Fungrim entry: d0d03b

\frac{d^{r}}{{d s}^{r}} \zeta\!\left(s, a\right) = \zeta^{(r)}\!\left(s, a\right)

Assumptions:

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; s \ne 1 \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) > 0 \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0}

TeX:

\frac{d^{r}}{{d s}^{r}} \zeta\!\left(s, a\right) = \zeta^{(r)}\!\left(s, a\right)

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; s \ne 1 \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) > 0 \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}(z)$	Real part
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("d0d03b"),
    Formula(Equal(ComplexDerivative(HurwitzZeta(s, a), For(s, s, r)), HurwitzZeta(s, a, r))),
    Variables(s, a, r),
    Assumptions(And(Element(s, CC), NotEqual(s, 1), Element(a, CC), Greater(Re(a), 0), Element(r, ZZGreaterEqual(0)))))

Topics using this entry

Hurwitz zeta function