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Module « scipy.special »

Fonction sph_harm_y - module scipy.special

Signature de la fonction sph_harm_y

def sph_harm_y(*args, **kwargs) 

Description

help(scipy.special.sph_harm_y)

sph_harm_y(n, m, theta, phi, *, diff_n=0)

    Spherical harmonics. They are defined as

    .. math::

        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 :math:`P_n^m` are the (unnormalized) associated Legendre polynomials.

    Parameters
    ----------
    n : ArrayLike[int]
        Degree of the harmonic. Must have ``n >= 0``. This is
        often denoted by ``l`` (lower case L) in descriptions of
        spherical harmonics.
    m : ArrayLike[int]
        Order of the harmonic.
    theta : ArrayLike[float]
        Polar (colatitudinal) coordinate; must be in ``[0, pi]``.
    phi : ArrayLike[float]
        Azimuthal (longitudinal) coordinate; must be in ``[0, 2*pi]``.
    diff_n : Optional[int]
        A non-negative integer. Compute and return all derivatives up
        to order ``diff_n``. Default is 0.

    Returns
    -------
    y : ndarray[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

    .. 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\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).

    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
    

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