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MathGroup Archive 1995

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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


















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