Talk:Hankel transform

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The Hankel transform of order n − 1 arises when you wish to perform the Fourier transform of a radially symmetric function (i.e. f(r) where r=|\mathbf{r}|) in n dimensions. I don't have good references for this, but it is referred to in the Integral transform section of The Encyclopedic Dictionary of Mathematics --Farmhouse121 03:46, 30 June 2006 (UTC)

[edit] in Mathematica

If you'd like to evaluate a Hankel Transform in Mathematica, it's relatively straightforward. Here's an example. FIrst we define the function to be transformed; in this case, let's use f(r) = r .

f[r_] := r;

and now call Mathematica's numerical integrator:

NIntegrate[ f[r]*r*BesselJ[0, k*r], {r,0,Infinity}, Method -> Oscillatory ]

which will give the output -1/k^3, although it will likely give you a warning that it's unhappy. Because it's numerical integration, you need to actually specify the value of k before the computation, not after. If you don't specify

Method -> Oscillatory

then it won't converge, and you'll probably get a meaningless answer below machine precision.

Note that this integral converges only because the Bessel Function is oscillatory. If we require that \int_0^\infty |f(r)|r^{1/2} dr < \infty , then f(r) = r violates this condition since it increases without bound. Lavaka 17:25, 11 April 2007 (UTC)