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On the resurgence properties of the uniform asymptotic expansion of the incomplete gamma function. (English) Zbl 0928.34039
Summary: The author examines the resurgence properties of the coefficients $$c_r(\eta)$$ appearing in a uniform asymptotic expansion of the incomplete gamma function. For the coefficients $$c_r(\eta)$$, he gives an asymptotic approximation as $$r\to\infty$$ that is a sum of two incomplete beta functions plus a simple asymptotic series in which the coefficients are again $$c_m(\eta)$$.
The method is based on the Borel-Laplace transform, which means that next to the asymptotic approximation of $$c_r(\eta)$$, one obtains an exponentially-improved asymptotic expansion for the incomplete gamma function.

##### MSC:
 34E05 Asymptotic expansions of solutions to ordinary differential equations 33B20 Incomplete beta and gamma functions (error functions, probability integral, Fresnel integrals)
##### Keywords:
incomplete gamma function; Borel-Laplace transform
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