Stirling's formula derived simply

TL;DR

This paper presents a straightforward derivation of Stirling's formula for factorials and gamma functions using recursion relations and normalization conditions, simplifying the understanding of its asymptotic behavior.

Contribution

It introduces a direct derivation method for Stirling's formula based on recursion equations and normalization, avoiding complex integral or series expansions.

Findings

Derivation of Stirling's formula from recursion relations

Simplified proof of gamma function asymptotics

Clarification of normalization role in asymptotic expansion

Abstract

Stirling's formula, the asymptotic expansion of $n!$ for $n$ large, or of $\Gamma(z)$ for $z\to \infty$, is derived directly from the recursion equation $\Gamma(z+1) =z \Gamma(s)$ and the normalization condition $\Gamma ({1/2}) =\sqrt{\pi}$.