Table of Integrals, Series, and Products
Eighth Edition
Table of
Integrals, Series, and Products Eighth Edition
I. S. Gradshteyn and I. M. Ryzhik
Daniel Zwillinger, Editor
Rensselaer Polytechnic Institute, USA
Victor Moll (Scientific Editor)
Tulane University, Department of Mathematics
Translated from Russian by Scripta Technica, Inc
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ISBN: 978-0-12-384933-5
Library of Congress Cataloging-in-Publication Data Gradshtein, I. S. (Izrail? Solomonovich)
[Tablitsy integralov, summ, riadov i proizvedenii. English]
Table of integrals, series, and products. – Eighth edition / Daniel Zwillinger.
pages cm
Includes bibliographical references and index.
ISBN 978-0-12-384933-5
1. Mathematics–Tables. I. Zwillinger, Daniel, 1957- II. Title.
QA55.G6613 2014 510.21–dc23
2014010276 British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
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Preface to the Eighth Edition
Gradshteyn and Ryzhik continues to be a resource greatly used by mathematicians, scientists, and engi- neers in the theoretical, applied, and computational sciences. Since the publication in 2007 of the revised seventh edition, users have continued to submit corrections and new results that improve the book, and make suggestions for changes that improve the presentation of the material. We regret that the structure of the book makes it impossible to acknowledge these users by their individual contributions so, as usual, their names have been added to the acknowledgment list at the front of the book.
This eighth edition includes corrections received since the publication of the seventh edition, together with a substantial amount of new material acquired from isolated sources. From among the many con- tributions we have included just those integrals that appear commonly in different contexts.
Following our previous conventions, an amended entry has a superscript “12” added to its entry reference number, where the equivalent superscript number for the seventh edition was “11”. Similarly, an asterisk on an entry’s reference number indicates a new result. When, for technical reasons, an entry in a previous edition has been removed, the entry numbers will jump. This preserves the continuity of numbering between the new and older editions. This edition has also removed chapters present in the 7th edition that were not aligned with integrals, series, and products (e.g., the chapters on matrices and norms).
We wish to express our gratitude to all who have been in contact with us with the object of improving and extending the book. Special thanks are extended to both Dr. Francis J. O’Brien, Jr. of the Naval Station in Newport, Rhode Island and Dr. Andrej Tenne-Sens of Ottowa, Ontario, Canada. They have each spent an unimaginably large number of hours helping with this edition.
Experience over many years has shown that each new edition of Gradshteyn and Ryzhik generates many suggestions for new entries and new errata. Hence, we do not expect this new edition to be free from errors. All users who identify errors, or who wish to propose new entries, are invited to contact the authors whose email addresses are listed below. Corrections will be posted on the web sitewww.mathtable.com/
gr/errata.
Finally, we mourn the passing of Professor Alan Jeffrey who, as Editor, guided this book from its 4th edition in 1965 to the 7th edition in 2007. Alan’s dedication to this book was evident both by his longevity—well over 40 years of improvements—and in the consideration he gave to all of our correspon- dents for whom he evaluated a huge number of integrals.
Daniel Zwillinger [email protected] Victor Moll [email protected]
xvii
Acknowledgments
The publisher and editors would like to take this opportunity to express their gratitude to the following users of the Table of Integrals, Series, and Products who either directly or through errata published in Mathematics of Computation have generously contributed corrections and addenda to the original printing.
