# 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

[1]

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()