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# 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 [R452] routine `j0`. It should not to be confused with the spherical Bessel functions (see `spherical_jn`).

  References

  [R452] | (1, 2) Cephes Mathematical Functions Library, http://www.netlib.org/cephes/index.html