# Homework Help: Bessel functions continuity

1. Mar 25, 2012

### dikmikkel

1. The problem statement, all variables and given/known data
I want to make sure that a solution to a differrential equation given by bessel functions of the first kind and second kind meet at a border(r=a), and it to be differenitable. So i shall determine the constants c_1 and c_2

I use notation from Schaums outlines
2. Relevant equations
The functions:
$R_1 = c_1J_\gamma(\kappa r),\hspace{8pt}r\in [0,a]\\ R_2 = c_2K_\gamma(\sigma r),\hspace{8pt}r\in[a,b]$

3. The attempt at a solution
For the solution to be continous at r=a:
$c_1J_\gamma(\kappa a) = c_2K_\gamma(\sigma a)$
For it to be differentiable:
$c_1J'_\gamma(\kappa a) = c_2K'_\gamma(\sigma a)$
I tried taking the determinant of the matrix describing the system at set it equal to zero, but i didn't seem to work. I also know from another part of the problem, that $\gamma\in\mathbb{Z}$
But is there a trick i've missed?

2. Mar 26, 2012

### sunjin09

Consider the extreme case of σ = $\kappa$, what you try to do is impossible. You can't expect a Bessel function to smoothly connect to a Neumann function, in the general case, you can only expect a Bessel function to connect smoothly with a linear combination of Bessel and Neumann functions. (Incident+Reflected=Transmitted)

3. Mar 28, 2012

### dikmikkel

Yeah my teacher said so too. He mentioned that of course the contuinity should be at the tangential field, not the radial. But thanks for the help.