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Fungrim entry: 0745ee

zeroszCPn ⁣(z)={xn,k:k{1,2,n}}\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:nZ0n \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
ZeroszerosP(x)f ⁣(x)\mathop{\operatorname{zeros}\,}\limits_{P\left(x\right)} f\!\left(x\right) Zeros (roots) of function
LegendrePolynomialPn ⁣(z)P_{n}\!\left(z\right) Legendre polynomial
CCC\mathbb{C} Complex numbers
SetBuilder{f ⁣(x):P ⁣(x)}\left\{ f\!\left(x\right) : P\!\left(x\right) \right\} Set comprehension
LegendrePolynomialZeroxn,kx_{n,k} Legendre polynomial zero
ZZBetween{a,a+1,b}\{a, a + 1, \ldots b\} Integers between a and b inclusive
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), 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