Modulus of Gamma Function of Imaginary Number

Theorem
Let $t$ be a real number, then:


 * $\displaystyle \left\vert \Gamma\left({it}\right) \right\vert = \sqrt{ \frac{\pi \operatorname{csch} \pi t} t }$

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

Proof
By Euler's Reflection Formula:


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

From Gamma Difference Equation, we have:


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

Then:

And:

So:


 * $\displaystyle \left\vert \Gamma\left({it}\right) \right\vert^2 = \frac {\pi \operatorname{csch} \left({\pi \left\vert t\right\vert }\right)} {\left\vert t\right\vert}$

As both sides are positive, we can write:


 * $\displaystyle \left\vert \Gamma\left({it}\right) \right\vert = \sqrt{ \frac {\pi \operatorname{csch} \left({\pi \left\vert t\right\vert }\right)} {\left\vert

t\right\vert} }$

However as, by Hyperbolic Sine Function is Odd:


 * $\displaystyle \frac{\pi \operatorname{csch}\left({-\pi t}\right)} {- t} = \frac{-\pi \operatorname{csch}\left({\pi t}\right)} {- t} = \frac {\pi \operatorname{csch}\left({\pi t}\right)} t$

We can remove the modulus and simply write:


 * $\displaystyle \left\vert \Gamma\left({it}\right) \right\vert = \sqrt{ \frac{\pi \operatorname{csch} \pi t} t }$