Formula:DLMF:25.11:E17

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∂ ∂ ⁡ a ⁡ \HurwitzZeta ⁢ @ ⁢ s ⁢ a = - s ⁢ \HurwitzZeta ⁢ @ ⁢ s + 1 ⁢ a partial-derivative 𝑎 \HurwitzZeta @ 𝑠 𝑎 𝑠 \HurwitzZeta @ 𝑠 1 𝑎 {\displaystyle{\displaystyle{\displaystyle\frac{\partial}{\partial a}% \HurwitzZeta@{s}{a}=-s\HurwitzZeta@{s+1}{a}}}}

Constraint(s)

s ≠ 0 , 1 𝑠 0 1 {\displaystyle{\displaystyle{\displaystyle s\neq 0,1}}} &
ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle{\displaystyle\Re{a}>0}}}


Proof

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Differentiate

\HurwitzZeta ⁢ @ ⁢ s ⁢ a = ∑ n = 0 ∞ 1 ( n + a ) s \HurwitzZeta @ 𝑠 𝑎 superscript subscript 𝑛 0 1 superscript 𝑛 𝑎 𝑠 {\displaystyle{\displaystyle{\displaystyle\HurwitzZeta@{s}{a}=\sum_{n=0}^{% \infty}\frac{1}{(n+a)^{s}}}}} {\displaystyle \HurwitzZeta@{s}{a} = \sum_{n=0}^\infty \frac{1}{(n+a)^s} } .


Symbols List

& : logical and
∂ ⁡ f ∂ ⁡ x 𝑓 𝑥 {\displaystyle{\displaystyle{\displaystyle\frac{\partial f}{\partial x}}}}  : partial derivative : http://dlmf.nist.gov/1.5#E3
ζ 𝜁 {\displaystyle{\displaystyle{\displaystyle\zeta}}}  : Hurwitz zeta function : http://dlmf.nist.gov/25.11#E1
ℜ ⁡ z 𝑧 {\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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