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.
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Table of Integrals, Series, and Products
Gradshteyn and Ryzhik, seventh English edition, 2007
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, English-only and Japanese versions. After several further editions, the Russian and German-English versions went out of print and have not been updated after the fall of the Iron Curtain, but the English version is still being actively maintained and refined by new editors, and it has recently been retranslated back into Russian as well.
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Overview
One of the valuable characteristics of Gradshteyn and Ryzhik compared to similar compilations is that most listed integrals are referenced. The literature list contains 92 main entries and 140 additional entries (in the eighth English 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 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. Earlier English editions were also criticized for their poor translation of mathematical terms[1][2][3] and mediocre print quality.[1][2][4][5]
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History
The work was originally compiled by the Russian mathematicians Iosif Moiseevich Ryzhik (Russian: Иосиф Моисеевич Рыжик, German: Jossif Moissejewitsch Ryschik)[6][nb 1] and Izrail Solomonovich Gradshteyn (Russian: Израиль Соломонович Градштейн, German: Israil Solomonowitsch Gradstein).[6][nb 2] While some contents were original, significant portions were collected from other previously existing integral tables like David Bierens de Haan's Nouvelles tables d'intégrales définies (1867),[6][7]Václav Jan Láska's Sammlung von Formeln der reinen und angewandten Mathematik (1888–1894)[6][8] or Edwin Plimpton Adams' and Richard Lionel Hippisley's Smithsonian Mathematical Formulae and Tables of Elliptic Functions (1922).[6][9]
The first edition, which contained about 5000 formulas,[10][11][nb 3] was authored by Ryzhik,[nb 1] who had already published a book on special functions in 1936[6][12] and died during World War II around 1941.[6] Not announcing this fact, his compilation was published posthumously[6][nb 1] in 1943, followed by a second corrected edition in his name in 1948.[nb 4]
The third edition (1951) was worked on by Gradshteyn,[13] who also introduced the chapter numbering system in decimal notation. Gradshteyn planned considerable expansion for the fourth edition, a work he could not finish due to his own death.[6][nb 2] Therefore, the fourth (1962/1963) and fifth (1971) editions were continued by Yuri Veniaminovich Geronimus (Russian: Юрий Вениаминович Геронимус, German: Juri Weniaminowitsch Geronimus)[6][nb 5] and Michail Yulyevich Tseytlin (Russian: Михаил Ю́льевич Цейтлин, German: Michael Juljewitsch Zeitlin).[nb 6] The fourth edition contained about 12000 formulas already.[14][nb 3]
"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.)
In 1963, it was followed by the second edition, a reprint edition with a four-page inlet listing corrections compiled by Eldon Robert Hansen.
Derived from the 1963 edition, but considerably expanded and split into two volumes, the third German-English edition by Ludwig Boll[nb 10] was finally published by MIR Moscow in 1981 (with permission of DVW and distributed through Verlag Harri Deutsch in the Western world); it incorporated the material of the fifth Russian edition (1971) as well.[nb 11]
Pending this third German-English edition an English-only edition by Alan Jeffrey[nb 12] was published in 1965. Lacking a clear designation by itself it was variously known as first, third or fourth English edition, as it was based on the then-current fourth Russian edition. The formulas were photographically reproduced and the text translated. This still held true for the expanded fourth English edition in 1980, which added chapters 10 to 17.[17]
Both of these editions saw multiple print runs each incorporating newly found corrections. Starting with the third printing, updated table entries were marked by adding a small superscript number to the entry number indicating the corresponding print run ("3" etc.), a convention carried over into later editions by continuing to increase the superscript number as kind of a revision number (no longer directly corresponding with actual print runs).
The fifth edition (1994), which contained close to 20000 formulas,[18][nb 3] was electronically reset[3] in preparation for a CD-ROM issue of the fifth edition (1996) and in anticipation of further editions. Since the sixth edition (2000), now corresponding with superscript number "10", Daniel Zwillinger[nb 13] started contributing as well. The last edition being edited by Jeffrey before his death[nb 12] was the seventh English edition published in 2007 (with superscript number "11").[19] This edition has been retranslated back into Russian as "seventh Russian edition" in 2011.[20][nb 11]
For the eighth edition (2014/2015, with superscript number "12") Zwillinger took over the role of the editor. He was assisted by Victor Hugo Moll.[21][nb 14] 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 50 pages in total were removed and some chapters renumbered (12 to 11, 17 to 12). This edition contains more than 10000 entries.[21][nb 3]
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Related projects
In 1995, Alan Jeffrey published his Handbook of Mathematical Formulas and Integrals.[22]
It was partially based on the fifth English edition of Gradshteyn and Ryzhik's Table of Integrals, Series, and Products and meant as an companion, but written to be more accessible for students and practitioners.[22] It went through four editions up to 2008.[22][23][24][25] The fourth edition also took advantage of changes incorporated into the seventh English edition of Gradshteyn and Ryzhik.[25]
Inspired by a 1988 paper in which Ilan Vardi[de] proved several integrals in Gradshteyn and Ryzhik,[26] Victor Hugo Moll with George Boros started a project to prove all integrals listed in Gradshteyn and Ryzhik and add additional commentary and references.[27] In the foreword of the book Irresistible Integrals (2004), they wrote:[28]
It took a short time to realize that this task was monumental.
