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Homework Help: Bessel function problem

  1. Jun 5, 2008 #1
    1. The problem statement, all variables and given/known data

    Show that:
    [tex]\int_0^xJ_0(t)dt=2\sum_{n=0}^{\infty}J_{2n+1}(x)[/tex]

    2. Relevant equations

    I know that
    [tex]J_0(t)=\sum_{s=0}^{\infty}\frac{(-1)^s}{s!s!}\frac{t^{2s}}{2^{2s}}[/tex]

    3. The attempt at a solution

    I tried to calculate the integral and i get :
    [tex]\sum_{s=0}^{\infty}\frac{(-1)^sx^{2s+1}}{s!s!2^{2s}(2s+1)} [/tex]
    I dont see how I can arrive to the reuslt, if I calclate :
    [tex]\sum_{n=0}^{\infty}J_{2n+1}(x)=\sum_{s,n=0}^{\infty}\frac{(-1)^sx^{2s+2n+1}}{s!(s+2n+1)!2^{2n+2s+1}} [/tex]
    I tried to arrive to the same result, but I cant, shall I use Legendre's duplicantion formula?
    But even if I use it I don't think that's the way
    Thank you!
     
    Last edited: Jun 5, 2008
  2. jcsd
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