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
(1)
(2)
(3)
(4)
(5)
(6)
The equation can also be derived using the integral definition of the factorial,
(7)
Note that the derivative of the logarithm of the integrand
can be written
(8)
The integrand is sharply peaked with the contribution important only near . Therefore, let where , and write
Plugging into the integral expression for then gives
(19)
(20)
Evaluating the integral gives
(21)
(22)
(Wells 1986, p. 45). Taking the logarithm of both
sides then gives
(23)
(24)
This is Stirling's series with only the first term retained and, for large , it reduces to Stirling's approximation
(25)
Taking successive terms of , where is the floor function,
gives the sequence 1, 2, 4, 10, 26, 64, 163, 416, 1067, 2755, ... (OEIS A055775).
Stirling's approximation can be extended to the double inequality
(26)
(Robbins 1955, Feller 1968).
Gosper has noted that a better approximation to (i.e., one which approximates the terms in Stirling's
series instead of truncating them) is given by
(27)
Considering
a real number so that , the equation (27) also
gives a much closer approximation to the factorial
of 0, ,
yielding
instead of 0 obtained with the conventional Stirling approximation.