scipy.special.kve#

scipy.special.kve(v, z, out=None) = <ufunc 'kve'>#

Exponentially scaled modified Bessel function of the second kind.

Returns the exponentially scaled, modified Bessel function of the second kind (sometimes called the third kind) for real order v at complex z:

kve(v, z) = kv(v, z) * exp(z)

Parameters:

varray_like of float: Order of Bessel functions
zarray_like of complex: Argument at which to evaluate the Bessel functions
outndarray, optional: Optional output array for the function results

Returns:

scalar or ndarray: The exponentially scaled modified Bessel function of the second kind.

See also

kv: This function without exponential scaling.
k0e: Faster version of this function for order 0.
k1e: Faster version of this function for order 1.

Notes

Wrapper for AMOS [1] routine zbesk. For a discussion of the algorithm used, see [2] and the references therein.

References

[1]

Donald E. Amos, “AMOS, A Portable Package for Bessel Functions of a Complex Argument and Nonnegative Order”, http://netlib.org/amos/

[2]

Donald E. Amos, “Algorithm 644: A portable package for Bessel functions of a complex argument and nonnegative order”, ACM TOMS Vol. 12 Issue 3, Sept. 1986, p. 265

Examples

In the following example kv returns 0 whereas kve still returns a useful finite number.

>>> import numpy as np
>>> from scipy.special import kv, kve
>>> import matplotlib.pyplot as plt
>>> kv(3, 1000.), kve(3, 1000.)
(0.0, 0.03980696128440973)

Evaluate the function at one point for different orders by providing a list or NumPy array as argument for the v parameter:

>>> kve([0, 1, 1.5], 1.)
array([1.14446308, 1.63615349, 2.50662827])

Evaluate the function at several points for order 0 by providing an array for z.

>>> points = np.array([1., 3., 10.])
>>> kve(0, points)
array([1.14446308, 0.6977616 , 0.39163193])

Evaluate the function at several points for different orders by providing arrays for both v for z. Both arrays have to be broadcastable to the correct shape. To calculate the orders 0, 1 and 2 for a 1D array of points:

>>> kve([[0], [1], [2]], points)
array([[1.14446308, 0.6977616 , 0.39163193],
       [1.63615349, 0.80656348, 0.41076657],
       [4.41677005, 1.23547058, 0.47378525]])

Plot the functions of order 0 to 3 from 0 to 5.

>>> fig, ax = plt.subplots()
>>> x = np.linspace(0., 5., 1000)
>>> for i in range(4):
...     ax.plot(x, kve(i, x), label=fr'$K_{i!r}(z)\cdot e^z$')
>>> ax.legend()
>>> ax.set_xlabel(r"$z$")
>>> ax.set_ylim(0, 4)
>>> ax.set_xlim(0, 5)
>>> plt.show()