scipy.special.legendre

scipy.special.legendre(n, monic=False)

Legendre polynomial.

Defined to be the solution of

\[\frac{d}{dx}\left[(1 - x^2)\frac{d}{dx}P_n(x)\right] + n(n + 1)P_n(x) = 0;\]

\(P_n(x)\) is a polynomial of degree \(n\).

Parameters:

n : int

Degree of the polynomial.

monic : bool, optional

If True, scale the leading coefficient to be 1. Default is False.

Returns:

P : orthopoly1d

Legendre polynomial.

Notes

The polynomials \(P_n\) are orthogonal over \([-1, 1]\) with weight function 1.

Examples

Generate the 3rd-order Legendre polynomial 1/2*(5x^3 + 0x^2 - 3x + 0):

>>> from scipy.special import legendre
>>> legendre(3)
poly1d([ 2.5,  0. , -1.5,  0. ])