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

Theorem

 * $\ds \map \Ln {\map \Gamma z} = \paren {z - \dfrac 1 2} \map \Ln z - z + \dfrac {\ln 2 \pi} 2 + \sum_{n \mathop = 1}^{d - 1} \frac {B_{2 n} } {2 n \paren {2 n - 1} z^{2 n - 1} } + \OO \paren {z^{1 - 2 d} }$

where:
 * $\Gamma$ is the Gamma function
 * $\Ln$ is the principal branch of the complex logarithm
 * $B_{2 n}$ is the $2n$th Bernoulli number
 * $\OO$ is Big-O notation.

Also see

 * Approximation to Stirling's Formula for Gamma Function