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

Theorem

 * $\displaystyle \operatorname{Ln} \Gamma \left({z}\right) = \left({z - \dfrac 1 2}\right) \operatorname{Ln} \left({z}\right) - z + \dfrac {\ln 2 \pi} 2 + \sum_{n \mathop = 1}^{d-1} \frac{B_{2n}} {2n \left({2 n - 1}\right) z^{2n-1} } + \mathcal O \left({z^{1 - 2 d}}\right)$

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

Also see

 * Approximation to Stirling's Formula for Gamma Function