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Bessel integral, solving strategy?

  1. Nov 18, 2009 #1
    Hello, I'm writing my bachelor project, and I've come across the following integral which according to the article I'm reading can be solved analytical.

    The integral is the following:
    [tex]R(E)= \int_{-\sqrt{2E}}^{\sqrt{2E}} dx J_{(0)}\left(\sqrt{ \frac{\pi}{2}} x \sqrt{E-\frac{1}{2} x^2 }\right)[/tex]

    The analytical solution is according to the article the following:
    [tex]R(E)= \sqrt{2 \pi} \sqrt{E/4} J_{(-1/4)} (E/4) J_{(1/4)}(E/4) [/tex]

    I'm guessing the way to solve this integral is by substitution.

    Here is what I have tried so far:

    [tex]R(E) = \int_{-\sqrt{2E}}^{\sqrt{2E}} dx J_0 \left( \sqrt{\pi/2} x \sqrt{E-1/2x^2} \right)[/tex]

    [tex]y &=\sqrt{\pi/2} x \sqrt{E-1/2x^2} [/tex]

    [tex]dy &=\sqrt{\pi/2} \left( dx \sqrt{E-1/2x^2} - \frac{x^2 dx}{2 \sqrt{E-1/2x^2}} \right)
    = dx \left( \frac{y}{x}- \frac{x^3 \pi}{4y} \right)[/tex]
    Which is where I hit a dead end.

    Another problem is that the borders with a shift like this will be zero so I will have to split the integral into atleast 2 pieces.
    [tex]x= \pm \sqrt{2E} \Rightarrow y=0 [/tex]
    [tex]x=0 \Rightarrow y=0[/tex]

    Any help would be greatly appreciated!

    Just for the sake of completeness the article I'm reading is the following:
    where the solution is equation (22) and the integral is giving in equation (18)
  2. jcsd
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