Formula:DLMF:25.5:E20

<< Formula:DLMF:25.5:E19

formula in Integral Representations

Formula:DLMF:25.5:E21 >>

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

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle s\neq 1,2,\dots}}}$ &

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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Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle\zeta}}}$ : Riemann zeta function : http://dlmf.nist.gov/25.2#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

Bibliography

Equation (20), Section 25.5 of DLMF.

URL links

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