# Multidimensional Fourier transform oddity

1. Sep 16, 2011

### muppet

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:
$$f(k)=\theta(1-k) k^{n-2}+\theta(k-1) k^{-2}$$
where n is some integer >2. A complicating factor is that k is really the modulus of a vector in 2 dimensions.
Evaluating
$$\int_{0}^{\infty} dk f(k)e^{ikb}$$
in Mathematica gives me something with a finite limit as b-> 0. However, trying to do the integral
$$\int d^{2}k f(k)e^{i\mathbf{k}\cdot\mathbf{b}}$$
gives me an integral that diverges as b->0; before passing the buck to Mathematica I've used an identity
$$\int d^n y e^{i\mathbf{x}\cdot\mathbf{y}}f(y)=\frac{(2\pi)^{n/2}}{x^{n/2-1}}\int_{0}^{\infty}y^{n/2}J_{n/2-1}(xy)f(y)$$
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. Sep 17, 2011

### HallsofIvy

??? 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.

3. Sep 20, 2011

### muppet

Thanks for your response, HallsofIvy- it prompted me to find the mistake I made when changing variables