Nevertheless, the efforts have meanwhile resulted in about 900 entries from Gradshteyn and Ryzhik discussed in a series of more than 30 articles[29][30][31] of which papers 1 to 28[a] have been published in issues 14 to 26 of Scientia, Universidad Técnica Federico Santa María (UTFSM), between 2007 and 2015[60] and compiled into a two-volume book series Special Integrals of Gradshteyn and Ryzhik: the Proofs (2014–2015) already.[61][62]
Рыжик, И. М. (1948). Таблицы интегралов, сумм, рядов и произведений (in Russian) (2ed.). Moscow: Gosudarstvennoe Izdatel'stvo Tehniko-Teoretičeskoj Literatury (Государственное издательство технико-теоретической литературы) (GITTL / ГИТТЛ) (Tehteoretizdat / Техтеоретиздат). 400 pages.[11]
Рыжик, И. М.; Градштейн, И. С. (1951). Таблицы интегралов, сумм, рядов и произведений (in Russian) (3ed.). Moscow: Gosudarstvennoe Izdatel'stvo Tehniko-Teoretičeskoj Literatury (Государственное издательство технико-теоретической литературы) (GITTL / ГИТТЛ) (Tehteoretizdat / Техтеоретиздат). LCCN52034158. 464 pages (+ errata inlet).[63][64][65]
Iosif Moiseevich Ryzhik (Иосиф Моисеевич Рыжик)[nb 15] (1918?–1941?). VIAF15286520. GND107340518, 1087809320. (NB. Some sources identify him as a sergeant (сержантом) born in 1918, originally from Vitebsk (Витебска), who was drafted into the army in 1939 from Chkalovsk (Чкаловска), Orenburg (Оренбург), and got missing in December 1941. However, since a birth year 1918 would have made him a very young author (18), this could also have been a namesake. In the foreword of the first edition of the book, Ryzhik thanked three mathematicians of the Moscow Mathematical Society for their suggestions and advice:
Vyacheslav Vassilievich Stepanov (Вячеслав Васильевич Степанов), Aleksei Ivanovich Markushevich (Алексей Иванович Маркушевич), and Ilya Nikolaevich Bronshtein (Илья Николаевич Бронштейн), suggesting that he must have been in some way associated with this group.)
The fact that Ryzhik's death was not announced before the third edition of the book in 1951 might indicate that his status was unclear for a number of years, or, in the case of the first edition, that typesetting had already started, but actual production of the book had to be delayed and was then finalized in his absence as a consequence of the war.
Michail Yulyevich Tseytlin (Михаил Ю́льевич Цейтлин), also as M. Yu. Ceitlin, Michael Juljewitsch Zeitlin, Michael Juljewitsch Zeitlein, Michael Juljewitsch Tseitlin, Mikhail Juljewitsch Tseitlin (?–).
Martin D. H. Strauss also as Martin D. H. Strauß (1907-03-18 Pillau, Baltijsk, Ostpreußen – 1978-05-17, East-Berlin, GDR), GND139569200, German physicist and philosopher.
The seventh Russian edition (2011) was named after the seventh English edition (2007), of which it was a retranslation. There was no sixth genuinely Russian edition. The English series of editions was originally (1965) based on the fourth Russian edition (1962/1963). It is unknown if any changes for the fifth Russian edition (1971) or the third German-English edition (1981), which did incorporate material from the fifth Russian edition, were reflected in any of the English editions in between (and thereby in the seventh Russian edition as well).
Wimp, Jet (April 1997). "Tables of Integrals, Series and Products By I. S. Gradshteyn and I. M. Ryzhik, edited by Alan Jeffrey". American Mathematical Monthly.
