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Asymptotics of zeros of incomplete gamma functions. (English) Zbl 0827.33001
The author is studying the complex zeros with respect to $z$ of the incomplete gamma functions $\gamma \left(a,z\right)$ and ${\Gamma }\left(a,z\right)$ with $a$ real and positive, moreover in the case when $a$ is large. The zeros are obtained from approximations that are computed by using uniform asymptotic expansions of the incomplete gamma functions. The complex zeros of the complementary error function are used as a first approximation. Applications are discussed for the zeros of the partial sums ${s}_{n}\left(z\right)={\sum }_{j=0}^{n}{z}^{j}/j!$ of $exp\left(z\right)$.
##### MSC:
 33B15 Gamma, beta and polygamma functions 33B20 Incomplete beta and gamma functions 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.) 65C99 Probabilistic methods, simulation and stochastic differential equations (numerical analysis)