Formula:DLMF:25.5:E9

<< Formula:DLMF:25.5:E8

formula in Integral Representations

Formula:DLMF:25.5:E10 >>

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

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle\Re{s}>1}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle\zeta}}}$ : 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
${\displaystyle{\displaystyle{\displaystyle\int}}}$ : integral : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle\mathrm{sinh}}}}$ : hyperbolic sine function : http://dlmf.nist.gov/4.28#E1
${\displaystyle{\displaystyle{\displaystyle\mathrm{d}^{n}x}}}$ : differential : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle\Re{z}}}}$ : real part : http://dlmf.nist.gov/1.9#E2

Bibliography

Equation (9), Section 25.5 of DLMF.

URL links

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