Hankel's contour integral is a unit (and nilpotent) for gamma function over $\mathbb{C}$ That is, $$\left(\frac{i}{2\pi}\int_\mathcal{C}(-t)^{-z}e^{-t}dt\right)\Gamma(z)=1, \qquad |z|<\infty.$$ Hankel's integral is holomorphic with simple zeros in $\mathbb{Z}_{\leq 0}$ Its path of integration starts on the positive real axis ad infinitum, rounds the origin counterclockwise and returns to $+\infty$ As an example of application of Hankel's integral, we have $$\frac{i}{2\pi}\int_\mathcal{C}(-t)^{-\frac{1}{2}}e^{-t}dt=\frac{1}{\sqrt{\pi}}\,,$$ where the path of integration is the one above mentioned.