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Bessel Function, Orthogonality and More

  1. Apr 20, 2010 #1
    I'm trying to show that

    Integral[x*J0(a*x)*J0(a*x), from 0 to 1] = 1/2 * J1(a)^2

    Here, (both) a's are the same and they are a root of J0(x). I.e., J0(a) = 0.

    I have found and can do the case where you have two different roots, a and b, and the integral evaluates to zero (orthogonality). How do I go about showing this relationship? I can't find details anywhere.
  2. jcsd
  3. Apr 20, 2010 #2


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    Try expanding J0 in a power series, collect terms in like powers, and integrate. Then you can also expand the right side in a power series and show the two are equal.
  4. Apr 21, 2010 #3
    Sorry for my ignorance, but if expanding into a power series don't we have two infinite sums multiplied together? I attempted it but wasn't able to get anywhere nicely (maybe it's beyond me)

    I was thinking something more along the lines of this:
    but I don't see the proper modifications that will give me my identity.

    Any further hints would be amazing!
  5. Apr 21, 2010 #4


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    Why isn't equation 15 of the link you sent exactly what you are looking for?
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