scipy.special.kvp#

scipy.special.kvp(v, z, n=1)[source]#

Compute derivatives of real-order modified Bessel function Kv(z)

Kv(z) is the modified Bessel function of the second kind. Derivative is calculated with respect to z.

Parameters:

varray_like of float: Order of Bessel function
zarray_like of complex: Argument at which to evaluate the derivative
nint, default 1: Order of derivative. For 0 returns the Bessel function kv itself.

Returns:

outndarray: The results

See also

kv

Notes

The derivative is computed using the relation DLFM 10.29.5 [2].

References

[1]

Zhang, Shanjie and Jin, Jianming. “Computation of Special Functions”, John Wiley and Sons, 1996, chapter 6. https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html

[2]

NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/10.29.E5

Examples

Compute the modified bessel function of the second kind of order 0 and its first two derivatives at 1.

>>> from scipy.special import kvp
>>> kvp(0, 1, 0), kvp(0, 1, 1), kvp(0, 1, 2)
(0.42102443824070834, -0.6019072301972346, 1.0229316684379428)

Compute the first derivative of the modified Bessel function of the second kind for several orders at 1 by providing an array for v.

>>> kvp([0, 1, 2], 1, 1)
array([-0.60190723, -1.02293167, -3.85158503])

Compute the first derivative of the modified Bessel function of the second kind of order 0 at several points by providing an array for z.

>>> import numpy as np
>>> points = np.array([0.5, 1.5, 3.])
>>> kvp(0, points, 1)
array([-1.65644112, -0.2773878 , -0.04015643])

Plot the modified bessel function of the second kind and its first three derivatives.

>>> import matplotlib.pyplot as plt
>>> x = np.linspace(0, 5, 1000)
>>> fig, ax = plt.subplots()
>>> ax.plot(x, kvp(1, x, 0), label=r"$K_1$")
>>> ax.plot(x, kvp(1, x, 1), label=r"$K_1'$")
>>> ax.plot(x, kvp(1, x, 2), label=r"$K_1''$")
>>> ax.plot(x, kvp(1, x, 3), label=r"$K_1'''$")
>>> ax.set_ylim(-2.5, 2.5)
>>> plt.legend()
>>> plt.show()