scipy.special.itj0y0

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