Fungrim entry: 4cfeac

$P_{n}\!\left(z\right) = \frac{1}{{2}^{n} n !} \left[ \frac{d^{n}}{{d t}^{n}} {\left({t}^{2} - 1\right)}^{n} \right]_{t = z}$
Assumptions:$n \in \mathbb{Z}_{\ge 0}$
Alternative assumptions:$z \in \mathbb{C}$
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
LegendrePolynomial$P_{n}\!\left(z\right)$ Legendre polynomial
Pow${a}^{b}$ Power
Factorial$n !$ Factorial
ComplexDerivative$\frac{d}{d z}\, f\!\left(z\right)$ Complex derivative
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
CC$\mathbb{C}$ 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.

2021-03-15 19:12:00.328586 UTC