Fungrim entry: ea4754

w_{n,k} = \frac{2}{\left(1 - {\left(x_{n,k}\right)}^{2}\right) {\left(\left[ \frac{d}{d t}\, P_{n}\!\left(t\right) \right]_{t = 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(\left[ \frac{d}{d t}\, P_{n}\!\left(t\right) \right]_{t = 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
`Derivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Derivative
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`ZZBetween`	$\{a, a + 1, \ldots b\}$	Integers between a and b inclusive

Source code for this entry:

Entry(ID("ea4754"),
    Formula(Equal(GaussLegendreWeight(n, k), Div(2, Mul(Sub(1, Pow(LegendrePolynomialZero(n, k), 2)), Pow(Derivative(LegendrePolynomial(n, t), Tuple(t, LegendrePolynomialZero(n, k), 1)), 2))))),
    Variables(n, k),
    Assumptions(And(Element(n, ZZGreaterEqual(1)), Element(k, ZZBetween(1, n)))))

Fungrim entry: ea4754

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