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# scipy.special.kv

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

  Modified Bessel function of the second kind of real order v.

  Returns the modified Bessel function of the second kind for real order v at complex z.

  These are also sometimes called functions of the third kind, Basset functions, or Macdonald functions. They are defined as those solutions of the modified Bessel equation for which,

  \[K_v(x) \sim \sqrt{\pi/(2x)} \exp(-x)\]

  as \(x \to \infty\) [3].

  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 results. Note that input must be of complex type to get complex output, e.g. `kv(3, -2+0j)` instead of `kv(3, -2)`.

  See also

  `kve`

    This function with leading exponential behavior stripped off.

  `kvp`

    Derivative of this function

  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

  [3]

  NIST Digital Library of Mathematical Functions, Eq. 10.25.E3. https://dlmf.nist.gov/10.25.E3

  Examples

  Plot the function of several orders for real input:

      >>> import numpy as np
      >>> from scipy.special import kv
      >>> import matplotlib.pyplot as plt
      >>> x = np.linspace(0, 5, 1000)
      >>> for N in np.linspace(0, 6, 5):
      ...     plt.plot(x, kv(N, x), label='$K_{{{}}}(x)$'.format(N))
      >>> plt.ylim(0, 10)
      >>> plt.legend()
      >>> plt.title(r'Modified Bessel function of the second kind $K_\nu(x)$')
      >>> plt.show()

  Calculate for a single value at multiple orders:

      >>> kv([4, 4.5, 5], 1+2j)
      array([ 0.1992+2.3892j,  2.3493+3.6j   ,  7.2827+3.8104j])