Formula:DLMF:25.11:E27

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Failed to parse (unknown function "\HurwitzZeta"): {\displaystyle {\displaystyle \HurwitzZeta@{s}{a} = \frac{1}{2} a^{-s} + \frac{a^{1-s}}{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^{ax}} \diff{x} }}

Constraint(s)

Failed to parse (unknown function "\realpart"): {\displaystyle {\displaystyle \realpart{s} > -1}} &
&
Failed to parse (unknown function "\realpart"): {\displaystyle {\displaystyle \realpart{a} > 0}}


Proof

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Argue as in

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} }} .


Symbols List

& : logical and
 : Hurwitz zeta function : http://dlmf.nist.gov/25.11#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 (27), Section 25.11 of DLMF.

URL links

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