An approximation for the factorial, n!, of a number n. The precise form of Stirling’s approximation is:

This approximation is valid for large values of n, being accurate for n greater than about 10. A simplified version of Stirling’s expression is log_e n!=nlog_e n – n, which is an approximation to the precise form, derived by dropping all those terms that do not increase at least as quickly as n. An important application of Stirling’s approximation is in the derivation of the Boltzmann distribution (see Boltzmann equation). In this application n is very much greater than 10, enabling the simplified version of Stirling’s approximation to be used. The approximation is named after the Scottish mathematician James Stirling (1692–1770).