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.