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Fungrim entry: db2b0a

solutionsxC[Tn ⁣(x)=1]={cos ⁣(2k1nπ):k{1,2,,n+12}}\mathop{\operatorname{solutions}\,}\limits_{x \in \mathbb{C}} \left[T_{n}\!\left(x\right) = -1\right] = \left\{ \cos\!\left(\frac{2 k - 1}{n} \pi\right) : k \in \{1, 2, \ldots, \left\lfloor \frac{n + 1}{2} \right\rfloor\} \right\}
Assumptions:nZ1n \in \mathbb{Z}_{\ge 1}
TeX:
\mathop{\operatorname{solutions}\,}\limits_{x \in \mathbb{C}} \left[T_{n}\!\left(x\right) = -1\right] = \left\{ \cos\!\left(\frac{2 k - 1}{n} \pi\right) : k \in \{1, 2, \ldots, \left\lfloor \frac{n + 1}{2} \right\rfloor\} \right\}

n \in \mathbb{Z}_{\ge 1}
Definitions:
Fungrim symbol Notation Short description
SolutionssolutionsxSQ(x)\mathop{\operatorname{solutions}\,}\limits_{x \in S} Q(x) Solution set
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("db2b0a"),
    Formula(Equal(Solutions(Brackets(Equal(ChebyshevT(n, x), -1)), ForElement(x, CC)), Set(Cos(Mul(Div(Sub(Mul(2, k), 1), n), Pi)), ForElement(k, Range(1, Floor(Div(Add(n, 1), 2))))))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(1))))

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2019-12-11 23:01:54.699850 UTC