gamma functions $\Gamma(a,z)$ and $\gamma(a,z)$ for large $a$ and $z$. They write $\Gamma(a,z)$ or $\gamma(a,z)$ as an exponential factor times another factor. This second factor is expanded at the asymptotically relevant point of the exponential factor (saddle point or end point). They require four different expansions to cover the region $(a,z)\in C\times (C-R^{-})$ depending on the value of $a-z$. Three of them valid away from the transition point $a=z$ in the regions (i) $R(a)<0$, (ii) $R(a)>-1$, $R(a)>R(z)$ and (iii) $R(a)>-1$, $R(a)<R(z)$ correspondingly and the fourth one valids around the transition point $a=z$.
[Chrysoula G. Kokologiannaki (Patras)]

MSC 2000:: *33B20 Incomplete beta and gamma functions
33F05 Numerical approximation of special functions
41A60 Asymptotic problems in approximation

Keywords: incomplete gamma functions; asymptotic expansions

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