It is well known that the solution of Bessel differential equation can be:
1) Bessel first kind, J, and Neumann, Y, (Bessel 2nd kind) or
2) two Hankel functions or
3) Bessel 1st kind, J, and Hankel, H (not commonly used )
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In the attached paper, they used option 1:
i.e. had an equation B*J+C*Y
if we use B*J+C*H values of B and C should change so the equation would remain
but when I tried to change to option 3. Results are not the same! why should not they be the same? did I do a mistake? if there is a mistake where?
I have the program written using both Neumann and Hankel to compare what went wrong?
The tasks will be to check:
1) Mathematical correctness
2) Compare programs
This is very important, the two programs that I need to compare contain only 12 lines of differences each is just a change from Neumann to Hankel
The real objective of the project is much simple:
1) If I can write the solution of a Bessel differential equation as B*J+C*Y where B, C are constants and J, Y are Bessel 1st kind and 2nd kind or Neumann then it can be written as D*J+E*H since H=J+1i*Y
in that case D*J+E*(J+1i*Y) is related so B=D+E and C=1i*E
2) Since number 1 is true then why the program does not work. It is very important to verify that my assumption in 1 is true Mathematically. The paper is just an example since it is very simple and all it matrices are written in appendix.
The project should not take a real Mathematical professional more than 30 minute.