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Bessel function equivalence

  1. Dec 18, 2009 #1
    1. The problem statement, all variables and given/known data
    Known formula:[tex]J_0(k\sqrt{\rho^2+\rho'^2-\rho\rho'\cos\phi})=\sum e^{im\phi}J_m(k\rho)J_m(k\rho')[/tex]
    I can't derive to next equation which is [tex]e^{ik\rho\cos\phi}=\sum i^me^{im\phi}J_m(k\rho)[/tex]

    2. Relevant equations
    Can anyone help me? Thanks a lot!!

    3. The attempt at a solution
  2. jcsd
  3. Dec 18, 2009 #2


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    Gold Member

    The formula for the first bessel function won't help. I believe this can be proven by looking at the expansion of [tex]e^{ik\rho sin(\phi)}[/tex] in terms of bessel functions.
  4. Dec 30, 2009 #3
    You can use the basic Bessel relation, i.e;

    let [tex]k\rho=x[/tex]

    [tex]e^{(x/2)(t-1/t)}=\sum J_n(x) t^n[/tex]

    then make the transformation [tex]t=e^{i\alpha}[/tex] st. [tex]\alpha=\theta+\pi/2[/tex]

    and then substitute them all in the Bessel relation, then you can obtain the given result.
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