Modulus of Gamma Function of One Half plus Imaginary Number

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


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

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:


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