Formula:DLMF:25.5:E19

<< Formula:DLMF:25.5:E18

formula in Integral Representations

Formula:DLMF:25.5:E20 >>

${\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{m+s}=(-1)^{m-1}\frac{\Gamma\left(s\right)\sin\left(\pi s\right)}{\pi\Gamma\left(m+s\right)}\*\int_{0}^{\infty}{\psi^{(m)}}\left(1+x\right)x^{-s}\mathrm{d}x}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle m=1,2,3,\dots}}}$ &
${\displaystyle{\displaystyle{\displaystyle 0<\Re{s}<1}}}$

Proof

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Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle\zeta}}}$ : Riemann zeta function : http://dlmf.nist.gov/25.2#E1
${\displaystyle{\displaystyle{\displaystyle(-1)}}}$ : negative unity to an integer power : http://dlmf.nist.gov/5.7.E7
${\displaystyle{\displaystyle{\displaystyle\Gamma}}}$ : Euler's gamma function : http://dlmf.nist.gov/5.2#E1
${\displaystyle{\displaystyle{\displaystyle\mathrm{sin}}}}$ : sine function : http://dlmf.nist.gov/4.14#E1
${\displaystyle{\displaystyle{\displaystyle\pi}}}$ : ratio of a circle's circumference to its diameter : http://dlmf.nist.gov/5.19.E4
${\displaystyle{\displaystyle{\displaystyle\int}}}$ : integral : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle\psi}}}$ : psi (or digamma) function : http://dlmf.nist.gov/5.2#E2
${\displaystyle{\displaystyle{\displaystyle\mathrm{d}^{n}x}}}$ : differential : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle\Re{z}}}}$ : real part : http://dlmf.nist.gov/1.9#E2

Bibliography

Equation (19), Section 25.5 of DLMF.

URL links

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<< Formula:DLMF:25.5:E18

formula in Integral Representations

Formula:DLMF:25.5:E20 >>