# Fungrim entry: 83065e

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

## 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