Formula:DLMF:25.5:E3

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formula in Integral Representations

Formula:DLMF:25.5:E4 >>

${\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{s}=\frac{1}{(1-2^{1-s})\Gamma\left(s\right)}\int_{0}^{\infty}\frac{x^{s-1}}{{\mathrm{e}^{x}}+1}\mathrm{d}x}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle\Re{s}>0}}}$ &
${\displaystyle{\displaystyle{\displaystyle s\neq 1}}}$

Proof

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Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle\zeta}}}$ : Riemann zeta function : http://dlmf.nist.gov/25.2#E1
${\displaystyle{\displaystyle{\displaystyle\Gamma}}}$ : Euler's gamma function : http://dlmf.nist.gov/5.2#E1
${\displaystyle{\displaystyle{\displaystyle\int}}}$ : integral : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}$ : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
${\displaystyle{\displaystyle{\displaystyle\mathrm{d}^{n}x}}}$ : differential : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle\Re{z}}}}$ : real part : http://dlmf.nist.gov/1.9#E2

Bibliography

Equation (3), Section 25.5 of DLMF.

URL links

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