1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    Pengwuino

    User Avatar
    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Bessel function equivalence
  1. Bessel Functions (Replies: 0)

Loading...