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

Chebyshev polynomials