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...
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1$\begingroup$ I allow myself to point out the question I've asked recently, mathoverflow.net/questions/80562/… since I'm interested by any continuation on the answers I got from there. I'm a bit skeptical about Wolfram Alpha result, at least that such a result would hold with no condition on $x$, but maybe I'm wrong ... $\endgroup$– Adrien HardyCommented Nov 14, 2011 at 12:40
3 Answers
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.)
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$\begingroup$ Yes, I am aware of why this is defined, what I wonder about is how does one actually go about computing these things. $\endgroup$ Commented Nov 14, 2011 at 15:23
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1$\begingroup$ Ah, the edit is very illuminating! $\endgroup$ Commented Nov 14, 2011 at 16:44
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1$\begingroup$ I disagree with Paul Garrett answer: the Fourier transform of $\vert x\vert^s$ on $\mathbb R^n$ is a constant multiple of $$ \vert x\vert^{-s-n}, $$ not $\vert x\vert^{s-n}$ as written in his answer. To have both sides locally integrable, we need $$ -n<\Re s<0. $$ Bazin. $\endgroup$– BazinCommented Apr 22, 2012 at 14:00
Only a small addendum to the excellent answer by Paul Garrett: A place where the Fourier transform is worked out explicitly (in 1d) is this preprint by Burnol. See in particular Page 13.
To find the Fourier transform of this and many other functions I enthusiastically recommend volume 1 of the magnificent treatise Generalized Functions, by Gelfand and coauthors.
This monograph contains so many mathematical gems and it pains me to notice that it is quasi - invisible to the Internet generation (By definition, you belong to the Internet generation, if you do no have a vivid memory of an era without E-mail.)
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2$\begingroup$ Given the answers to this very question, I conjecture that physicists must be reading the book (since no one has suggested any other reference, and it seems to be physics bread and butter... $\endgroup$ Commented Apr 23, 2012 at 13:01
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$\begingroup$ @Igor: Hum...I thought I did suggest another reference. $\endgroup$ Commented Apr 23, 2012 at 16:12
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1$\begingroup$ @Abdelmalek: you did, but this is a number theory preprint, which computes an example (a family, if you want to be generous). Somehow, I don't see crowds of physics grads students reading this... $\endgroup$ Commented Apr 23, 2012 at 16:51