scipy.special.k0e#

scipy.special.k0e(x, out=None) = <ufunc 'k0e'>#

Exponentially scaled modified Bessel function K of order 0.

Defined as:

k0e(x) = exp(x) * k0(x).

Parameters:

xarray_like: Argument (float)
outndarray, optional: Optional output array for the function values

Returns:

Kscalar or ndarray: Value of the exponentially scaled modified Bessel function K of order 0 at x.

See also

kv: Modified Bessel function of the second kind of any order
k0: 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 k0e. k0e is useful for large arguments: for these, k0 easily underflows.

Array API Standard Support

k0e has support for Python Array API Standard compatible backends in addition to NumPy. The following combinations of backend and device (or other capability) are supported.

Library	CPU	GPU
NumPy	✅	n/a
CuPy	n/a	✅
PyTorch	✅	✅
JAX	✅	⛔
Dask	✅	n/a

For the NumPy backend, this function supports all NumPy ufunc keyword arguments. Other backends may support out, but none of the other ufunc kwargs. out is typically supported for CuPy and PyTorch, but not currently in cases where SciPy relies on a generic Array API implementation or, for PyTorch on CPU, falls back to the NumPy backend. out is never supported for JAX because JAX arrays are immutable.

See Support for the array API standard for more information.

References

[1]

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

Examples

In the following example k0 returns 0 whereas k0e still returns a useful finite number:

>>> from scipy.special import k0, k0e
>>> k0(1000.), k0e(1000)
(0., 0.03962832160075422)

Calculate the function at several points by providing a NumPy array or list for x:

>>> import numpy as np
>>> k0e(np.array([0.5, 2., 3.]))
array([1.52410939, 0.84156822, 0.6977616 ])

Plot the function from 0 to 10.

>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> x = np.linspace(0., 10., 1000)
>>> y = k0e(x)
>>> ax.plot(x, y)
>>> plt.show()