SciPy, x) = <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.


n : array_like of int

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

z : array_like of float

Argument at which to evaluate the Bessel functions


out : ndarray

The results

See also

Same function, but accepts real order and complex argument
Derivative of this function


Wrapper for AMOS [R465] routine zbesk. For a discussion of the algorithm used, see [R466] and the references therein.


[R465](1, 2) Donald E. Amos, “AMOS, A Portable Package for Bessel Functions of a Complex Argument and Nonnegative Order”,
[R466](1, 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


Plot the function of several orders for real input:

>>> 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 $K_n(x)$')

(Source code)


Calculate for a single value at multiple orders:

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

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