Formula:DLMF:25.5:E6

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Failed to parse (unknown function "\RiemannZeta"): {\displaystyle {\displaystyle \RiemannZeta@{s} = \frac{1}{2} + \frac{1}{s-1} + \frac{1}{\EulerGamma@{s}} \int_0^\infty \left( \frac{1}{\expe^x-1} - \frac{1}{x} + \frac{1}{2} \right) \frac{x^{s-1}}{\expe^x} \diff{x} }}

Constraint(s)

Failed to parse (unknown function "\realpart"): {\displaystyle {\displaystyle \realpart{s} > -1}} &


Proof

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Comes from

Failed to parse (unknown function "\RiemannZeta"): {\displaystyle {\displaystyle \RiemannZeta@{s} = \frac{1}{\EulerGamma@{s}} \int_0^\infty \frac{x^{s-1}}{\expe^x-1} \diff{x} }}
by using the identity Failed to parse (unknown function "\expe"): {\displaystyle {\displaystyle \expe^{-x} = (1-\expe^{-x})/(\expe^x-1)}}
in the integral Failed to parse (unknown function "\EulerGamma"): {\displaystyle {\displaystyle \EulerGamma@{s} = \int_0^{\infty} \expe^{-x} x^{s-1} \diff{x}}}
(see
Failed to parse (unknown function "\EulerGamma"): {\displaystyle {\displaystyle \EulerGamma@{z} = \int_0^\infty \expe^{-t} t^{z-1} \diff{t} }}
) together with

Failed to parse (unknown function "\EulerGamma"): {\displaystyle {\displaystyle \EulerGamma@{z+1} = z \EulerGamma@{z} }} .


Symbols List

& : logical and
 : Riemann zeta function : http://dlmf.nist.gov/25.2#E1
 : Euler's gamma function : http://dlmf.nist.gov/5.2#E1
 : integral : http://dlmf.nist.gov/1.4#iv
 : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
 : differential : http://dlmf.nist.gov/1.4#iv
 : real part : http://dlmf.nist.gov/1.9#E2

Bibliography

Equation (6), Section 25.5 of DLMF.

URL links

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