Archiv der Mathematik

, Volume 95, Issue 2, pp 161–169 | Cite as

New asymptotic expansion for the Gamma function

Article

Abstract

Using a series transformation, the Stirling-De Moivre asymptotic series approximation to the Gamma function is converted into a new one with better convergence properties. The new formula is being compared with those of Stirling, Laplace, and Ramanujan for real arguments greater than 0.5 and turns out to be, for equal number of “correction” terms, numerically superior to all of them. As a side benefit, a closed-form approximation has turned up during the analysis which is about as good as 3rd order Stirling’s (maximum relative error smaller than 1e − 10 for real arguments greater or equal to 24).

Mathematics Subject Classification (2000)

33B15 41A60 

Keywords

Asymptotic approximations Asymptotic expansions Gamma function Stirling’s formula 

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Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  1. 1.Loránd Eötvös UniversityBudapestHungary

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