scipy.special.k0#

scipy.special.k0(x, out=None) = <ufunc 'k0'>#

Modified Bessel function of the second kind of order 0, \(K_0\).

This function is also sometimes referred to as the modified Bessel function of the third kind of order 0.

Parameters:
xarray_like

Argument (float).

outndarray, optional

Optional output array for the function values

Returns:
Kscalar or ndarray

Value of the modified Bessel function \(K_0\) at x.

See also

kv

Modified Bessel function of the second kind of any order

k0e

Exponentially scaled modified Bessel function of the second kind

Notes

The range is partitioned into the two intervals [0, 2] and (2, infinity). Chebyshev polynomial expansions are employed in each interval.

This function is a wrapper for the Cephes [1] routine k0.

k0 has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable SCIPY_ARRAY_API=1 and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported.

Library

CPU

GPU

NumPy

n/a

CuPy

n/a

PyTorch

JAX

⚠️ no JIT

Dask

n/a

See Support for the array API standard for more information.

References

[1]

Cephes Mathematical Functions Library, http://www.netlib.org/cephes/

Examples

Calculate the function at one point:

>>> from scipy.special import k0
>>> k0(1.)
0.42102443824070823

Calculate the function at several points:

>>> import numpy as np
>>> k0(np.array([0.5, 2., 3.]))
array([0.92441907, 0.11389387, 0.0347395 ])

Plot the function from 0 to 10.

>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> x = np.linspace(0., 10., 1000)
>>> y = k0(x)
>>> ax.plot(x, y)
>>> plt.show()
../../_images/scipy-special-k0-1.png