# 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



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()