       Integration problem

• To: mathgroup at christensen.cybernetics.net
• Subject: [mg1009] Integration problem
• From: f85-tno at filsun04.nada.kth.se (Tommy Nordgren)
• Date: Mon, 8 May 1995 05:58:30 -0400
• Organization: Royal Institute of Technology, Stockholm, Sweden

```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
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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