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

Corollary to Binet's Formula for Logarithm of Gamma Function: Formulation 1

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

Then:

$\ds \lim_{z \mathop \to \infty} \size {\Ln \map \Gamma z - \paren {z - \frac 1 2} \Ln z + z - \frac 1 2 \ln 2 \pi } \to 0$

where:

$\Gamma$ is the Gamma function

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

Proof

\(\ds \Ln \map \Gamma z\)

\(=\)

\(\ds \paren {z - \frac 1 2} \Ln z - z + \frac 1 2 \ln 2 \pi + \int_0^\infty \paren {\frac 1 2 - \frac 1 t + \frac 1 {e^t - 1} } \frac {e^{-t z} } t \rd t\)

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

\(\ds \leadsto \ \ \)

\(\ds \lim_{z \mathop \to \infty} \paren {\Ln \map \Gamma z - \paren {z - \frac 1 2} \Ln z + z - \frac 1 2 \ln 2 \pi}\)

\(=\)

\(\ds \lim_{z \mathop \to \infty} \paren {\int_0^\infty \paren {\frac 1 2 - \frac 1 t + \frac 1 {e^t - 1} } \frac {e^{-t z} } t \rd t }\)

\(\ds \leadsto \ \ \)

\(\ds \lim_{z \mathop \to \infty} \size {\Ln \map \Gamma z - \paren {z - \frac 1 2} \Ln z + z - \frac 1 2 \ln 2 \pi }\)

\(\to\)

\(\ds 0\)

Exponential Tends to Zero and Infinity

$\blacksquare$