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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?

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# Multidimensional Fourier transform oddity

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