James Stirling (1692-1770, Scotland)
Because of Euler-MacLaurin formula
where B1 = −1/2, B2 = 1/6, B3 = 0, B4 = −1/30, B5 = 0, B6 = 1/42, B7 = 0, B8 = −1/30, ... are the Bernoulli numbers, and R is an error term which is normally small for suitable values of p.
Then, for large N,
after some further manipulation one arrives at
|N||N! (exact)||N! (Stirling)||Error (%)|
As one usually deals with number of the order of the Avogadro constant () this formula is essentially exact. In computer simulations the number of atoms or molecules (N) is invariably greater than 100, where the percentage error is less than .