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Zbl 0921.33001
Dunster, T.M.; Paris, R.B.; Cang, S.
On the high-order coefficients in the uniform asymptotic expansion for the incomplete gamma function. (English)
[J] Methods Appl. Anal. 5, No.3, 223-247 (1998). ISSN 1073-2772

The authors study coefficients $c_k(\eta)$ in the asymptotic expansion of the normalized incomplete gamma function $Q(a,z)\equiv \Gamma(a,z)/\Gamma(a)$ as $a\rightarrow\infty$, as given by {\it N. M. Temme} [SIAM J. Math. Anal. 10, 757-766 (1979; Zbl 0412.33001) and ``Special functions'' (1996; Zbl 0856.33001)]: $$ Q(a,z)\sim {1\over 2}\text{erfc} \Biggl( \eta\sqrt{{1\over 2}a}\Biggr) + {e^{-a\eta^2/2}\over \sqrt{2\pi a}}\sum_{k=0}^{\infty} c_k(\eta)a^{-k} $$ where $$ \eta=\{2(\mu-\log{(1+\mu)}\}^{1/2},\qquad \mu=\lambda-1,\ \lambda={z\over a}. $$ \par First the asymptotic behavior of $c_k(\eta)$ as $k\rightarrow\infty$ is given (showing a different behavior on the left and right $\eta$ half plane) using the saddle point method on the two integrals appearing in an explicit expression for these coefficients. The asymptotic behavior is also derived using the MacLaurin expansion of the $c_k$'s. \par Finally some numerical results are discussed, showing a.o. the possibility to use the coefficients $c_k(\eta)$ for optimal truncation, needed to depict the accuracy obtained by the asymptotics for $Q(a,z)$ for fixed $| a| $ and $| z| $. \par A typical example of hard analysis.
[M.G.de Bruin (Delft)]

MSC 2000:: *33B20 Incomplete beta and gamma functions
30E15 Asymptotic representations in the complex domain

Keywords: incomplete gamma function; asymptotic expansion; resurgence

Citations: Zbl 0412.33001; Zbl 0856.33001

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