Stirling's approximation

Stirling's approximation is an approximate formula for n! := 1×2×3× … ×n (n factorial). The approximation is useful for very large values of the positive integer n. It is a series expansion with the first five terms given by:

$n!=e^{-n}n^{n+\tfrac{1}{2}}(2\pi)^{\tfrac{1}{2}}\left(1 +\frac{1}{12n} + +\frac{1}{288n^2}-\frac{139}{51840n^3}-\frac{571}{2488320n^4}+O(\frac{1}{n^5})\right).$

Here ex stands for the exponential function of x, n is a positive integer, and O(1/n5) is the big O notation for a rest term that falls off as M/n5 where M is a positive real constant.

A very common application for Stirling's approximation is in statistical thermodynamics, where n is usually on the order of Avogadro's number NA = 6.02×1023. In such applications it is usually the natural logarithm ln(n!) that appears and from the formula above follows in a one-term approximation:

$\ln n! = (n+\frac{1}{2})\ln n -n + \frac{1}{2}\ln(2\pi) \approx -n + n\ln n.$

The right-hand side is the approximation most commonly used in statistical thermodynamics.

The formula was first proposed by the Scottish mathematician James Stirling (1692–1770) in his 1730 work Methodus Differentialis.

Reference

D. E. Knuth, The Art of Computer Programming, Vol. 1, Addison-Wesley, Reading, Mass. (1969), p. 111.