Binet's Formula for Logarithm of Gamma Function

Theorem

Formulation 1

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

Then:

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

where:

$\Gamma$ is the Gamma function

$\Ln$ is the principal branch of the complex logarithm.

Formulation 2

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

Then:

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

where:

$\Gamma$ is the Gamma function

$\Ln$ is the principal branch of the complex logarithm.

Source of Name

This entry was named for Jacques Philippe Marie Binet.