# Fungrim entry: 40c3e2

$\frac{d^{r}}{{d a}^{r}} \zeta\!\left(s, a\right) = \left(1 - s - r\right)_{r} \zeta\!\left(s + r, a\right)$
Assumptions:$s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; s \ne 1 \;\mathbin{\operatorname{and}}\; s + r \ne 1 \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) > 0 \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0}$
TeX:
\frac{d^{r}}{{d a}^{r}} \zeta\!\left(s, a\right) = \left(1 - s - r\right)_{r} \zeta\!\left(s + r, a\right)

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; s \ne 1 \;\mathbin{\operatorname{and}}\; s + r \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
RisingFactorial$\left(z\right)_{k}$ Rising factorial
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("40c3e2"),
Formula(Equal(ComplexDerivative(HurwitzZeta(s, a), For(a, a, r)), Mul(RisingFactorial(Sub(Sub(1, s), r), r), HurwitzZeta(Add(s, r), a)))),
Variables(s, a, r),
Assumptions(And(Element(s, CC), NotEqual(s, 1), NotEqual(Add(s, r), 1), Element(a, CC), Greater(Re(a), 0), Element(r, ZZGreaterEqual(0)))))

## Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC