## Stirling's Series

The Asymptotic Series for the Gamma Function is given by

 (1)

(Sloane's A001163 and A001164). The series for 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 analytic expression

 (3) (4)

where is a Bernoulli Number.

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. Sequences A001163/M5400 and A001164/M4878 in 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.