# Fungrim entry: 0745ee

$\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} P_{n}\!\left(z\right) = \left\{ x_{n,k} : k \in \{1, 2, \ldots n\} \right\}$
Assumptions:$n \in \mathbb{Z}_{\ge 0}$
TeX:
\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} P_{n}\!\left(z\right) = \left\{ x_{n,k} : k \in \{1, 2, \ldots n\} \right\}

n \in \mathbb{Z}_{\ge 0}
Definitions:
Fungrim symbol Notation Short description
Zeros$\mathop{\operatorname{zeros}\,}\limits_{P\left(x\right)} f\!\left(x\right)$ Zeros (roots) of function
LegendrePolynomial$P_{n}\!\left(z\right)$ Legendre polynomial
CC$\mathbb{C}$ Complex numbers
SetBuilder$\left\{ f\!\left(x\right) : P\!\left(x\right) \right\}$ Set comprehension
LegendrePolynomialZero$x_{n,k}$ Legendre polynomial zero
ZZBetween$\{a, a + 1, \ldots b\}$ Integers between a and b inclusive
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
Source code for this entry:
Entry(ID("0745ee"),
Formula(Equal(Zeros(LegendrePolynomial(n, z), z, Element(z, CC)), SetBuilder(LegendrePolynomialZero(n, k), k, Element(k, ZZBetween(1, n))))),
Variables(n),
Assumptions(Element(n, 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.

2019-08-21 11:44:15.926409 UTC