# Fungrim entry: 2c26a1

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

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z}_{\ge 0}
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
Sqrt$\sqrt{z}$ Principal square root
Pow${a}^{b}$ Power
Pi$\pi$ The constant pi (3.14...)
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
Source code for this entry:
Entry(ID("2c26a1"),
Formula(Equal(Integral(Mul(Mul(ChebyshevT(n, x), ChebyshevT(m, x)), Div(1, Sqrt(Sub(1, Pow(x, 2))))), For(x, -1, 1)), Cases(Tuple(0, NotEqual(n, m)), Tuple(Pi, Equal(n, m, 0)), Tuple(Div(Pi, 2), And(Equal(n, m), NotEqual(n, 0)))))),
Variables(n, m),
Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(m, 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.

2021-03-15 19:12:00.328586 UTC