scipy.special.k1e

scipy.special.k1e(x, out=None) = <ufunc 'k1e'>

Exponentially scaled modified Bessel function K of order 1

Defined as:

k1e(x) = exp(x) * k1(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 1 at x.

See also

kv

Modified Bessel function of the second kind of any order

k1

Modified Bessel function of the second kind of order 1

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 k1e.

References

[1]

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

Examples

Calculate the function at one point:

>>> from scipy.special import k1e
>>> k1e(1.)
1.636153486263258

Calculate the function at several points:

>>> import numpy as np
>>> k1e(np.array([0.5, 2., 3.]))
array([2.73100971, 1.03347685, 0.80656348])

Plot the function from 0 to 10.

>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> x = np.linspace(0., 10., 1000)
>>> y = k1e(x)
>>> ax.plot(x, y)
>>> plt.show()

Exponentially scaled Bessel functions are useful for large arguments for which the unscaled Bessel functions are not precise enough. In the following example k1 returns zero whereas k1e still returns a useful floating point number.

>>> from scipy.special import k1
>>> k1(1000.), k1e(1000.)
(0., 0.03964813081296021)