Formula:DLMF:25.11:E17

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a ζ ( s , a ) = - s ζ ( s + 1 , a ) partial-derivative a Hurwitz-zeta s a s Hurwitz-zeta s 1 a {\displaystyle\frac{\partial}{\partial a}\mathop{\zeta\/}\nolimits\!\left(s,a% \right)=-s\mathop{\zeta\/}\nolimits\!\left(s+1,a\right)}


Constraint(s)


Failed to parse(LaTeXML Invalid response ('Error fetching URL: name lookup timed out') from server 'http://latexml.mathweb.org/convert':): {\displaystyle s \neq 0,1} & a > 0 𝑎 0 {\displaystyle\realpart{a}>0}


Proof


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Differentiate
ζ ( s , a ) = n = 0 1 ( n + a ) s Hurwitz-zeta s a superscript subscript n 0 1 superscript n a s {\displaystyle\mathop{\zeta\/}\nolimits\!\left(s,a\right)=\sum_{n=0}^{\infty}% \frac{1}{(n+a)^{s}}} .



Symbols List


ζ 𝜁 {\displaystyle\zeta}  : Hurwitz zeta function : http://dlmf.nist.gov/25.11#E1

a b fragments a fragments b {\displaystyle\frac{\mathopen{\partial}\hskip 0.0pta}{\mathopen{\partial}% \hskip 0.0ptb}}  : partial derivative : http://dlmf.nist.gov/1.5#E3

a 𝑎 {\displaystyle\Re{\,\hskip 0.0pta}}  : real part : http://dlmf.nist.gov/1.9#E2

Bibliography


Equation (1), Section 25.11 of DLMF.

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