# Fungrim entry: 27688e

$\left(1 - {z}^{2}\right) P''_{n}(z) - 2 z P'_{n}(z) + n \left(n + 1\right) P_{n}\!\left(z\right) = 0$
Assumptions:$n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}$
TeX:
\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}
Definitions:
Fungrim symbol Notation Short description
Pow${a}^{b}$ Power
ComplexDerivative$\frac{d}{d z}\, f\!\left(z\right)$ Complex derivative
LegendrePolynomial$P_{n}\!\left(z\right)$ Legendre polynomial
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
CC$\mathbb{C}$ Complex numbers
Source code for this entry:
Entry(ID("27688e"),
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))))

## 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