besselIx¶

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

Synopsis¶

besselIx (xnu, z, n)

Required Arguments¶

float xnu (Input): The lowest order desired. 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 \(I_v(z)\): is defined to be

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

For large arguments, z, Temme’s (1975) algorithm is used to find \(I_v(z)\). The \(I_v(z)\) values are recurred upward (if this is stable). This involves evaluating a continued fraction. If this evaluation fails to converge, the answer may not be accurate.

For moderate and small arguments, Miller’s method is used.

Example¶

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

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

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

Output¶

I sub 0.30 ((1.20,0.50)) = (1.163,0.396)
I sub 1.30 ((1.20,0.50)) = (0.447,0.332)
I sub 2.30 ((1.20,0.50)) = (0.082,0.127)
I sub 3.30 ((1.20,0.50)) = (0.006,0.029)