# scipy.special.kve

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

  Exponentially scaled modified Bessel function of the second kind.

  Returns the exponentially scaled, modified Bessel function of the second kind (sometimes called the third kind) for real order v at complex z:

      kve(v, z) = kv(v, z) * exp(z)

  Parameters:
    v array_like of float

      Order of Bessel functions

    z array_like of complex

      Argument at which to evaluate the Bessel functions

    out ndarray, optional

      Optional output array for the function results

  Returns:
    scalar or ndarray

      The exponentially scaled modified Bessel function of the second kind.

  See also

  `kv`

    This function without exponential scaling.

  `k0e`

    Faster version of this function for order 0.

  `k1e`

    Faster version of this function for order 1.

  Notes

  Wrapper for AMOS [1] routine zbesk. For a discussion of the algorithm used, see [2] and the references therein.

  References

  [1]

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

  [2]

  Donald E. Amos, “Algorithm 644: A portable package for Bessel functions of a complex argument and nonnegative order”, ACM TOMS Vol. 12 Issue 3, Sept. 1986, p. 265

  Examples

  In the following example `kv` returns 0 whereas `kve` still returns a useful finite number.

      >>> import numpy as np
      >>> from scipy.special import kv, kve
      >>> import matplotlib.pyplot as plt
      >>> kv(3, 1000.), kve(3, 1000.)
      (0.0, 0.03980696128440973)

  Evaluate the function at one point for different orders by providing a list or NumPy array as argument for the v parameter:

      >>> kve([0, 1, 1.5], 1.)
      array([1.14446308, 1.63615349, 2.50662827])

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

      >>> points = np.array([1., 3., 10.])
      >>> kve(0, points)
      array([1.14446308, 0.6977616 , 0.39163193])

  Evaluate the function at several points for different orders by providing arrays for both v for z. Both arrays have to be broadcastable to the correct shape. To calculate the orders 0, 1 and 2 for a 1D array of points:

      >>> kve([[0], [1], [2]], points)
      array([[1.14446308, 0.6977616 , 0.39163193],
             [1.63615349, 0.80656348, 0.41076657],
             [4.41677005, 1.23547058, 0.47378525]])

  Plot the functions of order 0 to 3 from 0 to 5.

      >>> fig, ax = plt.subplots()
      >>> x = np.linspace(0., 5., 1000)
      >>> for i in range(4):
      ...     ax.plot(x, kve(i, x), label=f'$K_{i!r}(z)\cdot e^z$')
      >>> ax.legend()
      >>> ax.set_xlabel(r"$z$")
      >>> ax.set_ylim(0, 4)
      >>> ax.set_xlim(0, 5)
      >>> plt.show()