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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
ZeroszerosP(x)f ⁣(x)\mathop{\operatorname{zeros}\,}\limits_{P\left(x\right)} f\!\left(x\right) Zeros (roots) of function
ChebyshevUUn ⁣(x)U_{n}\!\left(x\right) Chebyshev polynomial of the second 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("ce39ac"),
    Formula(Equal(Zeros(ChebyshevU(n, x), x, Element(x, CC)), SetBuilder(Cos(Mul(Div(k, Add(n, 1)), ConstPi)), k, Element(k, ZZBetween(1, n))))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(0))))

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2019-08-21 11:44:15.926409 UTC