# Are there extensive tables of Fourier transforms available online?

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.

• Just to check... have you also looked at Gradshteyn and Ryzhik to be sure? (But if you say you couldn't find what you needed in Erdelyi's, I guess things will be a bit difficult.) Commented Aug 26, 2010 at 17:47
• Well done tables of Fourier transforms are sorely missing, there are some in 1D but for higher dimension the best I know is the one-page table in the appendix of the Japanese Encyclopedia of Math... Commented Aug 26, 2010 at 19:14
• Maybe I'm looking in the wrong place, but in the last (7th) ed. of Gradshteyn-Ryzhik I only saw the table on pages 1118-1120. It doesn't contain the example I need. However, thank you for recommending the book! It might be useful in other situations. Commented Aug 27, 2010 at 21:11

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.

• The Erdelyi book is a good one, but rather hard to find nowadays. :( Commented Aug 26, 2010 at 17:31
• ...but OP also did say "I failed to find the answer not only online, but also in ... 'Tables of Integral Transforms' from the Bateman project". Commented Aug 26, 2010 at 17:45
• @J.M.isn'tamathematician authors.library.caltech.edu/43489 hope this is helpful Commented Nov 29, 2020 at 14:15

I tried entering "inverse fourier transform of (I-x)^3*Log[I-x]^4" at http://www.wolframalpha.com/ and it timed out. I think I read somewhere that Wolfram have ensured that Alpha knows all formulae from Gradshteyn and Ryzhik, as well as many other things. Similarly, Maple just gives me an answer in terms of the derivatives of the inverse Fourier transform of $\log(i-x)^4$, so I suspect that there is no explicit answer. I would certainly say that you are more likely to get a useful answer out of Maple or Mathematica than out of a book of tables.

I don't know of an good online source, but there is another book containing tables of Fourier Transforms that hasn't been mentioned yet:

Fourier Integrals for Practical Applications
Campbell & Foster
D. Van Nostrand, 1948
QA 404.C25 1948

Not sure whether it contains what you are looking for.

Oberhettinger, Tables of Fourier transforms and Fourier transforms of distributions, Springer 1990 (1957 reprint).