Stirling's Formula

Asymptotic approximation for n! and log n!, used for entropy and large-N counting in statistical mechanics.

Stirling’s Formula

Stirling’s formula is a basic asymptotic tool for combinatorial and phase-space counting, and it is frequently used when connecting microscopic counting to Boltzmann entropy and to information-theoretic quantities like Shannon entropy .

Statement

As $n\to\infty$ ,

n! \sim \sqrt{2\pi n}\,\Big(\frac{n}{e}\Big)^n.

Equivalently,

\log(n!) = n\log n - n + \tfrac12\log(2\pi n) + o(1).

A common quantitative refinement is: for every integer $n\ge 1$ there exists $\theta_n\in(0,1)$ such that

n! = \sqrt{2\pi n}\,\Big(\frac{n}{e}\Big)^n \exp\!\Big(\frac{\theta_n}{12n}\Big).

In particular,

\log(n!) = n\log n - n + \tfrac12\log(2\pi n) + O\!\Big(\frac{1}{n}\Big).

Key hypotheses and conclusions

Hypotheses

$n\in\mathbb{N}$ and $n\to\infty$ .

Conclusions

Accurate leading-order and next-order asymptotics for $n!$ and $\log(n!)$ .
Enables asymptotics for multinomial coefficients; e.g. leading terms produce entropy-like functionals.