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

Hurwitz zeta function