# Fungrim entry: 3d77ab

$\int_{-1}^{1} T_{n}\!\left(x\right) {x}^{m} \frac{1}{\sqrt{1 - {x}^{2}}} \, dx = 0$
Assumptions:$n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; m \in \{0, 1, \ldots, n - 1\}$
TeX:
\int_{-1}^{1} T_{n}\!\left(x\right) {x}^{m} \frac{1}{\sqrt{1 - {x}^{2}}} \, dx = 0

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; m \in \{0, 1, \ldots, n - 1\}
Definitions:
Fungrim symbol Notation Short description
Integral$\int_{a}^{b} f(x) \, dx$ Integral
ChebyshevT$T_{n}\!\left(x\right)$ Chebyshev polynomial of the first kind
Pow${a}^{b}$ Power
Sqrt$\sqrt{z}$ Principal square root
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
Range$\{a, a + 1, \ldots, b\}$ Integers between given endpoints
Source code for this entry:
Entry(ID("3d77ab"),
Formula(Equal(Integral(Mul(Mul(ChebyshevT(n, x), Pow(x, m)), Div(1, Sqrt(Sub(1, Pow(x, 2))))), For(x, -1, 1)), 0)),
Variables(n, m),
Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(m, Range(0, Sub(n, 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