Hankel contour integral

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.

Title	Hankel contour integral
Canonical name	HankelContourIntegral
Date of creation	2013-03-22 17:27:50
Last modified on	2013-03-22 17:27:50
Owner	perucho (2192)
Last modified by	perucho (2192)
Numerical id	5
Author	perucho (2192)
Entry type	Result
Classification	msc 30D30
Classification	msc 33B15