Manisero
Nov20-11, 08:03 AM
Hello,
In a quantummechanical problem I'm trying to solve, I came across this expression:
S(k,k') = \sumlPl(cos\phi1-\phi2)Jl(kR)Jl(k'R)
Where Pl are legendre polynomials, k,k' \phi1 and \phi2 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\phi1 and dk'=k''dkd\phi2 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...
In a quantummechanical problem I'm trying to solve, I came across this expression:
S(k,k') = \sumlPl(cos\phi1-\phi2)Jl(kR)Jl(k'R)
Where Pl are legendre polynomials, k,k' \phi1 and \phi2 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\phi1 and dk'=k''dkd\phi2 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...