Stirling's approximation

James Stirling (1692-1770, Scotland)

$\left.\ln N!\right.=\ln 1+\ln 2+\ln 3+...+\ln N=\sum _{k=1}^{N}\ln k.$

Because of Euler-MacLaurin formula

$\sum _{k=1}^{N}\ln k=\int _{1}^{N}\ln x\,dx+\sum _{k=1}^{p}{\frac {B_{2k}}{2k(2k-1)}}\left({\frac {1}{n^{2k-1}}}-1\right)+R,$

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,

$\ln N!\sim \int _{1}^{N}\ln x\,dx\sim N\ln N-N.$

Applications in statistical mechanics

• Ideal gas Helmholtz energy function