scipy.special.kve#

scipy.special.kve(v, z, out=None) = <wrapped_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.

Array API Standard Support

kve 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 typically 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. kve does not currently support out for the PyTorch backend.

See Support for the array API standard for more information.

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()