# scipy.special.yve

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

  Exponentially scaled Bessel function of the second kind of real order.

  Returns the exponentially scaled Bessel function of the second kind of real order v at complex z:

      yve(v, z) = yv(v, z) * exp(-abs(z.imag))

  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 exponentially scaled Bessel function.

  See also

  `yv`

    Unscaled Bessel function of the second kind of real order.

  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).

  Exponentially scaled Bessel functions are useful for large z: for these, the unscaled Bessel functions can easily under-or overflow.

  References

  [1]

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

  Examples

  Compare the output of `yv` and `yve` for large complex arguments for z by computing their values for order `v=1` at `z=1000j`. We see that `yv` returns nan but `yve` returns a finite number:

      >>> import numpy as np
      >>> from scipy.special import yv, yve
      >>> v = 1
      >>> z = 1000j
      >>> yv(v, z), yve(v, z)
      ((nan+nanj), (-0.012610930256928629+7.721967686709076e-19j))

  For real arguments for z, `yve` returns the same as `yv` up to floating point errors.

      >>> v, z = 1, 1000
      >>> yv(v, z), yve(v, z)
      (-0.02478433129235178, -0.02478433129235179)

  The function can be evaluated for several orders at the same time by providing a list or NumPy array for v:

      >>> yve([1, 2, 3], 1j)
      array([-0.20791042+0.14096627j,  0.38053618-0.04993878j,
             0.00815531-1.66311097j])

  In the same way, the function can be evaluated at several points in one call by providing a list or NumPy array for z:

      >>> yve(1, np.array([1j, 2j, 3j]))
      array([-0.20791042+0.14096627j, -0.21526929+0.01205044j,
             -0.19682671+0.00127278j])

  It is also possible to evaluate several orders at several points at the same time by providing arrays for v and z with broadcasting compatible shapes. Compute `yve` for two different orders v and three points z resulting in a 2x3 array.

      >>> v = np.array([[1], [2]])
      >>> z = np.array([3j, 4j, 5j])
      >>> v.shape, z.shape
      ((2, 1), (3,))

      >>> yve(v, z)
      array([[-1.96826713e-01+1.27277544e-03j, -1.78750840e-01+1.45558819e-04j,
              -1.63972267e-01+1.73494110e-05j],
             [1.94960056e-03-1.11782545e-01j,  2.02902325e-04-1.17626501e-01j,
              2.27727687e-05-1.17951906e-01j]])