This is probably something five-year-old physicists know, but here goes: Is there a standard methodology for computing Fourier transforms of things like $\log |x|$? Wolfram Alpha will happily give an answer (involving a delta function), but actually trying to do this yourself (by parts) gives horribly divergent-looking terms (the question which actually came up had $x$ be a vector in $\mathbb{R}^3,$ where the divergent terms are even more horrible than in the one-dimensional case (I am referring to the technique of just cutting off the function at some large $R;$ there are obviously other techniques, like weighting the integrand by an exponential weight (so you are computing a combination of Fourier and Laplace transforms), then computing the analytic continuation at $0,$ but all these should give the same answer,and there should be a not-totally-ad-hoc way of doing this, one should think...



Fourier transforms of functions not in $L^2.$
The umbrella legitimization of many such Fourier transforms is as *tempered* \_distributions\_ (where the sense of "distribution" is not the probability sense, but in the sense of Laurent Schwartz). The various "regularization" tricks amount to approaching the given distribution in the "weak \*-topology" on distributions, by more tractable functions. Fourier transform on tempered distributions is (provably) continuous, so we conclude that all these trick must yield the same outcome.


[Edit in response to comment:] The "how to compute" (once we know that *any* device succeeds) is non-trivial, insofar as it is not clear a-priori how explicit an outcome could be expected. The first volume of Gelfand-Graev-et-alia's "Generalized Functions" does many illuminating examples, mostly computed via meromorphic continuation.


The simplest family of examples is probably $|x|^s$. Here, the homogeneity and rotational symmetry, and the fact that Fourier transform respects these (in suitable senses), promise that the Fourier transform of $|x|^s$ on $\mathbb R^n$ is a constant multiple of $|x|^{-n-s}$, for $-n<\Re(s)<0$ to assure local integrability (of both). The constant multiple is determined (for example) by integrating against Gaussians.


Then use the fact that the derivative of $|x|^s$ in $s$ multiplies it by $\log|x|$, and set $s=0$. This is the nice way logarithms can arise. The implicit claim that we can do complex analysis with distribution-valued functions was legitimized by Schwartz, and is pervasive in Gelfand-et-alia.


Products of $|x|^s$ by harmonic polynomials can be treated almost identically, using the repn theory of the orthogonal group on harmonic polynomials.


That is, very often, some sort of \_unique\_characterization\_ of the tempered distribution, and of its image under Fourier Transform, reduce the computation to determination of the relevant constant!


Edit: oops, as Bazin notes, the exponent is not $n-s$ but $-n-s$, and adjust the local integrability assertion. (Adjusted above.)


