Formula:DLMF:25.11:E28

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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} + \sum_{k=1}^n \frac{\EulerGamma@{s+2k-1}}{\EulerGamma@{s}} \frac{\BernoulliB{2k}}{(2k)!} a^{-2k-s+1} + \frac{1}{\EulerGamma@{s}} \int_0^\infty \left( \frac{1}{\expe^x-1} - \frac{1}{x} + \frac{1}{2} - \sum_{k=1}^n \frac{\BernoulliB{2k}}{(2k)!} x^{2k-1} \right) x^{s-1} \expe^{-ax} \diff{x} }}

Constraint(s)

Failed to parse (unknown function "\realpart"): {\displaystyle {\displaystyle \realpart{s} > -(2n+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} + \sum_{m=1}^n \frac{\BernoulliB{2m}}{(2m)!} \frac{\EulerGamma@{s+2m-1}}{\EulerGamma@{s}} + \frac{1}{\EulerGamma@{s}} \int_0^\infty \left( \frac{1}{\expe^x-1} - \frac{1}{x} + \frac{1}{2} - \sum_{m=1}^n \frac{\BernoulliB{2m}}{(2m)!} x^{2m-1} \right) \frac{x^{s-1}}{\expe^x} \diff{x} }} .


Symbols List

& : logical and
 : Hurwitz zeta function : http://dlmf.nist.gov/25.11#E1
 : sum : http://drmf.wmflabs.org/wiki/Definition:sum
 : Euler's gamma function : http://dlmf.nist.gov/5.2#E1
 : Bernoulli polynomial : http://dlmf.nist.gov/24.2#i
 : 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 (28), Section 25.11 of DLMF.

URL links

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