legendre_polynomial_p

→Tensor

legendre_polynomial_p(x: Tensor, n: int)

source

Legendre polynomial $P_n(x)$ .

The Legendre polynomials are orthogonal on $[-1, 1]$ with uniform weight, and serve as the angular eigenfunctions of the Laplacian on the sphere (spherical harmonics with $m = 0$ ) and as a basis for Gauss-Legendre quadrature.

Parameters

xTensor

Argument tensor; any floating-point dtype. Customarily evaluated on

[-1, 1]

nint

Non-negative polynomial degree.

Returns

Tensor

$P_n(x)$ element-wise, same shape and dtype as x.

Notes

Bonnet recurrence:

P_0(x) = 1, \quad P_1(x) = x, \quad (k+1)\, P_{k+1}(x) = (2k + 1)\, x\, P_k(x) - k\, P_{k-1}(x).

On $[-1, 1]$ , $|P_n(x)| \le 1$ with boundary values $P_n(1) = 1$ , $P_n(-1) = (-1)^n$ . Raises ValueError for n < 0.

Examples

>>> import lucid
>>> from lucid.special import legendre_polynomial_p
>>> legendre_polynomial_p(lucid.tensor([-1.0, 0.0, 0.5, 1.0]), n=3)
Tensor([-1.0000, 0.0000, -0.4375, 1.0000])