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).

Array API Standard Support

j0 has support for Python Array API Standard compatible backends in addition to NumPy. The following combinations of backend and device (or other capability) are supported.

Library	CPU	GPU
NumPy	✅	n/a
CuPy	n/a	✅
PyTorch	✅	✅
JAX	✅	⛔
Dask	✅	n/a

For the NumPy backend, this function supports all NumPy ufunc keyword arguments. Other backends may support out, but none of the other ufunc kwargs. out is typically supported for CuPy and PyTorch, but not currently in cases where SciPy relies on a generic Array API implementation or, for PyTorch on CPU, falls back to the NumPy backend. out is never supported for JAX because JAX arrays are immutable.

See Support for the array API standard for more information.

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()