scipy.special.spherical_kn#

scipy.special.spherical_kn(n, z, derivative=False)[source]#

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:
nint, array_like

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

zcomplex or float, array_like

Argument of the Bessel function.

derivativebool, optional

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

Returns:
knndarray

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],

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

New in version 0.18.0.

References

[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 \([1, 2]\):

>>> from scipy.special import spherical_kn
>>> 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
>>> from scipy.special import spherical_kn
>>> 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()
../../_images/scipy-special-spherical_kn-1.png