# Fungrim entry: ea4754

$w_{n,k} = \frac{2}{\left(1 - {\left(x_{n,k}\right)}^{2}\right) {\left(P'_{n}(x_{n,k})\right)}^{2}}$
Assumptions:$n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; k \in \{1, 2, \ldots, n\}$
TeX:
w_{n,k} = \frac{2}{\left(1 - {\left(x_{n,k}\right)}^{2}\right) {\left(P'_{n}(x_{n,k})\right)}^{2}}

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; k \in \{1, 2, \ldots, n\}
Definitions:
Fungrim symbol Notation Short description
GaussLegendreWeight$w_{n,k}$ Gauss-Legendre quadrature weight
Pow${a}^{b}$ Power
LegendrePolynomialZero$x_{n,k}$ Legendre polynomial zero
ComplexDerivative$\frac{d}{d z}\, f\!\left(z\right)$ Complex derivative
LegendrePolynomial$P_{n}\!\left(z\right)$ Legendre polynomial
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
Range$\{a, a + 1, \ldots, b\}$ Integers between given endpoints
Source code for this entry:
Entry(ID("ea4754"),
Formula(Equal(GaussLegendreWeight(n, k), Div(2, Mul(Sub(1, Pow(LegendrePolynomialZero(n, k), 2)), Pow(ComplexDerivative(LegendrePolynomial(n, t), For(t, LegendrePolynomialZero(n, k), 1)), 2))))),
Variables(n, k),
Assumptions(And(Element(n, ZZGreaterEqual(1)), Element(k, Range(1, n)))))

## Topics using this entry

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

2021-03-15 19:12:00.328586 UTC