Fungrim entry: 3d25dd

$\mathop{\operatorname{solutions}\,}\limits_{x \in \mathbb{C}} \left[T_{n}\!\left(x\right) \in \left\{-1, 1\right\}\right] = \left\{ \cos\!\left(\frac{k}{n} \pi\right) : k \in \{0, 1, \ldots n\} \right\}$
Assumptions:$n \in \mathbb{Z}_{\ge 1}$
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
Solutions$\mathop{\operatorname{solutions}\,}\limits_{P\left(x\right)} Q\!\left(x\right)$ Solution set
ChebyshevT$T_{n}\!\left(x\right)$ Chebyshev polynomial of the first 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("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))))

Topics using this entry

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

2019-08-17 11:32:46.829430 UTC