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Homework Help: Expansion of free space Green function in Bessel function

  1. Dec 15, 2009 #1
    1. The problem statement, all variables and given/known data
    In Jackson 3.16 we have to prove the expansion [tex]\frac{1}{\left{|}\vec{x}-\vec{x'}\right{|}}=\sum_{m=-\infty}^{\infty}\int_{0}^{\infty}dke^{im(\phi-\phi')}J_m(k\rho)J_m(k\rho')e^{-k(z_{>}-z_{<})}[/tex]

    2. Relevant equations

    3. The attempt at a solution
    I tried to use the techniques in the text book but I only got [tex]G=\frac{1}{2}\sum e^{im(\phi-phi')}\int_0^{\infty}dke^{k(z-z'}N_m(k\rho_>)J_m(k\rho_<)[/tex]I just can't get the relationship of [tex]z_>[/tex] and [tex]z_<[/tex] in the expansion. Can anybody help me?
  2. jcsd
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