Modulus of Gamma Function of One Half plus Imaginary Number

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

Then:


 * $\cmod {\Gamma \paren {\dfrac 1 2 + i t} } = \sqrt{\pi \sech \paren {\pi t} }$

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

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


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