Re: Fourier transform in arbitrary dimension?
- To: mathgroup at smc.vnet.net
- Subject: [mg88105] Re: Fourier transform in arbitrary dimension?
- From: Roman <rschmied at gmail.com>
- Date: Fri, 25 Apr 2008 05:26:44 -0400 (EDT)
- References: <fumr2o$sh2$1@smc.vnet.net>
Hi Barrow, I think the easiest would be to go to hyperspherical coordinates. Calling "theta" the angle between Q and x, such that Q\cdot x = q*x*cos(theta), the transformation is d^Dq --> P(D) * q^{d-1} * sin(theta)^{d-2} dq dtheta with P(D) = 2 * pi^{(D-1)/2} / Gamma((D-1)/2) In Mathematica writing: substituting d for D because of pre-assigned meaning of D, P[d_] = 2*Pi^((d-1)/2)/Gamma[(d-1)/2]; A[d_] := Integrate[q^(d-1) Sin[th]^(d-2) P[d]/(2 Pi)^d*Exp[- I*q*x*Cos[th]] * 1/q^2, {q, 0, Infinity}, {th, 0, Pi}, Assumptions -> x > 0] Doing the angular integral first, you get A[d_] := Integrate[(q^(d/2)*x^(1-d/2)*BesselJ[d/2-1,q*x])/(2*Pi)^(d/2) * 1/q^2, {q, 0, Infinity}, Assumptions -> x > 0] where your function to be transformed, 1/q^2, is still recognizable. Mathematica cannot do this last step directly; it takes some manual fiddling with intermediate results to see that the angular integral is really just a Bessel function. Assuming that 1/q^2 is really the function you wish to transform, you can simplify by substituting z=q*x, getting A[d_] := (2*Pi)^(-d/2) * x^(2-d) * Integrate[z^(d/2-2)*BesselJ[d/ 2-1,z], {z, 0, Infinity}] This integral does not converge for d>=5. However, it is oscillatory, and by neglecting the oscillating part we can get a decent result: J[d_, Z_] = Integrate[z^(d/2-2)*BesselJ[d/2-1, z], {z, 0, Z}, Assumptions -> {Z > 0, d > 2}]; Series[J[d, Z], {Z, Infinity, 0}] // Normal // Expand You see that the result contains three terms: the first two are oscillatory, and the third one is constant in Z, equal to 2^(d/2-2)*Gamma[d/2-1]. Only the last one is relevant. Putting everything together, the result of the d-dimensional Fourier transform of 1/q^2 is A[d_] = Gamma[d/2-1]/(4*Pi^(d/2)*x^(d-2)) Since you knew this already (up to the prefactor), I hope you are more interested in the general solution. The last step, neglecting the divergent oscillatory terms in the integral for d>=5, is a bit icky, but seems to work. Maybe somebody else has a good explanation of what's happening here? Cheers! Roman. On Apr 23, 10:13 am, Barrow <GRsemi... at gmail.com> wrote: > Dear all, > > I would like to calculate a Fourier transform in arbitrary dimension > , say D, of the function 1/q^2, where q denotes the absolute value > of a D dimensional spatial vector. > The integral I have to perform is > > \int \frac{d^Dq}{(2\pi)^D}\exp(-iQ\cdot x)\frac{1}{q^2} > > where |Q| = q. > But I can't find a way to tell Mathematica to calculate this integral > "of dimension D." > PS. The answer is proportional to \Gamma(D/2 - 1)(x^2/4)^{1-D/2} > > Any ideas would be appreciated. > Sincerely Barrow