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_nderivatives.
 
 - Notes - There are different conventions for the meanings of the input arguments - thetaand- phi. In SciPy- thetais the polar angle and- phiis the azimuthal angle. It is common to see the opposite convention, that is,- thetaas the azimuthal angle and- phias 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