# Fungrim entry: e36542

$\int_{-1}^{1} P_{n}\!\left(x\right) P_{m}\!\left(x\right) \, dx = \frac{2}{2 n + 1} \delta_{(n,m)}$
Assumptions:$n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z}_{\ge 0}$
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
Integral$\int_{a}^{b} f(x) \, dx$ Integral
LegendrePolynomial$P_{n}\!\left(z\right)$ Legendre polynomial
KroneckerDelta$\delta_{(x,y)}$ Kronecker delta
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ 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)))))

## Topics using this entry

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

2021-03-15 19:12:00.328586 UTC