Formula:DLMF:25.11:E17

<< Formula:DLMF:25.11:E16

formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E18 >>

${\displaystyle{\displaystyle{\displaystyle\frac{\partial}{\partial a}\HurwitzZeta@{s}{a}=-s\HurwitzZeta@{s+1}{a}}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle s\neq 0,1}}}$ &
${\displaystyle{\displaystyle{\displaystyle\Re{a}>0}}}$

Proof

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Differentiate

${\displaystyle{\displaystyle{\displaystyle\HurwitzZeta@{s}{a}=\sum_{n=0}^{\infty}\frac{1}{(n+a)^{s}}}}}$.

Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle\frac{\partial f}{\partial x}}}}$ : partial derivative : http://dlmf.nist.gov/1.5#E3
${\displaystyle{\displaystyle{\displaystyle\zeta}}}$ : Hurwitz zeta function : http://dlmf.nist.gov/25.11#E1
${\displaystyle{\displaystyle{\displaystyle\Re{z}}}}$ : real part : http://dlmf.nist.gov/1.9#E2

Bibliography

Equation (17), Section 25.11 of DLMF.

URL links

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<< Formula:DLMF:25.11:E16

formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E18 >>