`scipy.special._multiufuncs:assoc_legendre_p`

source: build-install/usr/lib/python3.14/site-packages/scipy/special/_multiufuncs.py

Signature

def assoc_legendre_p(n, m, z, *, branch_cut = 2, norm = False, diff_n = 0)

Summary

Associated Legendre polynomial of the first kind.

Extended Summary

Defined as

P_{n}^{m} (z) = (- 1)^{m} (1 - z^{2})^{m /2} \frac{d ^{m}}{d z ^{m}} P_{n} (z)

where $P_{n}$ is the Legendre polynomial.

This definition includes the Condon-Shortley phase $(- 1)^{m}$ .

Parameters

n : array_like of ints: Degree of the associated Legendre polynomial. Must have n >= 0.
m : array_like of ints: Order of the associated Legendre polynomial.
z : array_like: Input value.
branch_cut : array_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
norm : bool, optional: If True, compute the normalized associated Legendre polynomial. Default is False.
diff_n : int, 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

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

Examples

Evaluate :math:`P_2^1(z)` on :math:`[-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