scipy.special.it2j0y0

scipy.special.it2j0y0(x, out=None) = <ufunc 'it2j0y0'>

Integrals related to Bessel functions of the first kind of order 0.

Computes the integrals

\[\begin{split}\int_0^x \frac{1 - J_0(t)}{t} dt \\ \int_x^\infty \frac{Y_0(t)}{t} dt.\end{split}\]

For more on \(J_0\) and \(Y_0\) see j0 and y0.

Parameters
xarray_like

Values at which to evaluate the integrals.

outtuple of ndarrays, optional

Optional output arrays for the function results.

Returns
ij0scalar or ndarray

The integral for j0

iy0scalar or ndarray

The integral for y0

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 it2j0y0
>>> int_j, int_y = it2j0y0(1.)
>>> int_j, int_y
(0.12116524699506871, 0.39527290169929336)

Evaluate the functions at several points.

>>> import numpy as np
>>> points = np.array([0.5, 1.5, 3.])
>>> int_j, int_y = it2j0y0(points)
>>> int_j, int_y
(array([0.03100699, 0.26227724, 0.85614669]),
 array([ 0.26968854,  0.29769696, -0.02987272]))

Plot the functions from 0 to 10.

>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> x = np.linspace(0., 10., 1000)
>>> int_j, int_y = it2j0y0(x)
>>> ax.plot(x, int_j, label=r"$\int_0^x \frac{1-J_0(t)}{t}\,dt$")
>>> ax.plot(x, int_y, label=r"$\int_x^{\infty} \frac{Y_0(t)}{t}\,dt$")
>>> ax.legend()
>>> ax.set_ylim(-2.5, 2.5)
>>> plt.show()
../../_images/scipy-special-it2j0y0-1.png