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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 Γ- function,

Γ(z)=2πexp( z - 1 2 ) log z - z k=0 n-1 a k z k + R N (z)aszin|arg(z)|<π·

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 Re(z)=π 2, the transition formula connecting the different asymptotic relations and the asymptotic behaviour of the coefficients a k as k.


MSC:
33B15Gamma, beta and polygamma functions
41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)