Hankel contour integral

Hankel’s contour integral is a unit (and nilpotent) for gamma function over $ℂ$. That is,

Hankel’s integral is holomorphic with simple zeros in ${\mathbb{Z}}_{\le 0}$. Its path of integration starts on the positive real axis ad infinitum, rounds the origin counterclockwise and returns to $+\mathrm{\infty }$. As an example of application of Hankel’s integral, we have

$$\frac{i}{2\pi }{\int }_{𝒞}{(-t)}^{-\frac{1}{2}}{e}^{-t}𝑑t=\frac{1}{\sqrt{\pi }},$$

where the path of integration is the one above mentioned.