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# scipy.special.jv

scipy.special.jv(v, z, out=None) = <ufunc 'jv'>

  Bessel function of the first kind of real order and complex argument.

  Parameters:
    v array_like

      Order (float).

    z array_like

      Argument (float or complex).

    out ndarray, optional

      Optional output array for the function values

  Returns:
    J scalar or ndarray

      Value of the Bessel function, \(J_v(z)\).

  See also

  `jve`

    \(J_v\) with leading exponential behavior stripped off.

  `spherical_jn`

    spherical Bessel functions.

  `j0`

    faster version of this function for order 0.

  `j1`

    faster version of this function for order 1.

  Notes

  For positive v values, the computation is carried out using the AMOS [1] zbesj routine, which exploits the connection to the modified Bessel function \(I_v\),

  \[ \begin{align}\begin{aligned}J_v(z) = \exp(v\pi\imath/2) I_v(-\imath z)\qquad (\Im z > 0)\\J_v(z) = \exp(-v\pi\imath/2) I_v(\imath z)\qquad (\Im z < 0)\end{aligned}\end{align} \]

  For negative v values the formula,

  \[J_{-v}(z) = J_v(z) \cos(\pi v) - Y_v(z) \sin(\pi v)\]

  is used, where \(Y_v(z)\) is the Bessel function of the second kind, computed using the AMOS routine zbesy. Note that the second term is exactly zero for integer v; to improve accuracy the second term is explicitly omitted for v values such that v = floor(v).

  Not to be confused with the spherical Bessel functions (see `spherical_jn`).

  References

  [1]

  Donald E. Amos, “AMOS, A Portable Package for Bessel Functions of a Complex Argument and Nonnegative Order”, http://netlib.org/amos/

  Examples

  Evaluate the function of order 0 at one point.

      >>> from scipy.special import jv
      >>> jv(0, 1.)
      0.7651976865579666

  Evaluate the function at one point for different orders.

      >>> jv(0, 1.), jv(1, 1.), jv(1.5, 1.)
      (0.7651976865579666, 0.44005058574493355, 0.24029783912342725)

  The evaluation for different orders can be carried out in one call by providing a list or NumPy array as argument for the v parameter:

      >>> jv([0, 1, 1.5], 1.)
      array([0.76519769, 0.44005059, 0.24029784])

  Evaluate the function at several points for order 0 by providing an array for z.

      >>> import numpy as np
      >>> points = np.array([-2., 0., 3.])
      >>> jv(0, points)
      array([ 0.22389078,  1.        , -0.26005195])

  If z is an array, the order parameter v must be broadcastable to the correct shape if different orders shall be computed in one call. To calculate the orders 0 and 1 for a 1D array:

      >>> orders = np.array([[0], [1]])
      >>> orders.shape
      (2, 1)

      >>> jv(orders, points)
      array([[ 0.22389078,  1.        , -0.26005195],
             [-0.57672481,  0.        ,  0.33905896]])

  Plot the functions of order 0 to 3 from -10 to 10.

      >>> import matplotlib.pyplot as plt
      >>> fig, ax = plt.subplots()
      >>> x = np.linspace(-10., 10., 1000)
      >>> for i in range(4):
      ...     ax.plot(x, jv(i, x), label=f'$J_{i!r}$')
      >>> ax.legend()
      >>> plt.show()