Formula:DLMF:25.11:E30

<< Formula:DLMF:25.11:E29

formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E31 >>

${\displaystyle{\displaystyle{\displaystyle\HurwitzZeta@{s}{a}=\frac{\Gamma\left(1-s\right)}{2\pi\mathrm{i}}\int_{-\infty}^{(0+)}\frac{{\mathrm{e}^{az}}z^{s-1}}{1-{\mathrm{e}^{z}}}\mathrm{d}z}}}$

Constraint(s)

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

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

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

Proof

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Assume ${\displaystyle{\displaystyle{\displaystyle\Re{s}>1}}}$, collapse the integration path onto the

real axis, apply
${\displaystyle{\displaystyle{\displaystyle\HurwitzZeta@{s}{a}=\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\frac{x^{s-1}{\mathrm{e}^{-ax}}}{1-{\mathrm{e}^{-x}}}\mathrm{d}x}}}$
and
${\displaystyle{\displaystyle{\displaystyle\Gamma\left(z\right)\Gamma\left(1-z\right)=\pi/\sin\left(\pi z\right)}}}$

followed by analytic continuation.

Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle\zeta}}}$ : Hurwitz zeta function : http://dlmf.nist.gov/25.11#E1
${\displaystyle{\displaystyle{\displaystyle\Gamma}}}$ : Euler's gamma function : http://dlmf.nist.gov/5.2#E1
${\displaystyle{\displaystyle{\displaystyle\pi}}}$ : ratio of a circle's circumference to its diameter : http://dlmf.nist.gov/5.19.E4
${\displaystyle{\displaystyle{\displaystyle\int}}}$ : integral : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}$ : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
${\displaystyle{\displaystyle{\displaystyle\mathrm{d}^{n}x}}}$ : differential : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle\Re{z}}}}$ : real part : http://dlmf.nist.gov/1.9#E2

Bibliography

Equation (30), Section 25.11 of DLMF.

URL links

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<< Formula:DLMF:25.11:E29

formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E31 >>