Stirling Series

📐 Definition

Stirling’s formula gives an asymptotic approximation for the factorial function as $n \to \infty$ :

n! \sim \sqrt{2\pi n}\, n^n e^{-n}

Equivalently, using the Gamma function with $n! = \Gamma(n+1)$ ,

\Gamma(z) \sim \sqrt{2\pi}\, z^{z-\frac12} e^{-z}

as $|z| \to \infty$ with $|\arg z| < \pi$ .

Taking logarithms yields a particularly useful representation:

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

This form is central in entropy calculations and information theory.

Stirling’s formula admits a full asymptotic expansion:

n! \sim \sqrt{2\pi n}\, n^n e^{-n} \left( 1 + \frac{1}{12n} + \frac{1}{288n^2} - \frac{139}{51840n^3} + \cdots \right)

The coefficients are expressed in terms of Bernoulli numbers.

Sharper versions provide explicit bounds on the remainder. For example,

\sqrt{2\pi}\, n^{n+\frac12} e^{-n} < n! < \sqrt{2\pi}\, n^{n+\frac12} e^{-n+\frac{1}{12n}}

for all positive integers $n$ .

For complex arguments, Stirling’s formula holds uniformly in sectors

|\arg z| < \pi - \varepsilon

for any fixed $\varepsilon > 0$ .

This restriction reflects the branch cut of the logarithm.

$n$	Exact $n!$	Stirling approximation	Relative error
$5$	$120$	$118.02$	$1.6%$
$10$	$3.63\times10^6$	$3.60\times10^6$	$0.8%$
$50$	$3.04\times10^{64}$	$3.04\times10^{64}$	$<10^{-3}$

The approximation becomes remarkably accurate even for moderate $n$ .

Gamma Function Stirling's series provides the asymptotic expansion of Γ(z)

Digamma Function Uses Stirling-series asymptotics for large arguments

Mortici Accelerated Series Modern acceleration techniques building on Stirling's work

📜 Original Discovery (18th Century)

James Stirling introduced the approximation in his 1730 work Methodus Differentialis, primarily as a tool for approximating factorials.
🔬 Later Refinements

Laplace and Gauss provided rigorous asymptotic justification and extended the approximation to complex arguments, connecting it to integral methods.
🌍 Modern Perspective

Today, Stirling’s formula is a cornerstone of asymptotic analysis, probability theory, and mathematical physics, appearing wherever factorial growth must be controlled.