Reciprocal of gamma function

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Definition

As the reciprocal of the gamma function

It is defined as:

$z\mapsto {\frac {1}{\Gamma (z)}}$

if $z$ is not a simple pole of the gamma function. If $z$ is a simple pole, it sends $z$ to 0.

In terms of Hankel's loop contour

Let $\gamma$ denote Hankel's loop contour. The reciprocal of the gamma function is then defined as:

$z\mapsto \int _{\gamma }e^{-w}w^{-z}\,dw$