Abstract: Using a series transformation, Stirling-De Moivre asymptotic series approximation to the Gamma function is converted into a new one with better convergence properties. The new formula is 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 10).
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Note: this article is an extended version of an older one to which it adds the estimate of the remainder.
Some cited references with links:
- - Copson E.T., Asymptotic Expansions, in Cambridge Tracts in Mathematics, Vol.55,
Paperback Edition, Cambridge University Press 2004. 1st Edition 1965. ISBN 0-521-60482-6.
- - Whittaker E.T., Watson G.N., A Course of Modern Analysis,
Cambridge University Press 1996. ISBN 0-521-58807-3.
- - Abramowitz M., Stegun I.A., Editors, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables,
2nd Edition, Dover Publications 1972, ISBN 0-486-61272-4.
- - Luschny P., An overview and comparison of different approximations of the factorial