Fungrim entry: 0745ee

\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} P_{n}\!\left(z\right) = \left\{x_{n,1}, x_{n,2}, \ldots, x_{n,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,1}, x_{n,2}, \ldots, x_{n,n}\right\}

n \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`Zeros`	$\mathop{\operatorname{zeros}\,}\limits_{x \in S} f(x)$	Zeros (roots) of function
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`CC`	$\mathbb{C}$	Complex numbers
`LegendrePolynomialZero`	$x_{n,k}$	Legendre polynomial zero
`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), ForElement(z, CC)), Set(LegendrePolynomialZero(n, k), For(k, 1, n)))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(0))))

Topics using this entry

Legendre polynomials
Gaussian quadrature