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
`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("db2b0a"),
    Formula(Equal(Solutions(Brackets(Equal(ChebyshevT(n, x), -1)), x, Element(x, CC)), SetBuilder(Cos(Mul(Div(Sub(Mul(2, k), 1), n), ConstPi)), k, Element(k, ZZBetween(1, Floor(Div(Add(n, 1), 2))))))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(1))))

Topics using this entry

Chebyshev polynomials