scipy.special.it2j0y0 — SciPy v1.15.2 Manual (original) (raw)

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()