Stirling's formula
(another Prime Pages' Glossary entries)
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Stirling found the following formula for approximating factorials:

n! ~ !=(n/e)^n sqrt(2 pi n)
and more precisely,
n!=(n/e)^n sqrt(2 pi n) e^(theta(n)/12n)
where 0 < theta < 1 (and e is the base of the natural logarithms). In terms of the gamma function Stirling's formula is
a ugly formula
another ugly one
where Bk is the kth Bernoulli number.


M. Abramowitz and I. Stegun editors, Handbook of mathematical functions--with formulas, graphs, and mathematical tables, Dover Pub., 1974.  New York, NY, pp. xiv+1046, ISBN 0-486-61272-4. MR 94b:00012

Chris K. Caldwell © 1999-2019 (all rights reserved)