Stirling's Series
The Asymptotic Series for the Gamma Function is given by
 | (1) |
is obtained by adding an additional factor of 
, |  |  | |
|  | (2) |
The expansion of
is what is usually called Stirling's series. It is given by the simple analyticexpression | |  | (3) |
|  | (4) |
where
is a Bernoulli Number.See also Bernoulli Number, K-Function, Stirling's Approximation
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 257, 1972. Arfken, G. ``Stirling's Series.'' §10.3 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 555-559, 1985. Conway, J. H. and Guy, R. K. ``Stirling's Formula.'' In The Book of Numbers. New York: Springer-Verlag, pp. 260-261, 1996. Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 443, 1953. Sloane, N. J. A. SequencesA001163/M5400and A001164/M4878in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.
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