Hankel's Loop Integral

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\Gamma(s)=-\frac{1}{2i\sin(\pi s)}\int_{C}(-t)^{s-1}e^{-t}dt

\Downarrow

\frac{1}{\Gamma(s)}=\frac{1}{2\pi i}\int_{-\infty}^{(0,+)}e^{t}t^{-s}dt

Hankel’s Loop Integral is a common contour integral that frequently arises in the study of the Gamma and Zeta functions.
It may initially seem like a difficult type of reasoning, but it is so widely used that it is worth spending the time to understand it.

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