# Formula:DLMF:25.11:E30

${\displaystyle{\displaystyle{\displaystyle\HurwitzZeta@{s}{a}=\frac{% \EulerGamma@{1-s}}{2\pi i}\int_{-\infty}^{(0+)}\frac{e^{az}z^{s-1}}{1-e^{z}}% \diff{z}}}}$

## Constraint(s)

${\displaystyle {\displaystyle s\neq 1}}$ &
${\displaystyle{\displaystyle{\displaystyle\realpart{a}>0}}}$ &

the integration contour is a loop around the negative real axis; it starts at ${\displaystyle {\displaystyle -\infty }}$, encircles the origin once in the positive direction without enclosing any of the points

${\displaystyle{\displaystyle{\displaystyle z=\pm 2\pi i}}}$, ${\displaystyle{\displaystyle{\displaystyle\pm 4\pi i,\ldots,}}}$ and returns to ${\displaystyle {\displaystyle -\infty }}$

## Proof

We ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space.

Assume ${\displaystyle{\displaystyle{\displaystyle\realpart{s}>1}}}$, collapse the integration path onto the followed by analytic continuation.

## Symbols List

& : logical and
: Hurwitz zeta function : http://dlmf.nist.gov/25.11#E1
: Euler's gamma function : http://dlmf.nist.gov/5.2#E1
: ratio of a circle's circumference to its diameter : http://dlmf.nist.gov/5.19.E4
: integral : http://dlmf.nist.gov/1.4#iv
: the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
: differential : http://dlmf.nist.gov/1.4#iv
: real part : http://dlmf.nist.gov/1.9#E2