`scipy.special._basic:kvp`

source: build-install/usr/lib/python3.14/site-packages/scipy/special/_basic.py:978

Signature

def kvp(v, z, n = 1)

Summary

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

Extended Summary

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

Parameters

v : array_like of float: Order of Bessel function
z : array_like of complex: Argument at which to evaluate the derivative
n : int, default 1: Order of derivative. For 0 returns the Bessel function kv itself.

Returns

out : ndarray: The results

Notes

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

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)

✗

(np.float64(0.42102443824070834), np.float64(-0.6019072301972346), np.float64(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([-0.602, -1.023, -3.852])

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([-1.656, -0.277, -0.04 ])

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()

✗

<matplotlib.legend.Legend object at 0x7f3f68aeeba0>

plt.show()

✓

Aliases

scipy.special.kvp