Module « scipy.special »

Fonction sph_harm - module scipy.special

Signature de la fonction sph_harm

Description

sph_harm.__doc__

sph_harm(x1, x2, x3, x4, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

sph_harm(m, n, theta, phi)

Compute spherical harmonics.

The spherical harmonics are defined as

.. math::

    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 :math:`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 by ``l`` (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 :math:`Y^m_n` sampled at ``theta`` and ``phi``.

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

.. math::

    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).

References
----------
.. [1] Digital Library of Mathematical Functions, 14.30.
       https://dlmf.nist.gov/14.30
.. [2] https://en.wikipedia.org/wiki/Spherical_harmonics#Condon.E2.80.93Shortley_phase