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

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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.


Proof


Source of Name

This entry was named for Jacques Philippe Marie Binet.


Sources