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
`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))))

Fungrim entry: 0745ee

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