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Fungrim entry: e36542

11Pn ⁣(x)Pm ⁣(x)dx=22n+1δ(n,m)\int_{-1}^{1} P_{n}\!\left(x\right) P_{m}\!\left(x\right) \, dx = \frac{2}{2 n + 1} \delta_{(n,m)}
Assumptions:nZ0andmZ0n \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, m \in \mathbb{Z}_{\ge 0}
TeX:
\int_{-1}^{1} P_{n}\!\left(x\right) P_{m}\!\left(x\right) \, dx = \frac{2}{2 n + 1} \delta_{(n,m)}

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
LegendrePolynomialPn ⁣(z)P_{n}\!\left(z\right) Legendre polynomial
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:
Entry(ID("e36542"),
    Formula(Equal(Integral(Mul(LegendrePolynomial(n, x), LegendrePolynomial(m, x)), For(x, -1, 1)), Mul(Div(2, Add(Mul(2, n), 1)), 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.

2019-11-19 15:10:20.037976 UTC