Stirling's approximation gives an approximate value for the Factorial function or the Gamma
Function for . The approximation can most simply be derived for an Integer by approximating the
sum over the terms of the Factorial with an Integral, so that
The equation can also be derived using the integral definition of the Factorial,

(2) 
Note that the derivative of the Logarithm of the integrand can be written

(3) 
The integrand is sharply peaked with the contribution important only near . Therefore, let where
, and write

(4) 
Now,
so
Taking the Exponential of each side then gives

(7) 
Plugging into the integral expression for then gives
Evaluating the integral gives
Taking the Logarithm of both sides then gives

(11) 
This is Stirling's Series with only the first term retained and, for large , it reduces to
Stirling's approximation

(12) 
Gosper notes that a better approximation to (i.e., one which approximates the terms in Stirling's Series
instead of truncating them) is given by

(13) 
This also gives a much closer approximation to the Factorial of 0, , yielding
instead of 0 obtained with the conventional Stirling approximation.
See also Stirling's Series
© 19969 Eric W. Weisstein
19990526