scipy.special.j0

scipy.special.j0(x) = <ufunc 'j0'>

Bessel function of the first kind of order 0.

Parameters:

x : array_like

Argument (float).

Returns:

J : 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 [R504] routine j0. It should not to be confused with the spherical Bessel functions (see spherical_jn).

References

[R504]

(1, 2) Cephes Mathematical Functions Library, http://www.netlib.org/cephes/index.html