`scipy.special._spherical_bessel:spherical_kn`

source: /scipy/special/_spherical_bessel.py :312

Signature

def spherical_kn ( n , z , derivative = False )

Summary

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

Extended Summary

Defined as ^[1],

k_{n} (z) = \frac{π}{2 z} 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],

k_{n}^{'} = - k_{n - 1} - \frac{n + 1}{z} k_{n} . k_{0}^{'} = - k_{1}

Examples

The modified spherical Bessel functions of the second kind :math:`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)

✗

type(spherical_kn(0, 3+5j))

✓

We can verify the relation for the derivative from the Notes for :math:`n=3` in the interval :math:`[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))

✓

The first few :math:`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()

✓

Aliases

scipy.special.spherical_kn