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Fungrim entry: 7a7d1d

zerosxCTn ⁣(x)={cos ⁣(2k12nπ):k{1,2,,n}}\mathop{\operatorname{zeros}\,}\limits_{x \in \mathbb{C}} T_{n}\!\left(x\right) = \left\{ \cos\!\left(\frac{2 k - 1}{2 n} \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}} T_{n}\!\left(x\right) = \left\{ \cos\!\left(\frac{2 k - 1}{2 n} \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
ChebyshevTTn ⁣(x)T_{n}\!\left(x\right) Chebyshev polynomial of the first 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("7a7d1d"),
    Formula(Equal(Zeros(ChebyshevT(n, x), ForElement(x, CC)), Set(Cos(Mul(Div(Sub(Mul(2, k), 1), Mul(2, n)), Pi)), ForElement(k, Range(1, n))))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(0))))

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

2019-11-19 15:10:20.037976 UTC