Anonymous
Dr. Artem G. Abanov Dr. A. Abbas
Dr. S. M. Abrarov Dr. P. B. Abraham Dr. Ari Abramson Dr. Jose Adachi Dr. R. J. Adler Dr. N. Agmon Dr. M. Ahmad Dr. S. A. Ahmad Dr. Rajai S. Alassar Dr. Luis Alvarez-Ruso Dr. Maarten H P Ambaum Dr. R. K. Amiet
Dr. L. U. Ancarani Dr. M. Antoine Dr. Marian Apostol Dr. C. R. Appledorn Dr. D. R. Appleton Dr. Mitsuhiro Arikawa Dr. Ir. Luk R. Arnaut Dr. Peter Arnold Dr. P. Ashoshauvati Dr. C. L. Axness Dr. Scott Baalrud Dr. E. Badralexe Dr. S. B. Bagchi Dr. L. J. Baker Dr. R. Ball Dr. Ingo Barth
Dr. M. P. Barnett Dr. Fabio Bernardoni Dr. Florian Baumann Dr. Norman C. Beaulieu Dr. Jerome Benoit Mr. V. Bentley Dr. Laurent Berger Dr. M. van den Berg Dr. N. F. Berk Dr. C. A. Bertulani Dr. J. Betancort-Rijo Dr. P. Bickerstaff
Dr. Iwo Bialynicki-Birula Dr. Chris Bidinosti Dr. G. R. Bigg Dr. Ian Bindloss Dr. L. Blanchet Dr. Mike Blaskiewicz Dr. R. D. Blevins Dr. Anders Blom Dr. L. M. Blumberg Dr. R. Blumel Dr. S. E. Bodner Dr. Simone Boi Dr. M. Bonsager Dr. George Boros Dr. S. Bosanac
Dr. Ruben Van Boxem Dr. Christoph Bruder Dr. Patrick Bruno Dr. B. Van den Bossche
Dr. A. Bostr¨om Dr. J. E. Bowcock Dr. T. H. Boyer Dr. K. M. Briggs Dr. D. J. Broadhurst Dr. Chris Van Den Broeck Dr. W. B. Brower
Dr. H. N. Browne Dr. Christoph Bruegger Dr. William J. Bruno Dr. Vladimir Bubanja Dr. D. J. Buch Dr. D. J. Bukman Dr. F. M. Burrows Dr. R. Caboz Dr. T. Calloway Dr. F. Calogero Dr. D. Dal Cappello Dr. David Cardon Dr. J. A. Carlson Gallos Dr. B. Carrascal
Dr. A. R. Carr Dr. Neal Carron Dr. Florian Cartarius Dr. S. Carter
Dr. Miguel Carvajal Dr. G. Cavalleri
Mr. W. H. L. Cawthorne Dr. Alexandre Caz´e Dr. A. Cecchini Dr. B. Chan
xix
xx Acknowledgments
Dr. M. A. Chaudhry Dr. Sabino Chavez-Cerda Dr. Julian Cheng
Dr. H. W. Chew Dr. D. Chin
Dr. Young-seek Chung Dr. S. Ciccariello Dr. N. S. Clarke Dr. R. W. Cleary Dr. A. Clement Dr. Alain Cochard Dr. P. Cochrane Dr. D. K. Cohoon Dr. Howard S. Cohl Dr. L. Cole
Dr. Filippo Colomo Donal Connon Dr. J. R. D. Copley Henry Corback, Esq.
Dr. Daniel Benevides da Costa Dr. Barry J. Cox
Dr. D. Cox Dr. J. Cox Dr. J. W. Criss Dr. A. E. Curzon Dr. D. Dadyburjor Dr. D. Dajaputra Dr. C. Dal Cappello Dr. P. Daly
Dr. S. Dasgupta
Dr. Charles E. Davidson Dr. John Davies
Dr. C. L. Davis Dr. A. Degasperis Dr. Gustav Delius Dr. B. C. Denardo Dr. R. W. Dent Dr. E. Deutsch Dr. D. deVries Dr. Eran Dgani Dr. Enno Diekema Dr. P. Dita
Dr. P. J. de Doelder Dr. Mischa Dohler Dr. G. Dˆome Dr. Shi-Hai Dong Dr. Balazs Dora
Dr. M. R. D’Orsogna Dr. Forrest Doss
Dr. Adrian A. Dragulescu Dr. Zvi Drezner
Dr. Eduardo Duenez Mr. Tommi J. Dufva Dr. Duc V. Duong Dr. E. B. Dussan, V Dr. Percy Dusek Dr. C. A. Ebner Dr. M. van der Ende Dr. Jonathan Engle Dr. G. Eng
