scipy.special.itj0y0 — SciPy v1.15.2 Manual (original) (raw)

scipy.special.itj0y0(x, out=None) = <ufunc 'itj0y0'>#

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

Computes the integrals

\[\begin{split}\int_0^x J_0(t) dt \\ \int_0^x Y_0(t) dt.\end{split}\]

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 of j0

iy0scalar or ndarray

The integral of 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 itj0y0 int_j, int_y = itj0y0(1.) int_j, int_y (0.9197304100897596, -0.637069376607422)

Evaluate the functions at several points.

import numpy as np points = np.array([0., 1.5, 3.]) int_j, int_y = itj0y0(points) int_j, int_y (array([0. , 1.24144951, 1.38756725]), array([ 0. , -0.51175903, 0.19765826]))

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 = itj0y0(x) ax.plot(x, int_j, label=r"$\int_0^x J_0(t),dt$") ax.plot(x, int_y, label=r"$\int_0^x Y_0(t),dt$") ax.legend() plt.show()