scipy.special.j0#
- scipy.special.j0(x, out=None) = <ufunc 'j0'>#
- Bessel function of the first kind of order 0. - Parameters:
- xarray_like
- Argument (float). 
- outndarray, optional
- Optional output array for the function values 
 
- Returns:
- Jscalar or ndarray
- Value of the Bessel function of the first kind of order 0 at x. 
 
 - See also - jv
- Bessel function of real order and complex argument. 
- spherical_jn
- spherical Bessel functions. 
 - 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() 