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Multidimensional Fourier transform oddity

  1. Sep 16, 2011 #1
    Hi all,

    I'm trying to compute the fourier transform of a slightly odd function, a pair of monomials in k cobbled together with heaviside theta functions:
    [tex]f(k)=\theta(1-k) k^{n-2}+\theta(k-1) k^{-2}[/tex]
    where n is some integer >2. A complicating factor is that k is really the modulus of a vector in 2 dimensions.
    Evaluating
    [tex]\int_{0}^{\infty} dk f(k)e^{ikb} [/tex]
    in Mathematica gives me something with a finite limit as b-> 0. However, trying to do the integral
    [tex]\int d^{2}k f(k)e^{i\mathbf{k}\cdot\mathbf{b}} [/tex]
    gives me an integral that diverges as b->0; before passing the buck to Mathematica I've used an identity
    [tex]\int d^n y e^{i\mathbf{x}\cdot\mathbf{y}}f(y)=\frac{(2\pi)^{n/2}}{x^{n/2-1}}\int_{0}^{\infty}y^{n/2}J_{n/2-1}(xy)f(y)[/tex]
    in which the J is a Bessel function. Mathematically this seems strange, as the function seems reasonably well-behaved; it's also physically strange in the context of the problem I'm working on. Can someone give me a clue as to what's happening here?
     
  2. jcsd
  3. Sep 17, 2011 #2

    HallsofIvy

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    Science Advisor

    ??? The Bessel function is well behaved! In particular, it has circular symmetry, which, since you are dealing with the modulus of a vector seems reasonable for this problem.
     
  4. Sep 20, 2011 #3
    Thanks for your response, HallsofIvy- it prompted me to find the mistake I made when changing variables :redface:
     
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