scipy.special.sph_legendre_p — SciPy v1.15.2 Manual (original) (raw)

scipy.special.sph_legendre_p(n, m, theta, *, diff_n=0) = <scipy.special._multiufuncs.MultiUFunc object>[source]#

Spherical Legendre polynomial of the first kind.

Parameters:

nArrayLike[int]

Degree of the spherical Legendre polynomial. Must have n >= 0.

mArrayLike[int]

Order of the spherical Legendre polynomial.

thetaArrayLike[float]

Input value.

diff_nOptional[int]

A non-negative integer. Compute and return all derivatives up to order diff_n. Default is 0.

Returns:

pndarray or tuple[ndarray]

Spherical Legendre polynomial with diff_n derivatives.

Notes

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

\[\sqrt{\frac{(2 n + 1) (n - m)!}{4 \pi (n + m)!}}\]

It is the same as the spherical harmonic \(Y_{n}^{m}(\theta, \phi)\)with \(\phi = 0\).