Stirling's approximation

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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

where

For example:

N N! (exact) N! (Stirling) Error (%)
5 120 118.019168 1.016
6 720 710.078185 1.014
7 5040 4980.39583 1.012
8 40320 39902.3955 1.010
9 362880 359536.873 1.009
10 3628800 3598695.62 1.008

As one usually deals with number of the order of the Avogadro constant () this formula is essentially exact.

Applications in statistical mechanics