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# scipy.special.yv

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

  Bessel function of the second 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 results

  Returns:
    Y scalar or ndarray

      Value of the Bessel function of the second kind, \(Y_v(x)\).

  See also

  `yve`

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

  `y0`

    faster implementation of this function for order 0

  `y1`

    faster implementation of this function for order 1

  Notes

  For positive v values, the computation is carried out using the AMOS [1] zbesy routine, which exploits the connection to the Hankel Bessel functions \(H_v^{(1)}\) and \(H_v^{(2)}\),

  \[Y_v(z) = \frac{1}{2\imath} (H_v^{(1)} - H_v^{(2)}).\]

  For negative v values the formula,

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

  is used, where \(J_v(z)\) is the Bessel function of the first kind, computed using the AMOS routine zbesj. 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).

  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 yv
      >>> yv(0, 1.)
      0.088256964215677

  Evaluate the function at one point for different orders.

      >>> yv(0, 1.), yv(1, 1.), yv(1.5, 1.)
      (0.088256964215677, -0.7812128213002889, -1.102495575160179)

  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:

      >>> yv([0, 1, 1.5], 1.)
      array([ 0.08825696, -0.78121282, -1.10249558])

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

      >>> import numpy as np
      >>> points = np.array([0.5, 3., 8.])
      >>> yv(0, points)
      array([-0.44451873,  0.37685001,  0.22352149])

  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)

      >>> yv(orders, points)
      array([[-0.44451873,  0.37685001,  0.22352149],
             [-1.47147239,  0.32467442, -0.15806046]])

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

      >>> import matplotlib.pyplot as plt
      >>> fig, ax = plt.subplots()
      >>> x = np.linspace(0., 10., 1000)
      >>> for i in range(4):
      ...     ax.plot(x, yv(i, x), label=f'$Y_{i!r}$')
      >>> ax.set_ylim(-3, 1)
      >>> ax.legend()
      >>> plt.show()