A calculus problem by gino varkey

The magnitude of the complex number $\Gamma(4+2i) = (3+2i)!$ can be expressed as $\sqrt{A \times B \times \text{cosech}(B)}$ , where $A$ is an integer, and $B$ is a real number.

Submit your answer as $\dfrac{A\times B}{\pi}$ .

Notations:

• $i = \sqrt{-1}$ .
• For real numbers $x$ and $y$ , the magnitude of $x+yi$ is $|x+yi| = \sqrt{x^2 + y^2}$ .
• $\text{cosech}(\cdot)$ represents the hyperbolic cosecant function, $\text{cosech}(z) = \dfrac2{e^z-e^{-z}}$ .