# Fungrim entry: db2b0a

$\mathop{\operatorname{solutions}\,}\limits_{x \in \mathbb{C}} \left[T_{n}\!\left(x\right) = -1\right] = \left\{ \cos\!\left(\frac{2 k - 1}{n} \pi\right) : k \in \{1, 2, \ldots, \left\lfloor \frac{n + 1}{2} \right\rfloor\} \right\}$
Assumptions:$n \in \mathbb{Z}_{\ge 1}$
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
Solutions$\mathop{\operatorname{solutions}\,}\limits_{x \in S} Q(x)$ Solution set
ChebyshevT$T_{n}\!\left(x\right)$ Chebyshev polynomial of the first kind
CC$\mathbb{C}$ Complex numbers
Cos$\cos(z)$ Cosine
Pi$\pi$ The constant pi (3.14...)
Range$\{a, a + 1, \ldots, b\}$ Integers between given endpoints
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ 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))))

## Topics using this entry

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

2021-03-15 19:12:00.328586 UTC