Modulus of Gamma Function of Imaginary Number

Theorem
Let $t \in \R$ be a real number.

Then:


 * $\cmod {\Gamma \paren {i t} } = \sqrt {\dfrac {\pi \csch \pi t} t}$

where:
 * $\Gamma$ is the Gamma function
 * $\csch$ is the hyperbolic cosecant function.

Proof
By Euler's Reflection Formula:


 * $\Gamma \paren {i t} \Gamma \paren {1 - i t} = \pi \csc \paren {\pi i t}$

From Gamma Difference Equation:


 * $-i t \Gamma \paren {i t} \Gamma \paren {-i t} = \pi \csc \paren {\pi i t}$

Then:

and:

So:


 * $\cmod {\Gamma \paren {i t} }^2 = \dfrac {\pi \csch \paren {\pi \cmod t} } {\cmod t}$

As $\cmod z \ge 0$ for all complex numbers $z$, we can take the non-negative square root and write:


 * $\cmod {\Gamma \paren {i t} } = \sqrt {\dfrac {\pi \csch \paren {\pi \cmod t} } {\cmod t} }$

However, by Hyperbolic Sine Function is Odd:


 * $\dfrac {\pi \csch \paren {-\pi t} } {- t} = \dfrac {-\pi \csch \paren {\pi t} } {- t} = \dfrac {\pi \csch \paren {\pi t} } t$

Hence we can remove the modulus and simply write:


 * $\cmod {\Gamma \paren {i t} } = \sqrt {\dfrac {\pi \csch \pi t} t}$

Also reported as
This result can also be seen reported as:


 * $\cmod {\Gamma \paren {i t} }^2 = \dfrac \pi {t \sinh \pi t}$