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Fungrim entry: 473c36

11Un ⁣(x)Um ⁣(x)1x2dx=π2δ(n,m)\int_{-1}^{1} U_{n}\!\left(x\right) U_{m}\!\left(x\right) \sqrt{1 - {x}^{2}} \, dx = \frac{\pi}{2} \delta_{(n,m)}
Assumptions:nZ0andmZ0n \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, m \in \mathbb{Z}_{\ge 0}
\int_{-1}^{1} U_{n}\!\left(x\right) U_{m}\!\left(x\right) \sqrt{1 - {x}^{2}} \, dx = \frac{\pi}{2} \delta_{(n,m)}

n \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, m \in \mathbb{Z}_{\ge 0}
Fungrim symbol Notation Short description
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...)
KroneckerDeltaδ(x,y)\delta_{(x,y)} Kronecker delta
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
    Formula(Equal(Integral(Mul(Mul(ChebyshevU(n, x), ChebyshevU(m, x)), Sqrt(Sub(1, Pow(x, 2)))), Tuple(x, -1, 1)), Mul(Div(ConstPi, 2), KroneckerDelta(n, m)))),
    Variables(n, m),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(m, ZZGreaterEqual(0)))))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-06-18 07:49:59.356594 UTC