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Fungrim entry: 27688e

(1z2)Pn(z)2zPn(z)+n(n+1)Pn ⁣(z)=0\left(1 - {z}^{2}\right) P''_{n}(z) - 2 z P'_{n}(z) + n \left(n + 1\right) P_{n}\!\left(z\right) = 0
Assumptions:nZ0  and  zCn \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}
\left(1 - {z}^{2}\right) P''_{n}(z) - 2 z P'_{n}(z) + n \left(n + 1\right) P_{n}\!\left(z\right) = 0

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}
Fungrim symbol Notation Short description
Powab{a}^{b} Power
ComplexDerivativeddzf ⁣(z)\frac{d}{d z}\, f\!\left(z\right) Complex derivative
LegendrePolynomialPn ⁣(z)P_{n}\!\left(z\right) Legendre polynomial
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
CCC\mathbb{C} Complex numbers
Source code for this entry:
    Formula(Equal(Add(Sub(Mul(Sub(1, Pow(z, 2)), ComplexDerivative(LegendrePolynomial(n, z), For(z, z, 2))), Mul(Mul(2, z), ComplexDerivative(LegendrePolynomial(n, z), For(z, z, 1)))), Mul(Mul(n, Add(n, 1)), LegendrePolynomial(n, z))), 0)),
    Variables(n, z),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(z, CC))))

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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