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scipy.special.

# spherical_yn

scipy.special.spherical_yn(n, z, derivative=False)

  Spherical Bessel function of the second kind or its derivative.

  Defined as [1],

  \[y_n(z) = \sqrt{\frac{\pi}{2z}} Y_{n + 1/2}(z),\]

  where \(Y_n\) is the Bessel function of the second kind.

  Parameters:
    n int, array_like

      Order of the Bessel function (n >= 0).

    z complex or float, array_like

      Argument of the Bessel function.

    derivative bool, optional

      If True, the value of the derivative (rather than the function itself) is returned.

  Returns:
    yn ndarray

  Notes

  For real arguments, the function is computed using the ascending recurrence [2]. For complex arguments, the definitional relation to the cylindrical Bessel function of the second kind is used.

  The derivative is computed using the relations [3],

  \[ \begin{align}\begin{aligned}y_n' = y_{n-1} - \frac{n + 1}{z} y_n.\\y_0' = -y_1\end{aligned}\end{align} \]

  Added in version 0.18.0.

  References

  [1]

  https://dlmf.nist.gov/10.47.E4

  [2]

  https://dlmf.nist.gov/10.51.E1

  [3]

  https://dlmf.nist.gov/10.51.E2

  [AS]

  Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972.

  Examples

  The spherical Bessel functions of the second kind \(y_n\) accept both real and complex second argument. They can return a complex type:

      >>> from scipy.special import spherical_yn
      >>> spherical_yn(0, 3+5j)
      (8.022343088587197-9.880052589376795j)
      >>> type(spherical_yn(0, 3+5j))
      <class 'numpy.complex128'>

  We can verify the relation for the derivative from the Notes for \(n=3\) in the interval \([1, 2]\):

      >>> import numpy as np
      >>> x = np.arange(1.0, 2.0, 0.01)
      >>> np.allclose(spherical_yn(3, x, True),
      ...             spherical_yn(2, x) - 4/x * spherical_yn(3, x))
      True

  The first few \(y_n\) with real argument:

      >>> import matplotlib.pyplot as plt
      >>> x = np.arange(0.0, 10.0, 0.01)
      >>> fig, ax = plt.subplots()
      >>> ax.set_ylim(-2.0, 1.0)
      >>> ax.set_title(r'Spherical Bessel functions $y_n$')
      >>> for n in np.arange(0, 4):
      ...     ax.plot(x, spherical_yn(n, x), label=rf'$y_{n}$')
      >>> plt.legend(loc='best')
      >>> plt.show()