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Fungrim entry: 93e149

(sC{1})    (ζ ⁣(s,a) is holomorphic on aC(,0])\left(s \in \mathbb{C} \setminus \left\{1\right\}\right) \implies \left(\zeta\!\left(s, a\right) \text{ is holomorphic on } a \in \mathbb{C} \setminus \left(-\infty, 0\right]\right)
TeX:
\left(s \in \mathbb{C} \setminus \left\{1\right\}\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
CCC\mathbb{C} Complex numbers
IsHolomorphicf(z) is holomorphic at z=cf(z) \text{ is holomorphic at } z = c Holomorphic predicate
HurwitzZetaζ ⁣(s,a)\zeta\!\left(s, a\right) Hurwitz zeta function
OpenClosedInterval(a,b]\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

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

2020-04-08 16:14:44.404316 UTC