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Incomplete gamma functions for large values of their variables. (English) Zbl 1076.33001
The authors derive simple asymptotic expansions of the incomplete gamma functions ${\Gamma }\left(a,z\right)$ and $\gamma \left(a,z\right)$ for large $a$ and $z$. They write ${\Gamma }\left(a,z\right)$ or $\gamma \left(a,z\right)$ 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 $\left(a,z\right)\in C×\left(C-{R}^{-}\right)$ depending on the value of $a-z$. Three of them valid away from the transition point $a=z$ in the regions (i) $R\left(a\right)<0$, (ii) $R\left(a\right)>-1$, $R\left(a\right)>R\left(z\right)$ and (iii) $R\left(a\right)>-1$, $R\left(a\right) correspondingly and the fourth one valids around the transition point $a=z$.
##### MSC:
 33B20 Incomplete beta and gamma functions 33F05 Numerical approximation and evaluation of special functions 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
##### Keywords:
incomplete gamma functions; asymptotic expansions