scipy.special.sph_harm¶

scipy.special.
sph_harm
(m, n, theta, phi) = <ufunc 'sph_harm'>¶ Compute spherical harmonics.
The spherical harmonics are defined as
\[Y^m_n(\theta,\phi) = \sqrt{\frac{2n+1}{4\pi} \frac{(nm)!}{(n+m)!}} e^{i m \theta} P^m_n(\cos(\phi))\]where \(P_n^m\) are the associated Legendre functions; see
lpmv
.Parameters: m : array_like
Order of the harmonic (int); must have
m <= n
.n : array_like
Degree of the harmonic (int); must have
n >= 0
. This is often denoted byl
(lower case L) in descriptions of spherical harmonics.theta : array_like
Azimuthal (longitudinal) coordinate; must be in
[0, 2*pi]
.phi : array_like
Polar (colatitudinal) coordinate; must be in
[0, pi]
.Returns: y_mn : complex float
The harmonic \(Y^m_n\) sampled at
theta
andphi
.Notes
There are different conventions for the meanings of the input arguments
theta
andphi
. In SciPytheta
is the azimuthal angle andphi
is the polar angle. It is common to see the opposite convention, that is,theta
as the polar angle andphi
as the azimuthal angle.Note that SciPy’s spherical harmonics include the CondonShortley phase [R521] 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
[R520] Digital Library of Mathematical Functions, 14.30. http://dlmf.nist.gov/14.30 [R521] (1, 2) https://en.wikipedia.org/wiki/Spherical_harmonics#Condon.E2.80.93Shortley_phase