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Table of Integrals, Series, and Products

I. S. Gradshteyn and I. M. Ryzhik
Academic Press
Publication Date: 
Number of Pages: 
Hardcover with CDROM
[Reviewed by
Allen Stenger
, on

This is the 7th edition of a famous handbook (GR for short) that focuses primarily on definite and indefinite integrals. It also has a lot of information about special functions, and handy information about other analysis topics such as inequalities, differential equations, integral transforms, and matrices and determinants. It is bound with a CD-ROM that has the whole book in MathML format.

Now that we have computers and the Internet, big books of tables have become less important in mathematics. Questions that come up regarding such a work include: Can I find things in this book? Could I find them more easily on the Internet, or using a computer algebra system (CAS) such as Mathematica ? How do we know the book (or the CAS) is accurate?

I tested "findability" by using JSTOR to look up some items in the American Mathematical Monthly Problem Section that used GR in their solution. I then checked whether I could independently find the item in GR. The selected problems had 9 definite integrals, 2 finite sums, and 1 infinite series. I spent about an hour with GR and was able to find 8 of these 12 items. I think this shows excellent ease of use — essentially I solved 8 Monthly problems in one hour! The formulas I found were the same ones referenced in the Monthly solutions, so they would already have been there if I were working the problems when the Monthly came out.

Searching for mathematical formulas on the Internet is a dismal affair and I've never had any luck with it. Mathematica 6.0 does a very good job of definite integrals and infinite series, but it was disappointing on these examples, finding only 5 of the 12 items. Handbooks still have an edge over computers! I think handbooks are also valuable for browsing: the exact result you want may not be known, but you may find a similar result whose proof you can adapt.

I tested the CD-ROM on a Macintosh using Firefox and Netscape Navigator 9.0b2 and had no trouble with the display. It does take a long time to load each chapter because they are enormous (several megabytes) and it takes time to render all the MathML.

The publisher advertises that the CD-ROM is "fully searchable" but that is an exaggeration. You can use the browser search feature to search for ordinary text, and you can search for the mathematical text in the sense that you can search for the formula with all the MathML tags removed. For example, suppose you seek the value of the integral from 0 to infinity of sin(x)/x . You can search for "sinxx" and find several relevant results, although you miss several other relevant results that use t for the variable of integration.

How do we know the stated results in a handbook are correct? The previous (6th) edition of GR has 64 pages of errata, so this is not a frivolous concern. For most of the results presented, especially definite integrals, there's no obvious way to verify them. GR (unlike a CAS) does give the source of each result, so at least in theory you can look at the original source and gain some confidence. But many of the sources for GR are old (from the 1800s) and probably not easily accessible, and it appears that some of them are earlier tables that may not have sources themselves.

George Boros and Victor Moll started several years ago an ambitious project to verify all the formulas in GR. So far this effort has produced a lot of corrections to GR, and a book "Irresistible Integrals" (Cambridge, 2004) full of interesting things, including proofs of a small portion of the GR formulas. The project continues and has a web page at

Allen Stenger is a math hobbyist, library propagandist, and retired computer programmer. He volunteers in his spare time at, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical analysis.

 0 Introduction; 1 Elementary Functions; 2 Indefinite Integrals of Elementary Functions; 3 Definite Integrals of Elementary Functions; 4.Combinations involving trigonometric and hyperbolic functions and power; 5 Indefinite Integrals of Special Functions; 6 Definite Integrals of Special Functions; 7.Associated Legendre Functions; 8 Special Functions; 9 Hypergeometric Functions; 10 Vector Field Theory; 11 Algebraic Inequalities; 12 Integral Inequalities; 13 Matrices and related results; 14 Determinants; 15 Norms; 16 Ordinary differential equations; 17 Fourier, Laplace, and Mellin Transforms; 18 The z-transform