Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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


    User Avatar
    Science Advisor

    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


    User Avatar
    Science Advisor

    Why isn't equation 15 of the link you sent exactly what you are looking for?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook