Binet's Formula for Logarithm of Gamma Function

Theorem
Let $z$ be a complex number with a positive real part, then:


 * $\displaystyle \operatorname{Ln} \Gamma\left({z}\right) = \left({z - \frac 1 2}\right) \operatorname{Ln} z - z + \frac 1 2 \ln 2\pi + \int_0^\infty \left({ \frac 1 2 - \frac 1 t + \frac 1 {e^t - 1} }\right) \frac { e^{-tz} } t \, \mathrm d t$

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