scipy.special.sph_harm_y#

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

Spherical harmonics. They are defined as

\[Y_n^m(\theta,\phi) = \sqrt{\frac{2 n + 1}{4 \pi} \frac{(n - m)!}{(n + m)!}} P_n^m(\cos(\theta)) e^{i m \phi}\]

where \(P_n^m\) are the (unnormalized) associated Legendre polynomials.

Parameters:
nArrayLike[int]

Degree of the harmonic. Must have n >= 0. This is often denoted by l (lower case L) in descriptions of spherical harmonics.

mArrayLike[int]

Order of the harmonic.

thetaArrayLike[float]

Polar (colatitudinal) coordinate; must be in [0, pi].

phiArrayLike[float]

Azimuthal (longitudinal) coordinate; must be in [0, 2*pi].

diff_nOptional[int]

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

Returns:
yndarray[complex] or tuple[ndarray[complex]]

Spherical harmonics with diff_n derivatives.

Notes

There are different conventions for the meanings of the input arguments theta and phi. In SciPy theta is the polar angle and phi is the azimuthal angle. It is common to see the opposite convention, that is, theta as the azimuthal angle and phi as the polar angle.

Note that SciPy’s spherical harmonics include the Condon-Shortley phase [2] because it is part of sph_legendre_p.

With SciPy’s conventions, the first several spherical harmonics are

\[\begin{split}Y_0^0(\theta, \phi) &= \frac{1}{2} \sqrt{\frac{1}{\pi}} \\ Y_1^{-1}(\theta, \phi) &= \frac{1}{2} \sqrt{\frac{3}{2\pi}} e^{-i\phi} \sin(\theta) \\ Y_1^0(\theta, \phi) &= \frac{1}{2} \sqrt{\frac{3}{\pi}} \cos(\theta) \\ Y_1^1(\theta, \phi) &= -\frac{1}{2} \sqrt{\frac{3}{2\pi}} e^{i\phi} \sin(\theta).\end{split}\]

References

[1]

Digital Library of Mathematical Functions, 14.30. https://dlmf.nist.gov/14.30

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