scipy.special.kn — SciPy v1.15.2 Manual (original) (raw)

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:

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 Kn(x)K_n(x)Kn​(x)') plt.show()

Calculate for a single value at multiple orders:

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