Approximation to Stirling's Formula for Gamma Function

Theorem
Let


 * $D_\epsilon = \{z \in \C : |\arg(z)| < \pi - \epsilon,\ |z| > 1\}$

Then for any $\epsilon > 0$ the gamma function satisfies:


 * $\displaystyle \Gamma(z) = \sqrt{\frac{2\pi}z}\left( \frac ze\right)^z\left(1 + \mathcal O\left( z^{-1} \right) \right)$

for all $z \in D_\epsilon$.