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Fungrim entry: 3d25dd

solutionsxC[Tn ⁣(x){1,1}]={cos ⁣(knπ):k{0,1,n}}\mathop{\operatorname{solutions}\,}\limits_{x \in \mathbb{C}} \left[T_{n}\!\left(x\right) \in \left\{-1, 1\right\}\right] = \left\{ \cos\!\left(\frac{k}{n} \pi\right) : k \in \{0, 1, \ldots n\} \right\}
Assumptions:nZ1n \in \mathbb{Z}_{\ge 1}
TeX:
\mathop{\operatorname{solutions}\,}\limits_{x \in \mathbb{C}} \left[T_{n}\!\left(x\right) \in \left\{-1, 1\right\}\right] = \left\{ \cos\!\left(\frac{k}{n} \pi\right) : k \in \{0, 1, \ldots n\} \right\}

n \in \mathbb{Z}_{\ge 1}
Definitions:
Fungrim symbol Notation Short description
SolutionssolutionsP(x)Q ⁣(x)\mathop{\operatorname{solutions}\,}\limits_{P\left(x\right)} Q\!\left(x\right) Solution set
ChebyshevTTn ⁣(x)T_{n}\!\left(x\right) Chebyshev polynomial of the first kind
CCC\mathbb{C} Complex numbers
SetBuilder{f ⁣(x):P ⁣(x)}\left\{ f\!\left(x\right) : P\!\left(x\right) \right\} Set comprehension
ConstPiπ\pi The constant pi (3.14...)
ZZBetween{a,a+1,b}\{a, a + 1, \ldots b\} Integers between a and b inclusive
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
Entry(ID("3d25dd"),
    Formula(Equal(Solutions(Brackets(Element(ChebyshevT(n, x), Set(-1, 1))), x, Element(x, CC)), SetBuilder(Cos(Mul(Div(k, n), ConstPi)), k, Element(k, ZZBetween(0, n))))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(1))))

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

2019-08-17 11:32:46.829430 UTC