PlanetMath: Hankel contour integral

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Hankel contour integral

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Hankel's contour integral is a unit (and nilpotent) for gamma function over $\mathbb{C}$ . That is,

$\displaystyle \left(\frac{i}{2\pi}\int_\mathcal{C}(-t)^{-z}e^{-t}dt\right)\Gamma(z)=1, \qquad \vert z\vert<\infty.$

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

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

where the path of integration is the one above mentioned.

"Hankel contour integral" is owned by perucho.

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Cross-references: origin, ad infinitum, real axis, positive, path, simple, holomorphic, integral, gamma function, nilpotent, unit

This is version 2 of Hankel contour integral, born on 2007-08-09, modified 2007-08-11.
Object id is 9847, canonical name is HankelContourIntegral.
Accessed 612 times total.

Classification:

AMS MSC:	33B15 (Special functions :: Elementary classical functions :: Gamma, beta and polygamma functions)
	30D30 (Functions of a complex variable :: Entire and meromorphic functions, and related topics :: Meromorphic functions, general theory)

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