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)))

Topics using this entry

Legendre polynomials