# scipy.special.kn

scipy.special.kn(n, x, out=None) = <ufunc 'kn'>

  Modified Bessel function of the second kind of integer order n

  Returns the modified Bessel function of the second kind for integer order n at real z.

  These are also sometimes called functions of the third kind, Basset functions, or Macdonald functions.

  Parameters:
    n array_like of int

      Order of Bessel functions (floats will truncate with a warning)

    x array_like of float

      Argument at which to evaluate the Bessel functions

    out ndarray, optional

      Optional output array for the function results.

  Returns:
    scalar or ndarray

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

  See also

  `kv`

    Same function, but accepts real order and complex argument

  `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

  Examples

  Plot the function of several orders for real input:

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

  Calculate for a single value at multiple orders:

      >>> kn([4, 5, 6], 1)
      array([   44.23241585,   360.9605896 ,  3653.83831186])