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Laplace transform

  1. Mar 9, 2012 #1
    1. The problem statement, all variables and given/known data
    I'm trying to find the Laplace transform of tJ''0(t), it's from bessels equation, but that doesn't matter too much at the moment, I just need to integrate (e^-st)*t*J''0(t) but am unsure how to go about this with the J''0(t) in there.
     
  2. jcsd
  3. Mar 9, 2012 #2

    Ray Vickson

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    Maple 14 gets an answer:
    t*diff(BesselJ(0,t),t$2): <---second derivative of J0(t), times t

    f:=expand(%);
    f := -t BesselJ(0, t) + BesselJ(1, t)

    L:=int(f*exp(-s*t),t=0..infinity);

    L:= [1 + 2s^2 + s^4 - (s^2 + 2s)*sqrt(1 + s^2)]/ (1+s^2)^2

    Note: Maple used the DE for J0 and recursions to express J0'' in terms of J0 and J1, then it integrated that.

    RGV
     
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