Formula:DLMF:25.5:E10

<< Formula:DLMF:25.5:E9

formula in Integral Representations

Formula:DLMF:25.5:E11 >>

${\displaystyle{\displaystyle{\displaystyle\zeta\left(s\right)=\frac{2^{s-1}}{1-2^{1-s}}\int_{0}^{\infty}\frac{\cos\!\left(s\mathrm{arctan}x\right)}{(1+x^{2})^{s/2}\cosh\!\left(\frac{1}{2}\cpi x\right)}\diffd x}}}$

Constraint(s)

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

Proof

We ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space.

Symbols List

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \zeta}}  : Riemann zeta function : http://dlmf.nist.gov/25.2#E1
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mathrm{d}^nx}}  : differential : http://dlmf.nist.gov/1.4#iv

Bibliography

Equation (10), Section 25.5 of DLMF.

URL links

We ask users to provide relevant URL links in this space.

<< Formula:DLMF:25.5:E9

formula in Integral Representations

Formula:DLMF:25.5:E11 >>