Fungrim entry: 473c36

\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:

n \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
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`ChebyshevU`	$U_{n}\!\left(x\right)$	Chebyshev polynomial of the second 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("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, NotEqual(n, m)), Tuple(Div(Pi, 2), Equal(n, m))))),
    Variables(n, m),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(m, ZZGreaterEqual(0)))))

Topics using this entry

Chebyshev polynomials