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

Published online by Cambridge University Press:  03 June 2015

Gergő Nemes
Gergő Nemes*
Affiliation:
Department of Mathematics and Its Applications, Central European University, Nádor utca 9, 1051 Budapest, Hungary, ([email protected])
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Abstract

In 1994 Boyd derived a resurgence representation for the gamma function, exploiting the 1991 reformulation of the method of steepest descents by Berry and Howls. Using this representation, he was able to derive a number of properties of the asymptotic expansion for the gamma function, including explicit and realistic error bounds, the smooth transition of the Stokes discontinuities and asymptotics for the late coefficients. The main aim of this paper is to modify Boyd’s resurgence formula, making it suitable for deriving better error estimates for the asymptotic expansions of the gamma function and its reciprocal. We also prove the exponentially improved versions of these expansions complete with error terms. Finally, we provide new (formal) asymptotic expansions for the coefficients appearing in the asymptotic series and compare their numerical efficacy with the results of earlier authors.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 145 , Issue 3 , June 2015 , pp. 571 - 596
Copyright
Copyright © Royal Society of Edinburgh 2015 

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