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()
../../_images/scipy-special-ivp-1.png