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The resurgence properties of the incomplete gamma function. II. (English) Zbl 1326.33004
Summary: In this paper, we derive a new representation for the incomplete gamma function, exploiting the reformulation of the method of steepest descents by C. J. Howls [Proc. R. Soc. Lond., Ser. A 439, No. 1906, 373–396 (1992; Zbl 0773.30040)]. Using this representation, we obtain numerically computable bounds for the remainder term of the asymptotic expansion of the incomplete gamma function $$\Gamma(-a,\lambda a)$$ with large $$a$$ and fixed positive $$\lambda$$, and an asymptotic expansion for its late coefficients. We also give a rigorous proof of Dingle’s formal result regarding the exponentially improved version of the asymptotic series of $$\Gamma(-a,\lambda a)$$.
For Part I see [the author, “The resurgence properties of the incomplete gamma function. I”, Preprint, arXiv:1408.0674].

##### MSC:
 33B20 Incomplete beta and gamma functions (error functions, probability integral, Fresnel integrals)
DLMF
Full Text:
##### References:
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