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Fungrim entry: 0745ee

zeroszCPn ⁣(z)={xn,1,,xn,n}\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} P_{n}\!\left(z\right) = \left\{x_{n,1}, \ldots, x_{n,n}\right\}
Assumptions:nZ0n \in \mathbb{Z}_{\ge 0}
TeX:
\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} P_{n}\!\left(z\right) = \left\{x_{n,1}, \ldots, x_{n,n}\right\}

n \in \mathbb{Z}_{\ge 0}
Definitions:
Fungrim symbol Notation Short description
ZeroszerosxSf(x)\mathop{\operatorname{zeros}\,}\limits_{x \in S} f(x) Zeros (roots) of function
LegendrePolynomialPn ⁣(z)P_{n}\!\left(z\right) Legendre polynomial
CCC\mathbb{C} Complex numbers
LegendrePolynomialZeroxn,kx_{n,k} Legendre polynomial zero
ZZGreaterEqualZn\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))))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-10-05 13:11:19.856591 UTC