# scipy.special.it2i0k0#

scipy.special.it2i0k0(x, out=None) = <ufunc 'it2i0k0'>#

Integrals related to modified Bessel functions of order 0.

Computes the integrals

$\begin{split}\int_0^x \frac{I_0(t) - 1}{t} dt \\ \int_x^\infty \frac{K_0(t)}{t} dt.\end{split}$
Parameters:
xarray_like

Values at which to evaluate the integrals.

outtuple of ndarrays, optional

Optional output arrays for the function results.

Returns:
ii0scalar or ndarray

The integral for i0

ik0scalar or ndarray

The integral for k0

References



S. Zhang and J.M. Jin, “Computation of Special Functions”, Wiley 1996

Examples

Evaluate the functions at one point.

>>> from scipy.special import it2i0k0
>>> int_i, int_k = it2i0k0(1.)
>>> int_i, int_k
(0.12897944249456852, 0.2085182909001295)


Evaluate the functions at several points.

>>> import numpy as np
>>> points = np.array([0.5, 1.5, 3.])
>>> int_i, int_k = it2i0k0(points)
>>> int_i, int_k
(array([0.03149527, 0.30187149, 1.50012461]),
array([0.66575102, 0.0823715 , 0.00823631]))


Plot the functions from 0 to 5.

>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> x = np.linspace(0., 5., 1000)
>>> int_i, int_k = it2i0k0(x)
>>> ax.plot(x, int_i, label=r"$\int_0^x \frac{I_0(t)-1}{t}\,dt$")
>>> ax.plot(x, int_k, label=r"$\int_x^{\infty} \frac{K_0(t)}{t}\,dt$")
>>> ax.legend()
>>> ax.set_ylim(0, 10)
>>> plt.show()