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Fourier transform of the product of bessel function with an exponential

  1. Nov 20, 2011 #1
    Hello,
    In a quantummechanical problem I'm trying to solve, I came across this expression:

    S(k,k') = [itex]\sum[/itex]lPl(cos[itex]\phi[/itex]1-[itex]\phi[/itex]2)Jl(kR)Jl(k'R)

    Where Pl are legendre polynomials, k,k' [itex]\phi[/itex]1 and [itex]\phi[/itex]2 are two sets of pole coordinates and Jl are Bessel functions of the first kind (R is a constant).
    Now I need to transform this to real space with a double fourier transform:

    S(r,r') = Fk,-k'[S(k,k')] = ∫∫S(k,k')eikre-ik'r'dkdk'

    With dk=kdkd[itex]\phi[/itex]1 and dk'=k''dkd[itex]\phi[/itex]2 I can do the integration of the legendre polynomials over the angles, but then I'm stuck integrating the remaining, beeing integrals of the form:

    ∫keikrJl(kR)dk

    I searched a lot in tables for this kind of integral, but I didn't find anything usefull so far. I hope anyone here can help me...
     
  2. jcsd
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