Modulus of Gamma Function of One Half plus Imaginary Number

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

Then:


 * $\left\vert{\Gamma \left({\dfrac 1 2 + i t}\right)}\right\vert = \sqrt{\pi \operatorname{sech} \left({\pi t}\right)}$

where:
 * $\Gamma$ is the Gamma function
 * $\operatorname{sech}$ is the hyperbolic secant function.

Proof
As both sides of the equation are positive for all $t$, we can take the non-negative square root and write:


 * $\left\vert{\Gamma \left({\dfrac 1 2 + i t}\right)}\right\vert = \sqrt{\pi \operatorname{sech}\left({\pi t}\right)}$