Fungrim home page

Fungrim entry: 473c36

11Un ⁣(x)Um ⁣(x)1x2dx={0,nmπ2,n=m\int_{-1}^{1} U_{n}\!\left(x\right) U_{m}\!\left(x\right) \sqrt{1 - {x}^{2}} \, dx = \begin{cases} 0, & n \ne m\\\frac{\pi}{2}, & n = m\\ \end{cases}
Assumptions:nZ0andmZ0n \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, m \in \mathbb{Z}_{\ge 0}
TeX:
\int_{-1}^{1} U_{n}\!\left(x\right) U_{m}\!\left(x\right) \sqrt{1 - {x}^{2}} \, dx = \begin{cases} 0, & n \ne m\\\frac{\pi}{2}, & n = m\\ \end{cases}

n \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, m \in \mathbb{Z}_{\ge 0}
Definitions:
Fungrim symbol Notation Short description
Integralabf(x)dx\int_{a}^{b} f(x) \, dx Integral
ChebyshevUUn ⁣(x)U_{n}\!\left(x\right) Chebyshev polynomial of the second kind
Sqrtz\sqrt{z} Principal square root
Powab{a}^{b} Power
ConstPiπ\pi The constant pi (3.14...)
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
Entry(ID("473c36"),
    Formula(Equal(Integral(Mul(Mul(ChebyshevU(n, x), ChebyshevU(m, x)), Sqrt(Sub(1, Pow(x, 2)))), For(x, -1, 1)), Cases(Tuple(0, Unequal(n, m)), Tuple(Div(ConstPi, 2), Equal(n, m))))),
    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.

2019-10-05 13:11:19.856591 UTC