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Fungrim entry: ce39ac

zerosxCUn ⁣(x)={cos ⁣(kn+1π):k{1,2,,n}}\mathop{\operatorname{zeros}\,}\limits_{x \in \mathbb{C}} U_{n}\!\left(x\right) = \left\{ \cos\!\left(\frac{k}{n + 1} \pi\right) : k \in \{1, 2, \ldots, n\} \right\}
Assumptions:nZ0n \in \mathbb{Z}_{\ge 0}
TeX:
\mathop{\operatorname{zeros}\,}\limits_{x \in \mathbb{C}} U_{n}\!\left(x\right) = \left\{ \cos\!\left(\frac{k}{n + 1} \pi\right) : k \in \{1, 2, \ldots, n\} \right\}

n \in \mathbb{Z}_{\ge 0}
Definitions:
Fungrim symbol Notation Short description
ZeroszerosxSf(x)\mathop{\operatorname{zeros}\,}\limits_{x \in S} f(x) Zeros (roots) of function
ChebyshevUUn ⁣(x)U_{n}\!\left(x\right) Chebyshev polynomial of the second kind
CCC\mathbb{C} Complex numbers
Piπ\pi The constant pi (3.14...)
Range{a,a+1,,b}\{a, a + 1, \ldots, b\} Integers between given endpoints
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
Entry(ID("ce39ac"),
    Formula(Equal(Zeros(ChebyshevU(n, x), ForElement(x, CC)), Set(Cos(Mul(Div(k, Add(n, 1)), Pi)), ForElement(k, Range(1, n))))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(0))))

Topics using this entry

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

2019-12-30 15:00:46.909060 UTC