Statement of Problem 2
Solution to the Problem
The Bessel functions should satisfy the recursion relation (equation 3.87 in Jackson) below.
O (x) + O (x) = 2 * v * O (x) / x v-1 v+1 v
where O is Omega, and v is nu. This can be checked by graphing the left hand side (LHS) of the equation, graphing the right hand side (RHS) of the equation, and then superimposing the two graphs to see if the two lines are one and the same. Below is a list of my results. Click on LHS to see the graph of the left hand side of the equation with respective n. Click on RHS to see the graph of the right hand side of the equation with respective n. Click on SUPER to see the graph of the superposition of the two sides of the equation with respective orders, n.
For n=-11: LHS RHS SUPER
For n=1: LHS RHS SUPER
For n=6: LHS RHS SUPER
code for LHS
code for RHS
code for SUPERIMPOSED Note: I was unable to get the code to function properly.
Since the LHS and the RHS graphs of the above equations are the same for arbitrary values of n, the above relation must be correct for the bessel functions.
There is a phase relation between the Bessel functions of different order of v for the large x (x >> 1, v) limit according to equation 3.91 in Jackson.
J (x) -> sqrt(2/(Pi*x)) * cos(x-((v*Pi)/2)-(Pi/4)) v
where v is assumed to be real and nonnegative.
For n=10 to 20 and x=500.1 to 510.1: plot nu versus phase
For n=10 to 20 and x=500.1 to 510.1: plot nu versus the bessel function
for n=10 to 20 and x=500.1 to 510.1: plot the phase versus the bessel functioncode for the above plots
The nu versus phase plot shows that as the n varies the phase remains constant.
The nu versus the bessel function plot shows that a somewhat linear relation between the value of nu and the result of the equation 3.91. It is not too interesting of a result.
The phase versus the bessel function plot shows that there is an oscillation. As the phase increases, the bessel function oscillates periodically. Nifty!