Dr. E. S. Erck Dr. Grant Erdmann Dr. Jan Erkelens Dr. Olivier Espinosa Dr. G. A. Est´evez Dr. K. Evans Dr. G. Evendon Dr. Valery I. Fabrikant Dr. L. A. Falkovsky Dr. Kambiz Farahmand Dr. Richard J. Fateman Dr. G. Fedele
Dr. A. R. Ferchmin Dr. P. Ferrant Dr. Andr´e Ferrari Dr. H. E. Fettis Dr. W. B. Fichter Dr. George Fikioris Mr. J. C. S. S. Filho Dr. L. Ford
Dr. Nicolao Fornengo Dr. J. France
Dr. B. Frank Dr. S. Frasier
Dr. Stefan Fredenhagen Dr. A. J. Freeman Dr. A. Frink
Dr. Jason M. Gallaspy Dr. J. A. C. Gallas Dr. J. A. Carlson Gallas Dr. G. R. Gamertsfelder Dr. Jianliang Gao Dr. T. Garavaglia
Dr. Jaime Zaratiegui Garcia
Dr. C. G. Gardner Dr. D. Garfinkle Dr. P. N. Garner Dr. F. Gasser Dr. E. Gath Dr. P. Gatt Dr. D. Gay Dr. M. P. Gelfand Dr. M. R. Geller Dr. Ali I. Genc Dr. Vincent Genot Dr. M. F. George Dr. Teschl Gerald Dr. P. Germain
Dr. Ing. Christoph Gierull Dr. S. P. Gill
Dr. Federico Girosi Dr. E. A. Gislason Dr. M. I. Glasser Dr. P. A. Glendinning Dr. L. I. Goldfischer Dr. Denis Golosov Dr. I. J. Good Dr. J. Good Mr. L. Gorin Dr. Martin G¨otz Dr. R. Govindaraj Dr. M. De Grauf Dr. Gabriele Gradoni Dr. L. Green
Mr. Leslie O. Green Dr. R. Greenwell Dr. K. D. Grimsley Dr. Albert Groenenboom Dr. V. Gudmundsson Dr. J. Guillera Dr. K. Gunn Dr. D. L. Gunter
Dr. Julio C. Guti´errez-Vega Dr. Roger Haagmans Dr. Howard Haber Dr. H. van Haeringen Dr. B. Hafizi
Dr. Bahman Hafizi Dr. T. Hagfors Dr. M. J. Haggerty Dr. Timo Hakulinen
Acknowledgments xxi
Dr. S. E. Hammel Dr. E. Hansen Dr. Wes Harker Dr. T. Harrett Dr. D. O. Harris Dr. Frank Harris Mr. Mazen D. Hasna Dr. Peter Hawkins Dr. Joel G. Heinrich Dr. Adam Dade Henderson Dr. Franck Hersant
Dr. Sten Herlitz Dr. Chris Herzog Dr. A. Higuchi Dr. R. E. Hise Dr. Henrik Holm Dr. N. Holte Dr. R. W. Hopper Dr. P. N. Houle Dr. C. J. Howard Jie Hu
Dr. Ben Yu-Kuang Hu Dr. J. H. Hubbell Dr. Felix Huber Dr. J. R. Hull Dr. W. Humphries Dr. Jean-Marc Hur´e Dr. Jamal A. Hussein Dr. Y. Iksbe
Dr. Philip Ingenhoven Mr. L. Iossif
Dr. Sean A. Irvine Dr. ´Ottar ´Isberg Dr. Kazuhiro Ishida Dr. Cyril-Daniel Iskander Dr. S. A. Jackson Dr. John David Jackson Dr. Francois Jaclot Dr. B. Jacobs Dr. Pierre Jacobs Dr. E. C. James Dr. B. Jancovici Dr. D. J. Jeffrey Dr. H. J. Jensen Dr. Bin Jiang
Dr. Edwin F. Johnson Dr. I. R. Johnson
Dr. Steven Johnson Dr. Joel T. Johnson Dr. Fredrik Johansson Dr. I. Johnstone Dr. Y. P. Joshi Dr. Jae-Hun Jung Dr. Damir Juric Dr. Florian Kaempfer Dr. S. Kanmani Dr. Z. Kapal
Dr. Peter Karadimas Dr. Dave Kasper Dr. M. Kaufman Dr. Eduardo Kausel Dr. B. Kay
Louis Kempeneers Dr. Jack Kerlin Dr. Avinash Khare Dr. Karen T. Kohl Dr. Ilki Kim Dr. Youngsun Kim Dr. S. Klama Dr. L. Klingen Dr. C. Knessl Dr. M. J. Knight Dr. Yannis Kohninos Dr. D. Koks
Dr. L. P. Kok Dr. K. S. K¨olbig Dr. Yannis Komninos Dr. D. D. Konowalow Dr. Z. Kopal
