besselJx¶

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

Synopsis¶

besselJx (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 \(J_v(z)\) is defined to be

\[\begin{split}\begin{array}{l} j_v(z) = \tfrac{1}{\pi} \int_0^{\pi} \cos(z \sin \theta - v \theta) d \theta - \tfrac{\sin(v \pi)}{\pi} \int_0^{\infty} e^{z \sinh t-vt} dt \\ \text{for } |\arg z| < \tfrac{\pi}{2} \\ \end{array}\end{split}\]

This function is based on the code BESSCC of Barnett (1981) and Thompson and Barnett (1987). This code computes \(J_v(z)\) from the modified Bessel function \(I_v(z)\), using the following relation, with \(\rho= e^{ip/2}\):

\[\begin{split}Y_v(z) = \begin{cases} \rho I_v (z/\rho) & \text{for } -\pi/2 < \arg z \leq \pi \\ \rho^3 I_v \left(\rho^3 z\right) & \text{for } -\pi < \arg z \leq \pi/2 \\ \end{cases}\end{split}\]

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.besselJx import besselJx

n = 4
xnu = 0.3
z = 1.2 + 0.5j
sequence = besselJx(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)) = (0.774,-0.107)
I sub 1.30 ((1.20,0.50)) = (0.400,0.159)
I sub 2.30 ((1.20,0.50)) = (0.087,0.092)
I sub 3.30 ((1.20,0.50)) = (0.008,0.024)

Fatal Errors¶

IMSL_BESSEL_CONT_FRAC Continued fractions have failed to converge. The double precision version of this function provides the most accurate solution.