Are there extensive tables of Fourier transforms available online? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T11:43:58Z http://mathoverflow.net/feeds/question/36780 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/36780/are-there-extensive-tables-of-fourier-transforms-available-online Are there extensive tables of Fourier transforms available online? senti_today 2010-08-26T16:43:35Z 2010-11-16T17:17:25Z <p>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.</p> <p>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 $b$?</p> <p>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.</p> <p>$\mathbf{EDIT.}$ 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.</p> http://mathoverflow.net/questions/36780/are-there-extensive-tables-of-fourier-transforms-available-online/36783#36783 Answer by David Corwin for Are there extensive tables of Fourier transforms available online? David Corwin 2010-08-26T17:21:08Z 2010-08-26T17:21:08Z <p>You may have to consult a textbook. See <a href="http://www.alibris.com/search/books/qwork/6523303/used/Tables%2520of%2520Integral%2520Transforms" rel="nofollow">Tables of Integral Transforms</a> by Arthur Erdelyi or the appendix of <a href="http://books.google.com/books?id=znP-ADtE8ZQC&amp;printsec=frontcover&amp;dq=a+first+course+in+fourier+analysis+kammler&amp;source=bl&amp;ots=1KBedvVtb3&amp;sig=AGAkP-dDkS_MTKTcbL1Zmh3fzAE&amp;hl=en&amp;ei=wqF2TO_DG4a8lQf78KnxCg&amp;sa=X&amp;oi=book_result&amp;ct=result&amp;resnum=2&amp;ved=0CB0Q6AEwAQ#v=onepage&amp;q&amp;f=false" rel="nofollow">A First Course in Fourier Analysis</a> by Kammler.</p> http://mathoverflow.net/questions/36780/are-there-extensive-tables-of-fourier-transforms-available-online/46255#46255 Answer by Dan Becker for Are there extensive tables of Fourier transforms available online? Dan Becker 2010-11-16T16:43:24Z 2010-11-16T16:43:24Z <p>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:</p> <p>Fourier Integrals for Practical Applications<br> Campbell &amp; Foster<br> D. Van Nostrand, 1948<br> QA 404.C25 1948 </p> <p>Not sure whether it contains what you are looking for.</p> http://mathoverflow.net/questions/36780/are-there-extensive-tables-of-fourier-transforms-available-online/46263#46263 Answer by Neil Strickland for Are there extensive tables of Fourier transforms available online? Neil Strickland 2010-11-16T17:17:25Z 2010-11-16T17:17:25Z <p>I tried entering "inverse fourier transform of (I-x)^3*Log[I-x]^4" at <a href="http://www.wolframalpha.com/" rel="nofollow">http://www.wolframalpha.com/</a> 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.</p>