besselYx¶

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

Synopsis¶

besselYx (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 \(Y_v(z)\) is defined to be

\[\begin{split}\begin{array}{l} Y_\nu(z) = \tfrac{1}{\pi} \int_{0}^{\pi} \sin (z \sin \theta - \nu \theta) d \theta - \tfrac{1}{\pi} \int_{0}^{\infty} \left[ e^{\nu t} + e^{-\nu t} \cos(\nu \pi) \right] e^{-z \sinh t} dt \\ \text{for } | \arg z | < \tfrac{\pi}{2} \end{array}\end{split}\]

This function is based on the code BESSCC of Barnett (1981) and Thompson and Barnett (1987). This code computes \(Y_v(z)\) from the modified Bessel functions \(I_v(z)\) and \(K_v(z)\), using the following relation:

\[Y_v\left(ze^{\pi i/2}\right) = e^{(v+1)\pi i/2} I_v(z) - \tfrac{2}{\pi} e^{-v \pi i/2} K_v(z) \text{ for } -\pi < \arg z \leq \tfrac{\pi}{2}\]

Example¶

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

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

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

Output¶

Y sub 0.30 ((1.20,0.50)) = (-0.013,0.380)
Y sub 1.30 ((1.20,0.50)) = (-0.716,0.338)
Y sub 2.30 ((1.20,0.50)) = (-1.048,0.795)
Y sub 3.30 ((1.20,0.50)) = (-1.625,3.684)