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In the past weeks, I have struggled with finding suitable tables for integral indentities for Beta functions, Chebyshev polynomials and their like.

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.

An example is the Bateman Manuscript Project, from which 3 out of 5 books are freely available:

Btw, is there way to search effectively for an integral identity?

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Two canonical online references are:

  • The Digital Library of Mathematical Functions (
    This is the official successor of the venerable Handbook of Mathematical Functions by Abramowitz and Stegun.

  • The Wolfram Functions Site (
    An expansive collection of identities and properties of special functions amassed and neatly categorized by Wolfram Research.

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Gradshteyn and Ryzhik

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 anyway. No proofs, just thousands of statements, each with its external reference.

N.N. Lebedev's classic "Special Functions..." is a small Dover reprint which gives proofs of many of the basic identities, which is sometimes the point of interest.

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).

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!!!

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 anything, 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!

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.

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...

(Vilenkin's book on special fcns and group repns is very interesting, but it, too, is very much caught up in a certain context...)


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Two copies?! Did you loan one out to people or did you often need to consult two parts of it simultaneously? – Qiaochu Yuan Jun 8 '12 at 0:02
I love the top review on Amazon for Gradshteyn-Ryzhik. – Matt Young Jun 8 '12 at 0:29
Andrews G.E., Askey R., Roy R. Special functions. А very good book, unofficial continuation of Harry Bateman books. – Sergei Aug 23 '14 at 11:57
And also 5 volumes of Prudnikov, Brychkov, Marichev - Integrals and Series. They are not only a source for tables and formulas but also a concise source of sp. func. identities and other properties. – Sergei Aug 23 '14 at 12:02

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