Re: Integration problem

*To*: mathgroup at christensen.cybernetics.net*Subject*: [mg1032] Re: Integration problem*From*: chen at fractal.eng.yale.edu (Richard Q. Chen)*Date*: Wed, 10 May 1995 06:05:28 -0400*Organization*: Yale University

In article <3o9ms8$b92 at news0.cybernetics.net>, f85-tno at filsun04.nada.kth.se (Tommy Nordgren) writes: > >I have a tough integration problem that I need useful tips on: >Given the definition > f[x_,y_,z_,k_]:= Exp[-a^2 x^2-I k y-a^2 z^2] > >I need to integrate > f[x1,y1,z1,k1] f[y2,x2,z2,k2] * > Exp[-l Sqrt[(x1-x2)^2+(y1-y2)^2+(z1-z2)^2]]/ > Sqrt[(x1-x2)^2+(y1-y2)^2+(z1-z2)^2]] > >with respect to x1,y1,z1,x2,y2,z2 . over R6. > >Do anyone have any good ideas on how to solve this integral? > >I've tried: >Writing some of the factors as the inverse transform of their own fourier >transforms, and then reduce the complexity of the resulting 12-dimensional >integral by replacing exprssions of the general (MATHEMATICAL) form >g[u,v,w,...] Exp[ I w x] with 2 Pi g[u,v,w,...] DiracDelta[ w] (thereby >reducing the dimension). >I've had to do this by cut-and-paste editing. Do anyone know of a package >that can integrate expressions containing Exp[I w x] or Cos[ w x] with respect >to x over infinite space, by introducing DiracDelta[ w] > >I've tried making the problem more tractable by changing coordinates: >Do anyone know of any good utility packages that is useful for handling >coordinate tarnsformations. >Generally, my transformations will be in the >form of transformation rules generated by Solve. >These rules will generally not be in the simplest possible form. >Do any-one have good ideas on how to simplify the resulting rules, >and then simplify the transformed integrand efficiently. > >------------------------------------------------------------------------- >Tommy Nordgren "Home is not where you are born, >Royal Institute of Technology but where your heart finds peace." >Stockholm Tommy Nordgren - The dying old crone >f85-tno at nada.kth.se >-------------------------------------------------------------------------- > > I doubt any CAS can do this integral automatically without the user's help. The integral involves a convolution. This means Fourier transform is very useful. Indeed, after using Fourier transform, the integral becomes a 3D integral in k-space (Fourier space). The integrand is a product of three Fourier transforms. Only two of them are essentially different. The transform for g = E^(I r)/r (r = Sqrt[x^2+y^2+z^2] can be most easily obtained from the Helmholtz equation: Laplacian of g + g = -4*Pi DiracDelta[r] The other Fourier transform is triavial to obtain. In short, the 3D integral in k-space can be carried out explicitly which involve functions like ExponentialIntegral[1,z] and DiracDelta[k1-k2]. The final result is not that compact so I will not reproduce it here. But one should be able to obtain the answer for himself/herself by following the above procedure or better, invent a new one. -- Richard Q. Chen chen at fractal.eng.yale.edu