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# scipy.special.j1

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

  Bessel function of the first kind of order 1.

  Parameters:
    x array_like

      Argument (float).

    out ndarray, optional

      Optional output array for the function values

  Returns:
    J scalar 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()