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

1. Nov 20, 2011

### Manisero

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

S(k,k') = $\sum$lPl(cos$\phi$1-$\phi$2)Jl(kR)Jl(k'R)

Where Pl are legendre polynomials, k,k' $\phi$1 and $\phi$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$\phi$1 and dk'=k''dkd$\phi$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...