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:


 * $\Gamma \paren {i t} \Gamma \left({1 - i t}\right) = \pi \csc \left({\pi i t}\right)$

From Gamma Difference Equation:


 * $-i t \Gamma \paren {i t} \Gamma \left({-i t}\right) = \pi \csc \left({\pi i t}\right)$

Then:

and:

So:


 * $\cmod {\Gamma \left({i t}\right)}^2 = \dfrac {\pi \csch \left({\pi \cmod t}\right)} {\cmod t}$

As both sides are positive, we can write:


 * $\cmod {\Gamma \left({i t}\right)} = \sqrt {\dfrac {\pi \csch \left({\pi \cmod t}\right)} {\cmod t} }$

However, by Hyperbolic Sine Function is Odd:


 * $\dfrac {\pi \csch \left({-\pi t}\right)} {- t} = \dfrac {-\pi \csch \left({\pi t}\right)} {- t} = \dfrac {\pi \csch \left({\pi t}\right)} t$

Hence we can remove the modulus and simply write:


 * $\cmod {\Gamma\left({i t}\right)} = \sqrt {\dfrac {\pi \csch \pi t} t}$

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


 * $\cmod {\Gamma\left({i t}\right)}^2 = \dfrac \pi {t \sinh \pi t}$