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Bessel functions continuity

  1. Mar 25, 2012 #1
    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:
    [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]

    3. The attempt at a solution
    For the solution to be continous 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?
     
  2. jcsd
  3. Mar 26, 2012 #2
    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)
     
  4. Mar 28, 2012 #3
    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.
     
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