Stirling's formula

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., New York, NY, 1974.  pp. xiv+1046, ISBN 0-486-61272-4. MR 94b:00012
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