Formula:DLMF:25.5:E13

<< Formula:DLMF:25.5:E12

formula in Integral Representations

Formula:DLMF:25.5:E15 >>

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

Substitution(s)

${\displaystyle{\displaystyle{\displaystyle{\displaystyle\omega(x)=\sum_{n=1}^{\infty}{\mathrm{e}^{-n^{2}\pi x}}=\frac{1}{2}\left(\JacobiThetaTau{3}@{0}{\mathrm{i}x}-1\right)}}}}$

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
${\displaystyle{\displaystyle{\displaystyle\pi}}}$ : ratio of a circle's circumference to its diameter : http://dlmf.nist.gov/5.19.E4
${\displaystyle{\displaystyle{\displaystyle\Gamma}}}$ : Euler's gamma function : http://dlmf.nist.gov/5.2#E1
${\displaystyle{\displaystyle{\displaystyle\int}}}$ : integral : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle\mathrm{d}^{n}x}}}$ : differential : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}$ : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
${\displaystyle{\displaystyle{\displaystyle\theta_{p}}}}$ : Jacobi theta ${\displaystyle{\displaystyle{\displaystyle\tau}}}$ function ${\displaystyle{\displaystyle{\displaystyle\theta_{j}}}}$ : http://dlmf.nist.gov/20.2#i
${\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}$ : imaginary unit : http://dlmf.nist.gov/1.9.i

Bibliography

Equation (13), Section 25.5 of DLMF.

URL links

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formula in Integral Representations

Formula:DLMF:25.5:E15 >>