# Fungrim entry: ce39ac

$\mathop{\operatorname{zeros}\,}\limits_{x \in \mathbb{C}} U_{n}\!\left(x\right) = \left\{ \cos\!\left(\frac{k}{n + 1} \pi\right) : k \in \{1, 2, \ldots n\} \right\}$
Assumptions:$n \in \mathbb{Z}_{\ge 0}$
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
Zeros$\mathop{\operatorname{zeros}\,}\limits_{P\left(x\right)} f\!\left(x\right)$ Zeros (roots) of function
ChebyshevU$U_{n}\!\left(x\right)$ Chebyshev polynomial of the second kind
CC$\mathbb{C}$ Complex numbers
SetBuilder$\left\{ f\!\left(x\right) : P\!\left(x\right) \right\}$ Set comprehension
ConstPi$\pi$ The constant pi (3.14...)
ZZBetween$\{a, a + 1, \ldots b\}$ Integers between a and b inclusive
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ 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))))

## Topics using this entry

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

2019-08-21 11:44:15.926409 UTC