Formula:DLMF:25.5:E21

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formula in Integral Representations

Formula:DLMF:25.6:E1:SE1 >>

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

Constraint(s)

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

The contour here is any loop that encircles the origin in the positive direction

not enclosing any of the points ${\displaystyle{\displaystyle{\displaystyle\pm\pi\mathrm{i}}}}$, ${\displaystyle{\displaystyle{\displaystyle\pm 3\pi\mathrm{i},\ldots}}}$

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
${\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}$ : imaginary unit : http://dlmf.nist.gov/1.9.i
${\displaystyle{\displaystyle{\displaystyle\int}}}$ : 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 (21), Section 25.5 of DLMF.

URL links

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