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()