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:

narray_like of int: Order of Bessel functions (floats will truncate with a warning)
xarray_like of float: Argument at which to evaluate the Bessel functions
outndarray, 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:

>>> importnumpyasnp
>>> fromscipy.specialimport kn
>>> importmatplotlib.pyplotasplt
>>> 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()

../../_images/scipy-special-kn-1_00_00.png

Calculate for a single value at multiple orders:

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