Formula:DLMF:25.11:E26

<< Formula:DLMF:25.11:E25

formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E27 >>

${\displaystyle{\displaystyle{\displaystyle\HurwitzZeta@{s}{a}=-s\int_{-a}^{\infty}\frac{x-\left\lfloor x\right\rfloor-\frac{1}{2}}{(x+a)^{s+1}}\mathrm{d}x}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle-1<\Re{s}<0}}}$ &
${\displaystyle{\displaystyle{\displaystyle 0<a\leq 1}}}$

Proof

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Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle\zeta}}}$ : Hurwitz zeta function : http://dlmf.nist.gov/25.11#E1
${\displaystyle{\displaystyle{\displaystyle\int}}}$ : integral : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle\left\lfloor a\right\rfloor}}}$ : floor : http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r16
${\displaystyle{\displaystyle{\displaystyle\mathrm{d}^{n}x}}}$ : differential : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle\Re{z}}}}$ : real part : http://dlmf.nist.gov/1.9#E2

Bibliography

Equation (26), Section 25.11 of DLMF.

URL links

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<< Formula:DLMF:25.11:E25

formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E27 >>