Dr. I. Kostyukov Dr. R. A. Krajcik Dr. Vincent Krakoviack Dr. Stefan Kr¨amer Dr. Tobias Kramer Dr. Hermann Krebs Dr. Chethan Krishnan Dr. J. W. Krozel Dr. E. D. Krupnikov Dr. Kun-Lin Kuo Dr. E. A. Kuraev Dr. Heinrich Kuttler Dr. Konstantinos Kyritsis Dr. Velimir Labinac
Dr. Javier Navarro Laboulais
Dr. A. D. J. Lambert Dr. A. Lambert Dr. A. Larraza Dr. K. D. Lee Dr. M. Howard Lee Dr. M. K. Lee Dr. P. A. Lee Dr. Todd Lee Dr. J. Legg Dr. Xianfu Lei Dr. Yefim Leifman Dr. Remigijus Leipus Dr. Armando Lemus Dr. Elvio Leonardo Dr. Eugene Lepelaars Dr. S. L. Levie Dr. D. Levi Dr. Michael Lexa Dr. Liang Li Dr. Kuo Kan Liang Dr. Sergey Liflandsky Dr. B. Linet
Dr. Andrej Likar Dr. M. A. Lisa Dr. Donald Livesay Dr. H. Li
Dr. Xian-Fang Li Dr. Georg Lohoefer Dr. I. M. Longman Dr. D. Long Dr. Sylvie Lorthois Dr. Wenzhou Lu Dr. Phil Lucht Dr. Stephan Ludwig Dr. Arpad Lukacs Dr. Y. L. Luke Dr. W. Lukosz Dr. T. Lundgren Dr. E. A. Luraev Dr. R. Lynch Dr. K. B. Ma Dr. Ilari Maasilta Dr. R. Mahurin Dr. R. Mallier Dr. G. A. Mamon Dr. A. Mangiarotti Dr. I. Manning
xxii Acknowledgments
Dr. Greg Marks Dr. J. Marmur Dr. A. Martin
Dr. Carmelo P. Martin Sr. Yuzo Maruyama Dr. Richard Marthar Dr. David J. Masiello Dr. Richard J. Mathar Dr. H. A. Mavromatis Dr. M. Mazzoni Dr. P. McCullagh Dr. J. H. McDonnell Dr. J. R. McGregor Dr. Kim McInturff Dr. N. McKinney
Dr. David McA McKirdy Dr. Andrew J. McHutchon Dr. Rami Mehrem
Dr. W. N. Mei Dr. Angelo Melino Mr. Jos´e Ricardo Mendes Dr. Zvi Mendlowitz Dr. Andy Mennim Dr. Sarah Messer Dr. J. P. Meunier Dr. Haixing Miao Dr. Krys A. Michalski Dr. Gerard P. Michon Dr. D. F. R. Mildner Dr. D. L. Miller Dr. Steve Miller Dr. P. C. D. Milly Dr. S. P. Mitra Dr. K. Miura Dr. N. Mohankumar Dr. M. Moll
Dr. Victor H. Moll Dr. D. Monowalow Mr. Tony Montagnese Dr. Thierry Montagu Dr. Jim Morehead Dr. J. Morice Dr. J. Guy Morgan Dr. W. Mueck Dr. C. Muhlhausen Dr. S. Mukherjee Dr. R. R. M¨uller
Dr. Pablo Parmezani Munhoz Dr. Frank Namin
Dr. Paul Nanninga Dr. A. Natarajan Dr. Sven Peter N¨asholm Dr. Javier Navarro Dr. Christian Netzel Dr. Stefan Neumeier Dr. C. T. Nguyen Dr. A. C. Nicol Dr. M. M. Nieto Dr. P. Noerdlinger Dr. Andrew N. Norris Dr. K. H. Norwich Dr. A. H. Nuttall Dr. F. O’Brien Dr. R. P. O’Keeffe Dr. A. Ojo
Dr. O. Olendski Dr. Oleg Olendski Dr. P. Olsson Dr. Gilad Oren Dr. M. Ortner Dr. Matthew Orton Dr. S. Ostlund Dr. J. Overduin Dr. J. Pachner
Mr. Robert A. Padgug Dr. D. Papadopoulos Dr. F. J. Papp Mr. Man Sik Park Dr. Jong-Do Park Dr. B. Patterson Dr. R. F. Pawula Dr. D. W. Peaceman Dr. Vittorio Peano Dr. Vincent Pegoraro Dr. D. Pelat
Dr. L. Peliti Dr. Y. P. Pellegrini Dr. Thiago S. Pereira Dr. G. J. Pert
Dr. Nicola Pessina Dr. J. B. Peterson Dr. Rickard Petersson Dr. Arnaud Pierens Dr. Emilio Pisanty
