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Fungrim entry: 83065e

ddaζ ⁣(s,a)=sζ ⁣(s+1,a)\frac{d}{d a}\, \zeta\!\left(s, a\right) = -s \zeta\!\left(s + 1, a\right)
Assumptions:sC  and  s{0,1}  and  aC  and  Re(a)>0s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; s \notin \left\{0, 1\right\} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) > 0
TeX:
\frac{d}{d a}\, \zeta\!\left(s, a\right) = -s \zeta\!\left(s + 1, a\right)

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; s \notin \left\{0, 1\right\} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) > 0
Definitions:
Fungrim symbol Notation Short description
ComplexDerivativeddzf ⁣(z)\frac{d}{d z}\, f\!\left(z\right) Complex derivative
HurwitzZetaζ ⁣(s,a)\zeta\!\left(s, a\right) Hurwitz zeta function
CCC\mathbb{C} Complex numbers
ReRe(z)\operatorname{Re}(z) Real part
Source code for this entry:
Entry(ID("83065e"),
    Formula(Equal(ComplexDerivative(HurwitzZeta(s, a), For(a, a)), Neg(Mul(s, HurwitzZeta(Add(s, 1), a))))),
    Variables(s, a),
    Assumptions(And(Element(s, CC), NotElement(s, Set(0, 1)), Element(a, CC), Greater(Re(a), 0))))

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2021-03-15 19:12:00.328586 UTC