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In (Boyd, Proc. R. Soc. Lond. A 447 (1994) 609--630), W. G. C. Boyd derived aresurgence representation for the gamma function, exploiting the reformulationof the method of steepest descents by M. Berry and C. Howls (Berry and Howls,Proc. R. Soc. Lond. A 434 (1991) 657--675). Using this representation, he wasable to derive a number of properties of the asymptotic expansion for the gammafunction, including explicit and realistic error bounds, the smooth transitionof the Stokes discontinuities, and asymptotics for the late coefficients. Themain aim of this paper is to modify the resurgence formula of Boyd making itsuitable for deriving better error estimates for the asymptotic expansions ofthe gamma function and its reciprocal. We also prove the exponentially improvedversions of these expansions complete with error terms. Finally, we provide new(formal) asymptotic expansions for the coefficients appearing in the asymptoticseries and compare their numerical efficacy with the results of earlierauthors.

Error bounds and exponential improvements for the asymptotic expansions of the gamma function and its reciprocal