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Integration of bessel function

  1. Feb 16, 2010 #1
    Hello Everyone trying to come up with a stratagey to solving this integral

    Int(x^3*J3(x),x) no limits

    Ive tried some integration by parts and tried breaking it down into J1 and J0's however i still get to a point where I have to integrate either : Int(x*J1(x),x) or Int(J6(x),x)
     
  2. jcsd
  3. Feb 16, 2010 #2
  4. Feb 16, 2010 #3

    Astronuc

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    Use the following recursion relationships.

    Start with the first one, and let n+1 = 3

    [tex]\frac{2n}{x}\,J_n(x)\,=\,J_{n-1}(x)\,+\,J_{n+1}(x)[/tex]

    [tex]2\frac{dJ_n(x)}{dx}\,=\,J_{n-1}(x)\,-\,J_{n+1}(x)[/tex]

    [tex]\frac{dJ_0(x)}{dx}\,=\,-J_1(x)[/tex]

    Of course, one could use the more general derivative identity

    [tex]\frac{d}{dx}[x^m J_m(x)]\,=\,x^m J_{m-1}(x)[/tex]

    but one should probably prove that.

    http://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html
     
    Last edited: Feb 18, 2010
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