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

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

Then:


 * $\displaystyle \operatorname{Ln} \Gamma \left({z}\right) = \left({z - \frac 1 2}\right) \operatorname{Ln} z - z + \frac 1 2 \ln 2\pi + 2 \int_0^\infty \frac {\arctan \left({t/z}\right)} {e^{2 \pi t} - 1} \rd t$

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