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scipy.special.

# spherical_kn

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

  Modified spherical Bessel function of the second kind or its derivative.

  Defined as [1],

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

  where \(K_n\) is the modified 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:
    kn ndarray

  Notes

  The function is computed using its definitional relation to the modified cylindrical Bessel function of the second kind.

  The derivative is computed using the relations [2],

  \[ \begin{align}\begin{aligned}k_n' = -k_{n-1} - \frac{n + 1}{z} k_n.\\k_0' = -k_1\end{aligned}\end{align} \]

  Added in version 0.18.0.

  References

  [1]

  https://dlmf.nist.gov/10.47.E9

  [2]

  https://dlmf.nist.gov/10.51.E5

  [AS]

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

  Examples

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

      >>> from scipy.special import spherical_kn
      >>> spherical_kn(0, 3+5j)
      (0.012985785614001561+0.003354691603137546j)
      >>> type(spherical_kn(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_kn(3, x, True),
      ...             - 4/x * spherical_kn(3, x) - spherical_kn(2, x))
      True

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

      >>> import matplotlib.pyplot as plt
      >>> x = np.arange(0.0, 4.0, 0.01)
      >>> fig, ax = plt.subplots()
      >>> ax.set_ylim(0.0, 5.0)
      >>> ax.set_title(r'Modified spherical Bessel functions $k_n$')
      >>> for n in np.arange(0, 4):
      ...     ax.plot(x, spherical_kn(n, x), label=rf'$k_{n}$')
      >>> plt.legend(loc='best')
      >>> plt.show()