Module « scipy.special »

Fonction kv - module scipy.special

Signature de la fonction kv

Description

kv.__doc__

kv(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

kv(v, z)

Modified Bessel function of the second kind of real order `v`

Returns the modified Bessel function of the second kind for real order
`v` at complex `z`.

These are also sometimes called functions of the third kind, Basset
functions, or Macdonald functions.  They are defined as those solutions
of the modified Bessel equation for which,

.. math::
    K_v(x) \sim \sqrt{\pi/(2x)} \exp(-x)

as :math:`x \to \infty` [3]_.

Parameters
----------
v : array_like of float
    Order of Bessel functions
z : array_like of complex
    Argument at which to evaluate the Bessel functions

Returns
-------
out : ndarray
    The results. Note that input must be of complex type to get complex
    output, e.g. ``kv(3, -2+0j)`` instead of ``kv(3, -2)``.

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

See Also
--------
kve : This function with leading exponential behavior stripped off.
kvp : Derivative of this function

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
.. [3] NIST Digital Library of Mathematical Functions,
       Eq. 10.25.E3. https://dlmf.nist.gov/10.25.E3

Examples
--------
Plot the function of several orders for real input:

>>> from scipy.special import kv
>>> import matplotlib.pyplot as plt
>>> x = np.linspace(0, 5, 1000)
>>> for N in np.linspace(0, 6, 5):
...     plt.plot(x, kv(N, x), label='$K_{{{}}}(x)$'.format(N))
>>> plt.ylim(0, 10)
>>> plt.legend()
>>> plt.title(r'Modified Bessel function of the second kind $K_\nu(x)$')
>>> plt.show()

Calculate for a single value at multiple orders:

>>> kv([4, 4.5, 5], 1+2j)
array([ 0.1992+2.3892j,  2.3493+3.6j   ,  7.2827+3.8104j])