TITLE: Fourier transforms of functions not in $L^2.$
QUESTION [11 upvotes]: 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...

REPLY [3 votes]: 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.)