scipy.special.assoc_legendre_p#

scipy.special.assoc_legendre_p(n, m, z, *, branch_cut=2, norm=False, diff_n=0) = <scipy.special._multiufuncs.MultiUFunc object>[source]#

Associated Legendre polynomial of the first kind.

Defined as

\[P_n^m(z) = (-1)^m (1 - z^2)^{m/2} \frac{d^m}{dz^m} P_n(z)\]

where \(P_n\) is the Legendre polynomial.

This definition includes the Condon-Shortley phase \((-1)^m\).

Parameters:

narray_like of ints: Degree of the associated Legendre polynomial. Must have n >= 0.
marray_like of ints: Order of the associated Legendre polynomial.
zarray_like: Input value.
branch_cutarray_like of ints, optional: Selects branch cut. Must be 2 (default) or 3. 2: cut on the real axis |z| > 1 3: cut on the real axis -1 < z < 1
normbool, optional: If True, compute the normalized associated Legendre polynomial. Default is False.
diff_nint, optional: A non-negative integer. Compute and return all derivatives up to order diff_n. Default is 0.

Returns:

ndarray or tuple of ndarray: Associated Legendre polynomial with diff_n derivatives.

Notes

The normalized counterpart of an (unnormalized) associated Legendre polynomial has the additional factor

\[\sqrt{\frac{(2 n + 1) (n - m)!}{2 (n + m)!}}\]

Examples

Evaluate \(P_2^1(z)\) on \([-1, 1]\):

>>> import numpy as np
>>> from scipy.special import assoc_legendre_p
>>> z = np.linspace(-1, 1, 11)
>>> expected = -3 * z * np.sqrt(1 - z**2)
>>> np.allclose(assoc_legendre_p(2, 1, z), expected)
True

Compute the normalized associated Legendre polynomial:

>>> scale = np.sqrt(5/12)
>>> np.allclose(assoc_legendre_p(2, 1, z, norm=True), scale * expected)
True