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Bessel functions

  1. Apr 26, 2010 #1
    1. The problem statement, all variables and given/known data

    Find the expansion of [itex] 1 - x^2 [/itex] on the interval [itex]0 < x < 1 [/itex] in terms of the Eigenfunctions
    [tex]J_0 ( \sqrt{ \lambda_k ^{(0)}} x) [/tex]

    (where [itex]\lambda_k ^{(0)}[/itex] denotes the kth root > 0 of [itex]J_0[/itex]) of

    [tex](x u')' + \lambda x u = 0 [/tex]
    [tex]u(1) = 0[/tex]
    u and u' bounded.

    2. Relevant equations

    Hint from the text: Use integration by parts and the following identity:
    [tex] \frac{d}{dt} [t^m J_m (t)] = t^m J_{m-1} (t) [/tex]

    3. The attempt at a solution

    Having never worked with Bessel functions before, I am a bit at a loss for what to do...
  2. jcsd
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