Approximation to Stirling's Formula for Gamma Function

Theorem
Let:


 * $D_\epsilon = \set {z \in \C : \cmod {\Arg z} < \pi - \epsilon,\ \cmod z > 1}$

where:
 * $\cmod {\Arg z}$ denotes the absolute value of the principal argument of $z$
 * $\cmod z$ denotes the modulus of $z$
 * $\epsilon \in \R_{>0}$.

Then for all $z \in D_\epsilon$, the gamma function of $z$ satisfies:


 * $\map \Gamma z = \sqrt {\dfrac {2 \pi} z} \paren {\dfrac z e}^z \paren {1 + \map \OO {z^{-1} } }$

where $\map \OO {z^{-1} }$ denotes big-O of $z^{-1}$.

Proof
From Gamma Function is Unique Extension of Factorial:

for $0 < y \le 1$ and $n \in \N$.

Let $x$ be given.

Let $n + 1$ be the largest natural number such that $n + 1 \le x$.

Let $x = y + n + 1$, and thus $0 < y \le 1$.

Then:

Similarly for the.

The result follows from Gamma Function Extends Factorial.

Also see

 * Logarithmic Approximation of Error Term of Stirling's Formula for Gamma Function