I hope this is suitable for MO… I was wondering if someone can suggest a website (or some online document) containing an $extensive$ table of Fourier transforms? When I try obvious Google searches, like "table of Fourier transforms", the several dozen top results give extremely short tables.
My question is in fact motivated by one concrete example (so if you know the answer to this one, please let me know!). Note that I failed to find the answer not only online, but also in the standard books with Fourier transform tables (such as "Tables of Integral Transforms" from the Bateman project). Suppose $a$ and $b$ are real numbers. The function $f(\xi)=(i-\xi)^a\cdot(\log(i-\xi))^b$ can be defined for all real $\xi$ (by choosing appropriate branches of $\log(i-\xi)$ and $\log(\log(i-\xi))$), and the inverse Fourier transform of $f(\xi)$ makes sense as a distribution on $\mathbb{R}$. Is there an explicit formula for it? Apparently, the answer is yes when $b$ is a nonnegative integer, but what about other values of $b$?
Ignoring this particular example, I think many people who work with Fourier transforms on a daily basis would benefit from having an easily accessible table of Fourier transforms of functions, especially ones that are quite nontrivial to compute explicitly.
$\mathbf{EDIT.}$ As was commented below, the Erdelyi book "Tables of Integral Transforms" is the same as the one I referred to above when I mentioned the Bateman project. I also checked the book "Table of Integrals, Series, and Products" by Gradshteyn and Ryzhik, and couldn't find the thing I'm looking for.
Best Answer
You may have to consult a textbook. See Tables of Integral Transforms by Arthur Erdelyi or the appendix of A First Course in Fourier Analysis by Kammler.