Stirling's Approximation

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

			(9)
			(10)

Now,

			(11)
			(12)

so

			(13)
			(14)
			(15)

Taking the exponential of each side then gives

			(16)
			(17)

Plugging into the integral expression for then gives

			(18)
			(19)

Evaluating the integral gives

			(20)
			(21)

(Wells 1986, p. 45). The factor is sometimes called the Stirling constant. Taking the logarithm of both sides then gives

			(22)
			(23)

This is Stirling's series with only the first term retained and, for large , it reduces to Stirling's approximation

(24)

Taking successive terms of , where is the floor function, gives the sequence 1, 2, 4, 10, 26, 64, 163, 416, 1067, 2755, ... (OEIS A055775).

For positive integers , Stirling's approximation can be extended to the double inequality

(25)

(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

(26)

Considering a real number so that , the equation (26) also gives a much closer approximation to the factorial of 0, , yielding instead of 0 obtained with the conventional Stirling approximation.