besselKx¶

Evaluates a sequence of modified Bessel functions of the second kind with real order and complex arguments.

Synopsis¶

besselKx (xnu, z, n)

Required Arguments¶

float xnu (Input): The lowest order desired. The argument xnu must be greater than −1/2.
complex z (Input): Argument for which the sequence of Bessel functions is to be evaluated.
int n (Input): Number of elements in the sequence.

Return Value¶

The n values of the function through the series. Element i contains the value of the Bessel function of order xnu + i for \(i=0, \ldots,n-1\).

Description¶

The Bessel function \(K_v(z)\) is defined to be

\[K_v(z) = \tfrac{\pi}{2} e^{v \pi i/2} \left[iJ_v\left(ze^{\pi i/2}\right) - Y_v\left(ze^{\pi i/2}\right)\right] \text{ for } -\pi < \arg z \leq \tfrac{\pi}{2}\]

This function is based on the code BESSCC of Barnett (1981) and Thompson and Barnett (1987).

For moderate or large arguments, z, Temme’s (1975) algorithm is used to find \(K_v(z)\). This involves evaluating a continued fraction. If this evaluation fails to converge, the answer may not be accurate. For small z, a Neumann series is used to compute \(K_v(z)\). Upward recurrence of the \(K_v(z)\) is always stable.

Example¶

In this example, \(K_{0.3+n-1}(1.2+0.5i)\), \(\nu=1,\ldots,4\) is computed and printed.

from __future__ import print_function
from numpy import *
from pyimsl.math.besselKx import besselKx

n = 4
xnu = 0.3
z = 1.2 + 0.5j
sequence = besselKx(xnu, z, n)
for i in range(0, n):
    print("K sub %4.2f ((%4.2f,%4.2f)) = (%5.3f,%5.3f)"
          % (xnu + i, z.real, z.imag, sequence[i].real, sequence[i].imag))

Output¶

K sub 0.30 ((1.20,0.50)) = (0.246,-0.200)
K sub 1.30 ((1.20,0.50)) = (0.336,-0.362)
K sub 2.30 ((1.20,0.50)) = (0.587,-1.126)
K sub 3.30 ((1.20,0.50)) = (0.719,-4.839)