Binet's Formula for Logarithm of Gamma Function/Formulation 2

Theorem
Let $z$ be a complex number with a positive real part.

Then:


 * $\displaystyle \Ln \map \Gamma z = \paren {z - \frac 1 2} \Ln z - z + \frac 1 2 \ln 2 \pi + 2 \int_0^\infty \frac {\map \arctan {t / z} } {e^{2 \pi t} - 1} \rd t$

where:
 * $\Gamma$ is the Gamma function
 * $\Ln$ is the principal branch of the complex logarithm.