scipy.special.ivp#

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

Compute derivatives of modified Bessel functions of the first kind.

Compute the nth derivative of the modified Bessel function Iv with respect to z.

Parameters:

varray_like or float: Order of Bessel function
zarray_like: Argument at which to evaluate the derivative; can be real or complex.
nint, default 1: Order of derivative. For 0, returns the Bessel function iv itself.

Returns:

scalar or ndarray: nth derivative of the modified Bessel function.

See also

iv

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 first kind of order 0 and its first two derivatives at 1.

>>> from scipy.special import ivp
>>> ivp(0, 1, 0), ivp(0, 1, 1), ivp(0, 1, 2)
(1.2660658777520084, 0.565159103992485, 0.7009067737595233)

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

>>> ivp([0, 1, 2], 1, 1)
array([0.5651591 , 0.70090677, 0.29366376])

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

>>> import numpy as np
>>> points = np.array([0., 1.5, 3.])
>>> ivp(0, points, 1)
array([0.        , 0.98166643, 3.95337022])

Plot the modified Bessel function of the first kind of order 1 and its first three derivatives.

>>> import matplotlib.pyplot as plt
>>> x = np.linspace(-5, 5, 1000)
>>> fig, ax = plt.subplots()
>>> ax.plot(x, ivp(1, x, 0), label=r"$I_1$")
>>> ax.plot(x, ivp(1, x, 1), label=r"$I_1'$")
>>> ax.plot(x, ivp(1, x, 2), label=r"$I_1''$")
>>> ax.plot(x, ivp(1, x, 3), label=r"$I_1'''$")
>>> plt.legend()
>>> plt.show()