Formula:DLMF:25.5:E16

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formula in Integral Representations

Formula:DLMF:25.5:E17 >>

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

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle 0<\Re{s}<2}}}$, ${\displaystyle{\displaystyle{\displaystyle s\neq 1}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle\zeta}}}$ : Riemann zeta function : http://dlmf.nist.gov/25.2#E1
${\displaystyle{\displaystyle{\displaystyle\mathrm{sin}}}}$ : 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 (16), Section 25.5 of DLMF.

URL links

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formula in Integral Representations

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