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

Legendre polynomials