Bessel Function, Orthogonality and More

1. Apr 20, 2010

Post

Hello,
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. Apr 20, 2010

phyzguy

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.

3. Apr 21, 2010

Post

Hi,
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:
http://physics.ucsc.edu/~peter/116C/bess_orthog.pdf
but I don't see the proper modifications that will give me my identity.

Any further hints would be amazing!

4. Apr 21, 2010

phyzguy

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