Hankel Representation of Riemann Zeta Function

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Theorem

Let $C$ be the Hankel contour.

Then for $s \in \C \setminus \Z_{>0}$:

$\displaystyle \zeta\left({s}\right) = \frac {i \Gamma \left({1 - s}\right)} {2 \pi} \oint_C \frac {\left({-z}\right)^{s - 1} } {e^z - 1} \, \mathrm d z$

where:

$\zeta$ is the Riemann Zeta function
$\Gamma$ is the Gamma function.


Proof


Source of Name

This entry was named for Hermann Hankel.


Sources