scipy.special.j0#

scipy.special.j0(x, out=None) = <ufunc 'j0'>#

Bessel function of the first kind of order 0.

Parameters:

Returns:

Jscalar or ndarray: Value of the Bessel function of the first kind of order 0 at x.

See also

Notes

The domain is divided into the intervals [0, 5] and (5, infinity). In the first interval the following rational approximation is used:

\[J_0(x) \approx (w - r_1^2)(w - r_2^2) \frac{P_3(w)}{Q_8(w)},\]

where \(w = x^2\) and \(r_1\), \(r_2\) are the zeros of \(J_0\), and \(P_3\) and \(Q_8\) are polynomials of degrees 3 and 8, respectively.

In the second interval, the Hankel asymptotic expansion is employed with two rational functions of degree 6/6 and 7/7.

This function is a wrapper for the Cephes [1] routine j0. It should not be confused with the spherical Bessel functions (see spherical_jn).

References

[1]

Cephes Mathematical Functions Library, http://www.netlib.org/cephes/

Examples

Calculate the function at one point:

>>> from scipy.special import j0
>>> j0(1.)
0.7651976865579665

Calculate the function at several points:

>>> import numpy as np
>>> j0(np.array([-2., 0., 4.]))
array([ 0.22389078,  1.        , -0.39714981])

Plot the function from -20 to 20.

>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> x = np.linspace(-20., 20., 1000)
>>> y = j0(x)
>>> ax.plot(x, y)
>>> plt.show()