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Fungrim entry: 4cfeac

Pn ⁣(z)=12nn![dndtn(t21)n]t=zP_{n}\!\left(z\right) = \frac{1}{{2}^{n} n !} \left[ \frac{d^{n}}{{d t}^{n}} {\left({t}^{2} - 1\right)}^{n} \right]_{t = z}
Assumptions:nZ0n \in \mathbb{Z}_{\ge 0}
Alternative assumptions:zCz \in \mathbb{C}
TeX:
P_{n}\!\left(z\right) = \frac{1}{{2}^{n} n !} \left[ \frac{d^{n}}{{d t}^{n}} {\left({t}^{2} - 1\right)}^{n} \right]_{t = z}

n \in \mathbb{Z}_{\ge 0}

z \in \mathbb{C}
Definitions:
Fungrim symbol Notation Short description
LegendrePolynomialPn ⁣(z)P_{n}\!\left(z\right) Legendre polynomial
Powab{a}^{b} Power
Factorialn!n ! Factorial
ComplexDerivativeddzf ⁣(z)\frac{d}{d z}\, f\!\left(z\right) Complex derivative
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
CCC\mathbb{C} Complex numbers
Source code for this entry:
Entry(ID("4cfeac"),
    Formula(Equal(LegendrePolynomial(n, z), Mul(Div(1, Mul(Pow(2, n), Factorial(n))), ComplexDerivative(Pow(Sub(Pow(t, 2), 1), n), For(t, z, n))))),
    Variables(n, z),
    Assumptions(And(Element(n, ZZGreaterEqual(0))), Element(z, CC)))

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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