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Gamma function asymptotics by an extension of the method of steepest descents. (English) Zbl 0829.33001
The paper deals with the usual asymptotic expansion of the $$\Gamma$$- function, $\Gamma(z)= \sqrt {2\pi} \exp \bigl\{ (z- {\textstyle {1\over 2}}) \log z-z \bigr\} \Biggl\{ \sum_{k=0}^{n-1} {\textstyle {a_k \over z^k}}+ R_N (z) \Biggr\} \qquad \text{as }z\to \infty \quad \text{in } |\arg (z)|<\pi.$ Based on the recent work of M. V. Berry and C. J. Howls [Proc. R. Soc., Lond., Ser. A 434, No. 1892, 657-675 (1991; Zbl 0764.30031)] the author derives integral formulas for the coefficients $$a_k$$ and the remainder term $$R_N (z)$$. As a result he obtains explicit and good bounds for the remainder terms of the Stokes line $$\text{Re} (z)= {\pi\over 2}$$, the transition formula connecting the different asymptotic relations and the asymptotic behaviour of the coefficients $$a_k$$ as $$k\to \infty$$.

MSC:
 33B15 Gamma, beta and polygamma functions 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
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