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 HankelContourIntegral 2013-03-22 17:27:50 2013-03-22 17:27:50 perucho (2192) perucho (2192) 5 perucho (2192) Result msc 30D30 msc 33B15