Gradshteyn and Ryzhik

Table of Integrals, Series, and Products

Gradshteyn and Ryzhik, seventh English edition, 2007
Author Gradshteyn, Ryzhik, et al.
Country Russia
Language Russian, German, Polish, English
Genre Math
Publisher Academic Press
Publication date
 1943

Gradshteyn and Ryzhik (GR) is the informal name of a comprehensive table of integrals originally compiled by the Russian mathematicians I. S. Gradshteyn and I. M. Ryzhik. Its full title today is Table of Integrals, Series, and Products.

Since its first publication in 1943, it was considerably expanded and it soon became a "classic" and highly regarded reference for mathematicians, scientists and engineers. After the deaths of the original authors, the work was maintained and further expanded by other editors.

At some stage a German and English dual-language translation became available, followed by Polish and English-only versions. After several further editions, the Russian and German-English versions went out of print and have not been updated after the German reunification, but the English-only version is still being actively maintained and refined by new editors, and it has recently being retranslated back into Russian as well.

Overview

One of the valuable characteristics of Gradshteyn and Ryzhik compared to similar compilations is that each listed integral is referenced. The literature list contains 86 main entries and 140 additional entries (in the 6. edition). The integrals are classified by numbers, which haven't changed from the fourth Russian up to the seventh English edition (the numbering in older editions as well as in the eighth English edition is not fully compatible). The book does not only contain the integrals, but also lists additional properties and related special functions. It also includes tables for integral transforms. Another advantage of the Gradshteyn and Ryzhik compared to computer algebra systems is the fact that all special functions and constants used in the evaluation of the integrals are listed in a registry as well, thereby allowing reverse lookup of integrals based on special functions or constants.

On the downsides, Gradshteyn and Ryzhik has become known to contain a relatively high number of typographical errors even in newer editions, which has repeatedly led to the publication of extensive errata lists.

History

The work was originally compiled by the Russian mathematicians Izrail Solomonovich Gradshteyn (Russian: Израиль Соломонович Градштейн, German: Israil Solomonowitsch Gradstein) and Iosif Moiseevich Ryzhik (Russian: Иосиф Моисеевич Рыжик, German: Jossif Moissejewitsch Ryschik).

The first two editions 1943 and 1948 were authored by Ryzhik; after his death the third edition was worked on by Gradshteyn, who planed considerable expansion for the fourth edition, a work he could not finish due to his own death. Therefore, the fourth and fifth editions were continued by Yuri Weneaminovich Geronimus (Russian: Юрий Венеаминович Геронимус, German: Juri Weneaminowitsch Geronimus) and M. Ju. Zeitlin.

Based on the third Russian edition (1951), the first German-English edition was published in 1957 by the East-German Deutscher Verlag der Wissenschaften (DVW) with German translations by Christa and Lothar Berg and the English texts by Martin Strauss. In 1963, it was followed by the second edition, a reprint edition with a four page inlet listing corrections compiled by Eldon Hansen. In Zentralblatt MATH Karl Prachar wrote:[1]

"Die sehr reichhaltigen Tafeln wurden nur aus dem Russischen ins Deutsche und Englische übersetzt."
(Translation: The very comprehensive tables were only translated into German and English language.)

Derived from the 1963 edition, but considerably expanded, the third German-English edition by Ludwig Boll was finally published in 1981; it incorporate the material of the fifth Russian edition (1971) as well.

An expanded English-only edition by Alan Jeffrey was published in 1965 as "third edition". It was based on the fourth Russian edition (1962). Since the sixth edition (2000), Daniel Zwillinger started contributing as well. The last edition being edited by Jeffrey before his death was the seventh English edition published in 2007.[2] This edition has been retranslated back into Russian as "seventh Russian edition" in 2011 as well.[3]

For the eighth edition (2014/2015) Zwillinger took over the role of the editor. He was assisted by Victor Hugo Moll.[4] In order to make room for additional information without increasing the size of the book significantly, the former chapters 11 (on algebraic inequalities), chapters 13 to 16 (on matrices and related results, determinants, norms, ordinary differential equations) and chapter 18 (on z-transforms) worth about 70 pages in total were removed and some chapters renumbered (12 to 11, 17 to 12).

Related projects

Victor Hugo Moll started a project to proof all integrals listed in the Gradshteyn and Ryzhik and add additional commentary and references. In the foreword of the book Irresistible Integrals (2004), he wrote:[5]

It took a short time to realize that this task was monumental.

Nevertheless, the efforts have meanwhile resulted in a two-volume book series Special Integrals of Gradshteyn and Ryzhik: The Proofs (2014-2015) already.[6][7]

Editions

Russian editions

German editions

Polish edition

English editions

See also

References

  1. ↑ Zbl 0080.33703 (1. German-English edition, 1957)
  2. ↑ Table of Integrals, Series, and Products. Academic Press, 2007. ISBN 978-0-12-373637-6
  3. 1 2 http://www.apmath.spbu.ru/ru/staff/maximov/publ/publ1.pdf
  4. ↑ Table of Integrals, Series, and Products. Academic Press, 2014/2015. ISBN 0-12-384933-0
  5. ↑ George Boros, Victor Hugo Moll: Irresistible Integrals. Symbolics, Analysis and Experiments in the Evaluation of Integrals. Cambridge University Press, 2004, ISBN 0521796369. (NB. This edition contains many typographical errors.)
  6. ↑ Victor Hugo Moll: Special Integrals of Gradshteyn and Ryzhik: the Proofs - Volume I (Series: Monographs and Research Notes in Mathematics). Chapman and Hall/CRC, 2014. ISBN 9781482256512.
  7. ↑ Victor Hugo Moll: Special Integrals of Gradshteyn and Ryzhik: the Proofs - Volume II (Series: Monographs and Research Notes in Mathematics). Chapman and Hall/CRC, 2015. ISBN 9781482256536.

External links

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