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Error bounds for the uniform asymptotic expansion of the incomplete gamma function. (English) Zbl 1013.65016
Summary: We derive simple, explicit error bounds for the uniform asymptotic expansion of the incomplete gamma function ${\Gamma }\left(a,z\right)$ valid for complex values of $a$ and $z$ as $|a|\to \infty$. Their evaluation depends on numerically pre-computed bounds for the coefficients ${c}_{k}\left(\eta \right)$ in the expansion of ${\Gamma }\left(a,z\right)$ taken along rays in the complex $\eta$ plane, where $\eta$ is a variable related to $z/a$. The bounds are compared with numerical computations of the remainder in the truncated expansion.
##### MSC:
 65D20 Computation of special functions, construction of tables 33F05 Numerical approximation and evaluation of special functions 33B20 Incomplete beta and gamma functions