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I'm trying to show that

Integral[x*J_{0}(a*x)*J_{0}(a*x), from 0 to 1] = 1/2 * J_{1}(a)^2

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

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# Bessel Function, Orthogonality and More

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