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MathGroup Archive 2007

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symmetric fourier integral problem

  • To: mathgroup at smc.vnet.net
  • Subject: [mg74389] symmetric fourier integral problem
  • From: "Paul" <paulbird at whsurf.net>
  • Date: Mon, 19 Mar 2007 22:03:36 -0500 (EST)

I am trying to work out this integral precisely:

f(|x|,|y|,|z|) = integral[   exp(i x.q + i y.k )/ ( (q.q) (k.k) (q+k).
(q+k) ) , k_1..4=-infinity..infinity, q_1..4=-infinity..infinity]

in terms of |x|, |y| and |z| ( or equivalently x.x, y.y and z.z) .
Perhaps as a series or as a special function.

where x,y and z are 4-vectors and z(x,y)=x-y. All I know about the
solution is that it is symmetric in |x|,|y| and |z|. It also
satisfies:

(d/dx . d/dx) (d/dy . d/dy) (d/dx + d/dy)^2 f(|x|,|y|,|z|) = 0  (for |
x|=/=0, |y|=/=0)

As an example I know the simpler integral:

f(|x|) = integral[ exp(i x.k)/ (k.k) ,k_1..4=-infinity..infinity ]   =
1/|x|^2

-----------------------------------------------
Also the more tricky integral:

g(|x|,|y|,|z|,|a|,|b|,|c|) = integral[   exp(i x.q + i y.k +iz.r)/
( (q.q) (k.k) (r.r) (q+k+r).(q+k+r) ) , k_1..4=-infinity..infinity,
q_1..4=-infinity..infinity, r_1..4=-infinity..infinity]

where a(x,y)=x-y, b(y,z)=y-z, c(x,z)=z-x.



------------------------
These integrals come from electrodynamics but I haven't seen them
worked out precisely anywhere.
---------------------
A useful identity:  2(x.y) = (z.z) - (x.x) - (y.y)
The metric doesn't make a difference to the solution.



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