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# scipy.special.sph_harm_y

scipy.special.sph_harm_y(n, m, theta, phi, *, diff_n=0) = <scipy.special._multiufuncs.MultiUFunc object>

  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:
    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

  \[\begin{split}Y_0^0(\theta, \phi) &= \frac{1}{2} \sqrt{\frac{1}{\pi}} \\ Y_1^{-1}(\theta, \phi) &= \frac{1}{2} \sqrt{\frac{3}{2\pi}} e^{-i\phi} \sin(\theta) \\ Y_1^0(\theta, \phi) &= \frac{1}{2} \sqrt{\frac{3}{\pi}} \cos(\theta) \\ Y_1^1(\theta, \phi) &= -\frac{1}{2} \sqrt{\frac{3}{2\pi}} e^{i\phi} \sin(\theta).\end{split}\]

  References

  [1]

  Digital Library of Mathematical Functions, 14.30. https://dlmf.nist.gov/14.30

  [2]

  https://en.wikipedia.org/wiki/Spherical_harmonics#Condon.E2.80.93Shortley_phase