Fungrim entry: 93e149

s \in \mathbb{C} \setminus \left\{1\right\} \;\implies\; \left(\zeta\!\left(s, a\right) \text{ is holomorphic on } a \in \mathbb{C} \setminus \left(-\infty, 0\right]\right)

TeX:

s \in \mathbb{C} \setminus \left\{1\right\} \;\implies\; \left(\zeta\!\left(s, a\right) \text{ is holomorphic on } a \in \mathbb{C} \setminus \left(-\infty, 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
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("93e149"),
    Formula(Implies(Element(s, SetMinus(CC, Set(1))), IsHolomorphic(HurwitzZeta(s, a), ForElement(a, SetMinus(CC, OpenClosedInterval(Neg(Infinity), 0)))))),
    Variables(s))

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