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

Theorem

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.

Proof

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Source of Name

This entry was named for Jacques Philippe Marie Binet.

Sources

• 1920: E.T. Whittaker and G.N. Watson: A Course of Modern Analysis (3rd ed.): $12.32$: Binet's second expression for $\log \Gamma \paren z$ in terms of an infinite integral