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Fungrim entry: ea4754

wn,k=2(1(xn,k)2)(Pn(xn,k))2w_{n,k} = \frac{2}{\left(1 - {\left(x_{n,k}\right)}^{2}\right) {\left(P'_{n}(x_{n,k})\right)}^{2}}
Assumptions:nZ1andk{1,2,,n}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
GaussLegendreWeightwn,kw_{n,k} Gauss-Legendre quadrature weight
Powab{a}^{b} Power
LegendrePolynomialZeroxn,kx_{n,k} Legendre polynomial zero
ComplexDerivativeddzf ⁣(z)\frac{d}{d z}\, f\!\left(z\right) Complex derivative
LegendrePolynomialPn ⁣(z)P_{n}\!\left(z\right) Legendre polynomial
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Range{a,a+1,,b}\{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)))))

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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