Wolfram, Stephen (2005-10-08). "The History and Future of Special Functions". Wolfram Technology Conference, Festschrift for Oleg Marichev, in honor of his 60th birthday (speech, blog post). Champaign, IL, USA: Stephen Wolfram, LLC. The story behind Gradshteyn-Ryzhik. Archived from the original on 2016-04-07. Retrieved 2016-04-06. […] In 1936 Iosif Moiseevich Ryzhik had a book entitled Special Functions published by the United Moscow-Leningrad Scientific-Technical Publisher. Ryzhik died in 1941, either during the siege of Leningrad, or fighting on the Russian front. In 1943, a table of formulas was published under Ryzhik's name by the Governmental Moscow-Leningrad Technical-Theoretical Publisher. The only thing the book seems to say about its origins is that it's responding to the shortage of books of formulas. It says that some integrals marked in it are original, but the others mostly come from three books—a French one from 1858, a German one from 1894, and an American one from 1922. It explains that effort went into the ordering of the integrals, and that some are simplified by using a new special function s equal to Gamma[x+y-1]/(Gamma[x]Gamma[y]). It then thanks three fairly prominent mathematicians from Moscow University. That's basically all we know about Ryzhik. […] Israil Solomonovitch Gradshteyn was born in 1899 in Odessa, and became a professor of mathematics at Moscow State University. But in 1948, he was fired as part of the Soviet attack on Jewish academics. To make money, he wanted to write a book. And so he decided to build on Ryzhik's tables. Apparently he never met Ryzhik. But he created a new edition, and by the third edition, the book was known as Gradshteyn-Ryzhik. […] Gradshteyn died of natural causes in Moscow in 1958. Though somehow there developed an urban legend that one of the authors of Gradshteyn-Ryzhik had been shot as a piece of anti-Semitism on the grounds that an error in their tables had caused an airplane crash. […] Meanwhile, starting around 1953, Yurii Geronimus, who had met Gradshteyn at Moscow State University, began helping with the editing of the tables, and actually added the appendices on special functions. Later on, several more people were involved. And when the tables were published in the West, there were arguments about royalties. But Geronimus [in 2005 was] still alive and well and living in Jerusalem, and Oleg phoned him […]
Moll, Victor Hugo; Vignat, Christophe. "The integrals in Gradshteyn and Ryzhik. Part 29: Chebyshev polynomials"(PDF). Scientia. Series A: Mathematical Sciences. Archived from the original on 2016-03-13. Retrieved 2016-03-13.{{cite journal}}: CS1 maint: unfit URL (link) (NB. This paper discusses 19 GR entries: 1.14.2.1, 1.320, 2.18.1.9, 3.753.2, 3.771.8, 6.611, 7.341.1, 7.341.2, 7.342, 7.343.1, 7.344.1, 7.344.2, 7.346, 7.348, 7.349, 7.355.1, 7.355.2, 8.411.1, 8.921. )
Градштейн, И. С.; Рыжик, И. М. (1971). "Errata in 4th edition". In Геронимус, Ю. В.; Цейтлин, М. Ю́. (eds.). Таблицы интегралов, сумм, рядов и произведений (in Russian) (5ed.). Nauka (Наука). pp.1101–1108. (NB. The 8-page errata list in later print runs of the fourth Russian edition affected 189 table entries.)
Ryshik-Gradstein: Summen-, Produkt- und Integral-Tafeln: Berichtigungen zur 1. Auflage (in German). Berlin, Germany: VEB Deutscher Verlag der Wissenschaften. 1962. MR0175273. (NB. This brochure was available free of charge from the publisher on request.)
Rosenblum, Marvin (October 1996). Koepf, Wolfram (ed.). "4. Table of Integrals, Series, and Products, CD-ROM Version 1.0 Edited by Alan Jeffrey"(PDF). Books and Journals: Review. Orthogonal Polynomials and Special Functions. Vol.7, no.1. SIAM Activity Group on Orthogonal Polynomials and Special Functions. pp.11–12. Archived(PDF) from the original on 2022-01-20. Retrieved 2022-01-23.
De Vos, Alexis (2020-11-09) [2009-03-19]. "Alexis De Vos". Universiteit Gent, Belgium. Archived from the original on 2021-06-13. Retrieved 2022-01-12. […] Finally, he is the proud discoverer of an error in equation 3.454.1 of the Gradshteyn and Ryzhik "Tables of integrals, series, and products". See errata for 6th edition by Alan Jeffrey and Daniel Zwillinger, pages 1 and 19. The error is now corrected in the 7th edition page 363 (with acknowledgement in page xxvi). […]
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