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).
Similar content being viewed by others
References
E.T. Copson, Asymptotic Expansions, Cambridge University Press, 1965.
E. Whittaker and G. Watson, A Course of Modern Analysis, Cambridge University Press, 1963, 251–253.
M. Abramowitz and I. A. Stegun (eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematical Series, 55, 9th printing, Dover, New York, 1972, 804.
P. Luschny, An overview and comparison of different approximations of the factorial function, http://www.luschny.de/math/factorial/approx/SimpleCases.html, 2007.
S. Ragahavan and S. S. Rangachari (eds.), S. Ramanujan: The lost notebook and other unpublished papers, Narosa Publ. House, Springer, New Delhi, Berlin, 1988.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Nemes, G. New asymptotic expansion for the Gamma function. Arch. Math. 95, 161–169 (2010). https://doi.org/10.1007/s00013-010-0146-9
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00013-010-0146-9