Resources for special functions, integral identities - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T12:07:13Z http://mathoverflow.net/feeds/question/99027 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/99027/resources-for-special-functions-integral-identities Resources for special functions, integral identities Marc Palm 2012-06-07T11:03:21Z 2012-06-08T01:43:28Z <p>In the past weeks, I have struggled with finding suitable tables for integral indentities for Beta functions, Chebyshev polynomials and their like. </p> <p>I would like to ask for online/offline resources and software, which collect tables of special functions and their properties, integral identities, tables of Fourier/Laplace transforms of special functions, and so forth.</p> <p>An example is the Bateman Manuscript Project, from which 3 out of 5 books are freely available: <a href="http://en.wikipedia.org/wiki/Bateman_Manuscript_Project" rel="nofollow">http://en.wikipedia.org/wiki/Bateman_Manuscript_Project</a></p> <p>Btw, is there way to search effectively for an integral identity?</p> http://mathoverflow.net/questions/99027/resources-for-special-functions-integral-identities/99031#99031 Answer by Igor Khavkine for Resources for special functions, integral identities Igor Khavkine 2012-06-07T12:20:03Z 2012-06-07T21:05:14Z <p>Two canonical online references are:</p> <ul> <li><p>The Digital Library of Mathematical Functions (<a href="http://dlmf.nist.gov/" rel="nofollow">http://dlmf.nist.gov/</a>) <br/> This is the official successor of the venerable Handbook of Mathematical Functions by Abramowitz and Stegun.</p></li> <li><p>The Wolfram Functions Site (<a href="http://functions.wolfram.com/" rel="nofollow">http://functions.wolfram.com/</a>) <br/> An expansive collection of identities and properties of special functions amassed and neatly categorized by Wolfram Research.</p></li> </ul> http://mathoverflow.net/questions/99027/resources-for-special-functions-integral-identities/99070#99070 Answer by paul garrett for Resources for special functions, integral identities paul garrett 2012-06-07T23:25:34Z 2012-06-08T01:43:28Z <p>Gradshteyn and Ryzhik </p> <p><a href="http://www.amazon.com/Table-Integrals-Series-Products-Edition/dp/0122947576/ref=sr_1_1?ie=UTF8&amp;qid=1339111210&amp;sr=8-1" rel="nofollow">http://www.amazon.com/Table-Integrals-Series-Products-Edition/dp/0122947576/ref=sr_1_1?ie=UTF8&amp;qid=1339111210&amp;sr=8-1</a></p> <p>is "the bible" for many people. At one point I owned two copies. It summarizes results from the Bateman project, with very precise references. It also has culled identities from many other sources. In a fairly recent edition, my collaborator Adrian Diaconu found a missing "n!", so one should not assume that all one-thousand pages are absolutely perfect, but of course that would have been foolish <em>anyway</em>. No proofs, just thousands of statements, each with its external reference.</p> <p>N.N. Lebedev's classic "Special Functions..." is a small Dover reprint which gives <em>proofs</em> of many of the basic identities, which is sometimes the point of interest.</p> <p>Whittaker and Watson's "Course of Modern [sic] Analysis" also gives nice illustrations of methods, and the exercises state many standard properties (and also Tripos-like arcana).</p> <p>Edit: I am greatly amused by Qiaochu Yuan's (possibly/presumably tongue-in-cheek) comment/question about whether I needed to have two copies to look at simultaneously!!! </p> <p>To respond to the innocent version of this question, which has some relevance for people, I think: back when there were no electronic versions of <em>anything</em>, but after the point that I had a little money, I would buy two copies of books that seemed important, so that I would not be carrying them back and forth to-and-from office-and-home. This was partly motivated by some poignant occasions on weekends in which I needed some basic idea but (pre-introblog) there was no way to get the information... Friggit!</p> <p>In fact, although I do still have the home-and-atwork copies of Lebedev, I only have the at-work copy of G-and-R, because I have come to a point ... note... where what I need/want to know is not quite so formulaic.</p> <p>In fact, I do also think that nearly everyone needs to go through a transitional stage om which reality is portrayed "in essence" by "formulas". A grounding. But/and, by this year (as opposed to 1890 or 1930) one finds that further progress is unintelligible in such terms...</p> <p>(Vilenkin's book on special fcns and group repns is very interesting, but it, too, is very much caught up in a certain context...)</p> <p>-pg</p>