scipy.special.sph_harm#
- scipy.special.sph_harm(m, n, theta, phi, out=None) = <ufunc 'sph_harm'>#
- Compute spherical harmonics. - The spherical harmonics are defined as \[Y^m_n(\theta,\phi) = \sqrt{\frac{2n+1}{4\pi} \frac{(n-m)!}{(n+m)!}} e^{i m \theta} P^m_n(\cos(\phi))\]- where \(P_n^m\) are the associated Legendre functions; see - lpmv.- Deprecated since version 1.15.0: This function is deprecated and will be removed in SciPy 1.17.0. Please use - scipy.special.sph_harm_yinstead.- Parameters:
- marray_like
- Order of the harmonic (int); must have - |m| <= n.
- narray_like
- Degree of the harmonic (int); must have - n >= 0. This is often denoted by- l(lower case L) in descriptions of spherical harmonics.
- thetaarray_like
- Azimuthal (longitudinal) coordinate; must be in - [0, 2*pi].
- phiarray_like
- Polar (colatitudinal) coordinate; must be in - [0, pi].
- outndarray, optional
- Optional output array for the function values 
 
- Returns:
- y_mncomplex scalar or ndarray
- The harmonic \(Y^m_n\) sampled at - thetaand- phi.
 
 - Notes - There are different conventions for the meanings of the input arguments - thetaand- phi. In SciPy- thetais the azimuthal angle and- phiis the polar angle. It is common to see the opposite convention, that is,- thetaas the polar angle and- phias the azimuthal angle.- Note that SciPy’s spherical harmonics include the Condon-Shortley phase [2] because it is part of - lpmv.- With SciPy’s conventions, the first several spherical harmonics are \[\begin{split}Y_0^0(\theta, \phi) &= \frac{1}{2} \sqrt{\frac{1}{\pi}} \\ Y_1^{-1}(\theta, \phi) &= \frac{1}{2} \sqrt{\frac{3}{2\pi}} e^{-i\theta} \sin(\phi) \\ Y_1^0(\theta, \phi) &= \frac{1}{2} \sqrt{\frac{3}{\pi}} \cos(\phi) \\ Y_1^1(\theta, \phi) &= -\frac{1}{2} \sqrt{\frac{3}{2\pi}} e^{i\theta} \sin(\phi).\end{split}\]- References [1]- Digital Library of Mathematical Functions, 14.30. https://dlmf.nist.gov/14.30