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In a quantummechanical problem I'm trying to solve, I came across this expression:

S(k,k') = [itex]\sum[/itex]_{l}P_{l}(cos[itex]\phi[/itex]_{1}-[itex]\phi[/itex]_{2})J_{l}(kR)J_{l}(k'R)

Where P_{l}are legendre polynomials, k,k' [itex]\phi[/itex]_{1}and [itex]\phi[/itex]_{2}are two sets of pole coordinates and J_{l}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') = F_{k,-k'}[S(k,k')] = ∫∫S(k,k')e^{ikr}e^{-ik'r'}dkdk'

Withdk=kdkd[itex]\phi[/itex]_{1}anddk'=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:

∫ke^{ikr}J_{l}(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...

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# Fourier transform of the product of bessel function with an exponential

Can you offer guidance or do you also need help?

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