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A uniform asymptotic expansion for the incomplete gamma function. (English) Zbl 1013.33002
A new uniform asymptotic expansion is given for the incomplete gamma function ${\Gamma }\left(a,z\right)$ valid for large $z$. The expansion is compared with one developed by the reviewer [SIAM J. Math. Anal. 10, 757-766 (1979; Zbl 0412.33001)]. The coefficients of the new expansion are simpler to evaluate near the transition point $a=z$, the domain of validity for the complex parameters $a$ and $z$ is smaller. In the new expansion the error function again gives the main approximation, higher terms are related with the error function (and with parabolic cylinder functions). Numerical examples are given to illustrate the accuracy of the expansion.
##### MSC:
 33B20 Incomplete beta and gamma functions