Binet's Formula for Logarithm of Gamma Function/Formulation 1

Theorem

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

Then:

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

where:

$\Gamma$ is the Gamma function

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

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.): $12.31$: Binet's first expression for $\log \Gamma \paren z$ in terms of an infinite integral