Formula:DLMF:25.11:E34

<< Formula:DLMF:25.11:E32

formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E35 >>

${\displaystyle{\displaystyle{\displaystyle n\int_{0}^{a}\HurwitzZeta^{\prime}@{1-n}{x}\mathrm{d}x=\HurwitzZeta^{\prime}@{-n}{a}-\RiemannZeta^{\prime}@{-n}+\frac{\BernoulliB{n+1}-\BernoulliB{n+1}@{a}}{n(n+1)}}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle n=1,2,\dots}}}$ &
${\displaystyle{\displaystyle{\displaystyle\Re{a}>0}}}$

Note(s)

primes on ${\displaystyle{\displaystyle{\displaystyle\HurwitzZeta}}}$ denote derivatives with respect to ${\displaystyle{\displaystyle{\displaystyle s}}}$

Proof

We ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space.

Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle\int}}}$ : integral : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle\zeta}}}$ : Hurwitz zeta function : http://dlmf.nist.gov/25.11#E1
${\displaystyle{\displaystyle{\displaystyle\mathrm{d}^{n}x}}}$ : differential : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle\zeta}}}$ : Riemann zeta function : http://dlmf.nist.gov/25.2#E1
${\displaystyle{\displaystyle{\displaystyle B_{n}}}}$ : Bernoulli polynomial : http://dlmf.nist.gov/24.2#i
${\displaystyle{\displaystyle{\displaystyle\Re{z}}}}$ : real part : http://dlmf.nist.gov/1.9#E2

Bibliography

Equation (34), Section 25.11 of DLMF.

URL links

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<< Formula:DLMF:25.11:E32

formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E35 >>