Formula:DLMF:25.5:E10

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formula in Integral Representations

Formula:DLMF:25.5:E11 >>

${\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{s}=\frac{2^{s-1}}{1-2^{1-s}}\int_{0}^{\infty}\frac{\cos\left(s\operatorname{arctan}x\right)}{(1+x^{2})^{s/2}\cosh\left(\frac{1}{2}\pi x\right)}\mathrm{d}x}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle s\neq 1}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle\zeta}}}$ : Riemann zeta function : http://dlmf.nist.gov/25.2#E1
${\displaystyle{\displaystyle{\displaystyle\int}}}$ : integral : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle\mathrm{cos}}}}$ : cosine function : http://dlmf.nist.gov/4.14#E2
${\displaystyle{\displaystyle{\displaystyle\mathrm{arctan}}}}$ : inverse tangent function : http://dlmf.nist.gov/4.23#SS2.p1
${\displaystyle{\displaystyle{\displaystyle\mathrm{cosh}}}}$ : hyperbolic cosine function : http://dlmf.nist.gov/4.28#E2
${\displaystyle{\displaystyle{\displaystyle\pi}}}$ : ratio of a circle's circumference to its diameter : http://dlmf.nist.gov/5.19.E4
${\displaystyle{\displaystyle{\displaystyle\mathrm{d}^{n}x}}}$ : differential : http://dlmf.nist.gov/1.4#iv

Bibliography

Equation (10), Section 25.5 of DLMF.

URL links

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