`scipy.special._spherical_bessel:spherical_kn`

source: build-install/usr/lib/python3.14/site-packages/scipy/special/_spherical_bessel.py:277

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: Value or derivative of modified spherical Bessel function of the second kind.

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

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)

✗

np.complex128(0.012985785614001563+0.0033546916031375468j)

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

✓

<class 'numpy.complex128'>

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))

✓

True

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')

✗

<matplotlib.legend.Legend object at 0x7f3f590a3620>

plt.show()

✓

Aliases

scipy.special.spherical_kn