k0e#

scipy.special.k0e(x,out=None)#

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 order0 atx.

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] routinek0e.k0e isuseful for large arguments: for these,k0 easily underflows.

Array API Standard Support

k0e has experimental support for Python Array API Standard compatiblebackends in addition to NumPy. Please consider testing these featuresby setting an environment variableSCIPY_ARRAY_API=1 and providingCuPy, PyTorch, JAX, or Dask arrays as array arguments. The followingcombinations 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

SeeSupport for the array API standard for more information.

References

[1]

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

Examples

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In the following examplek0 returns 0 whereask0e still returns auseful finite number:

>>>fromscipy.specialimportk0,k0e>>>k0(1000.),k0e(1000)(0., 0.03962832160075422)

Calculate the function at several points by providing a NumPy array orlist forx:

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

Plot the function from 0 to 10.

>>>importmatplotlib.pyplotasplt>>>fig,ax=plt.subplots()>>>x=np.linspace(0.,10.,1000)>>>y=k0e(x)>>>ax.plot(x,y)>>>plt.show()

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