Are Bessel Functions Differentiable at Boundary Conditions?

[dikmikkel]

Homework Statement

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

Homework Equations

The functions:
[itex]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][/itex]

The Attempt at a Solution

For the solution to be continuous at r=a:
[itex] c_1J_\gamma(\kappa a) = c_2K_\gamma(\sigma a)[/itex]
For it to be differentiable:
[itex] c_1J'_\gamma(\kappa a) = c_2K'_\gamma(\sigma a)[/itex]
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 [itex]\gamma\in\mathbb{Z}[/itex]
But is there a trick I've missed?

 

[sunjin09]

Consider the extreme case of σ = [itex]\kappa[/itex], 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)

 

[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.