scipy.special.j1#

scipy.special.j1(x, out=None) = <ufunc 'j1'>#

Bessel function of the first kind of order 1.

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 1 at x.

See also

jv: Bessel function of the first kind
spherical_jn: spherical Bessel functions.

Notes

The domain is divided into the intervals [0, 8] and (8, infinity). In the first interval a 24 term Chebyshev expansion is used. In the second, the asymptotic trigonometric representation is employed using two rational functions of degree 5/5.

This function is a wrapper for the Cephes [1] routine j1. It should not be confused with the spherical Bessel functions (see spherical_jn).

Array API Standard Support

j1 has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable SCIPY_ARRAY_API=1 and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported.

Library	CPU	GPU
NumPy	✅	n/a
CuPy	n/a	✅
PyTorch	✅	✅
JAX	⚠️ no JIT	⛔
Dask	✅	n/a

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 j1
>>> j1(1.)
0.44005058574493355

Calculate the function at several points:

>>> import numpy as np
>>> j1(np.array([-2., 0., 4.]))
array([-0.57672481,  0.        , -0.06604333])

Plot the function from -20 to 20.

>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> x = np.linspace(-20., 20., 1000)
>>> y = j1(x)
>>> ax.plot(x, y)
>>> plt.show()