Stirling's Series

The Asymptotic Series for the Gamma Function is given by

$$\Gamma(z)=e^{-z}z^{z-1/2}\sqrt{2\pi}\left({1+{1\over 12z}+{1... ...z^2}-{139\over 51840 z^3}-{571\over 2488320z^4}+\ldots}\right)$$ (1)

(Sloane's A001163 and A001164). The series for $z!$ is obtained by adding an additional factor of $z$,

$$z! = \Gamma(z+1)$$

$$= e^{-z}z^{z+1/2}\sqrt{2\pi}\left({1+{1\over 12z}+{1\over 288 z^2}-{139\over 51840 z^3}-{571\over 2488320z^4}+\ldots}\right).$$ (2)

The expansion of $\ln\Gamma(z)$ is what is usually called Stirling's series. It is given by the simple analytic expression

$$\ln \Gamma(z) = \sum_{n=1}^\infty {B_{2n}\over 2n(2n-1) z^{2n-1}}$$ (3)

$$= {\textstyle{1\over 2}}\ln(2\pi)+(z+{\textstyle{1\over 2}})\ln z-z+{1\over 12z}-{1\over 360z^3}+{1\over 1260z^5}-\ldots,$$ (4)

where $B_n$ 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. 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.