Fungrim entry: 26418b

a \in \mathbb{C} \;\implies\; \left(\zeta\!\left(s, a\right) \text{ is holomorphic on } s \in \left\{ t : t \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{Re}(t) < 0 \right\}\right)

TeX:

a \in \mathbb{C} \;\implies\; \left(\zeta\!\left(s, a\right) \text{ is holomorphic on } s \in \left\{ t : t \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{Re}(t) < 0 \right\}\right)

Definitions:

Fungrim symbol	Notation	Short description
`CC`	$\mathbb{C}$	Complex numbers
`IsHolomorphic`	$f(z) \text{ is holomorphic at } z = c$	Holomorphic predicate
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`Re`	$\operatorname{Re}(z)$	Real part

Source code for this entry:

Entry(ID("26418b"),
    Formula(Implies(Element(a, CC), IsHolomorphic(HurwitzZeta(s, a), ForElement(s, Set(t, ForElement(t, CC), Less(Re(t), 0)))))),
    Variables(a))

Topics using this entry

Hurwitz zeta function

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