Dr. Ralph Pixley Dr. Andrew Plumb Dr. Dror Porat Dr. E. A. Power Dr. E. Predazzi Dr. William S. Price Dr. Gunnar Pruessner Dr. Paul Radmore Dr. Carl E. Rasmussen Dr. F. Raynal
Dr. X. R. Resende Dr. Guy A. Reynolds Dr. J. M. Riedler Dr. Sjoerd Rienstra Dr. Thomas Richard Dr. E. Ringel Dr. T. M. Roberts Dr. N. I. Robinson Dr. P. A. Robinson Dr. Elvira Romera Dr. D. M. Rosenblum Dr. R. A. Rosthal Dr. J. R. Roth Dr. Klaus Rottbrand Dr. Bastien Roucaries Dr. D. Roy
Dr. E. Royer Dr. D. Rudermann Dr. Ahmad Rushdi Dr. Ali Rushdi Dr. Niall Ryan
Dr. Sanjib Sabhapandit Dr. C. T. Sachradja Dr. J. Sadiku Dr. A. Sadiq
Dr. Motohiko Saitoh Dr. Naoki Saito Dr. A. Salim Dr. Sherwood Samn Dr. J. H. Samson
Dr. Miguel A. Sanchis-Lozano Dr. J. A. Sanders
Dr. M. A. F. Sanjun Dr. P. Sarquiz Dr. Avadh Saxena Dr. Vito Scarola Dr. O. Sch¨arpf
Acknowledgments xxiii
Dr. A. Scherzinger Dr. B. Schizer Dr. Martin Schmid Dr. J. Scholes Dr. Mel Schopper Dr. H. J. Schulz Dr. Andreas Schulz Dr. Markus Schwarz Dr. G. J. Sears Dr. Kazuhiko Seki Dr. B. Seshadri Dr. Roger Sewell Dr. A. Shapiro Dr. Sihong Shao Dr. Masaki Shigemori Dr. J. S. Sheng Dr. Kenneth Ing Shing Dr. Tomohiro Shirai Dr. S. Shlomo Dr. D. Siegel
Dr. Matthew Stapleton Dr. Leo Stein
Dr. Michael L. Stein Dr. Steven H. Simon Dr. Ashok Kumar Singal Dr. Constantin Siriteanu Dr. C. Smith
Dr. Richard Smith Dr. G. C. C. Smith
Dr. Stefan Llewellyn Smith Dr. S. Smith
Dr. Sasha Sodin Dr. G. Solt Dr. J. Sondow Dr. A. Sørenssen Dr. Marcus Spradlin Dr. Andrzej Staruszkiewicz Dr. Philip C. L. Stephenson Dr. Edgardo Stockmeyer Dr. J. C. Straton Mr. H. Suraweera Dr. N. F. Svaiter Dr. V. Svaiter Dr. R. Szmytkowski
Dr. Sebastian S. Szyszkowicz Dr. S. Tabachnik
Dr. Erik Talvila Dr. G. Tanaka Dr. C. Tanguy Dr. G. K. Tannahill Dr. B. T. Tan Dr. C. Tavard Dr. Gon¸calo Tavares Dr. Aba Teleki Andrej Tenne-Sens Dr. Gerald Teschl
Dr. Arash Dahi Taleghani Dr. D. Temperley
Dr. A. J. Tervoort
Dr. Theodoros Theodoulidis Dr. D. J. Thomas
Dr. Michael Thorwart Dr. S. T. Thynell Dr. Tamer Tlas Dr. D. C. Torney Dr. R. Tough
Dr. Marwan Toutounji Dr. Dennis Trede Dr. B. F. Treadway Dr. Ming Tsai Dr. N. Turkkan Dr. Sandeep Tyagi Dr. J. J. Tyson Dr. Takahiro Ueda Dr. S. Uehara Dr. M. Vadacchino Dr. Stathis Vagenas Dr. O. T. Valls Dr. D. Vandeth
Dr. Klaas Vantournhout Mr. Andras Vanyolos Dr. D. Veitch
Mr. Jose Lopez Vicario Dr. Hari Vishnu Dr. K. Vogel
Dr. J. M. M. J. Vogels Dr. Alexis De Vos Dr. Emanuel Voto Dr. Stuart Walsh Dr. Haiming Wang Dr. J. J. Wang
Dr. Reinhold Wannemacher
Dr. S. Wanzura Dr. J. Ward Dr. S. I. Warshaw Dr. Alex Watson Dr. R. Weber
Dr. Steffen Weissmann Dr. Wei Qian
Dr. Detmar Welz Dr. Kyle Wendt Dr. D. H. Werner Dr. E. Wetzel
Dr. Robert Whittaker Dr. Peter Widerin Dr. D. T. Wilton Dr. C. Wiuf Dr. K. T. Wong
Dr. Rog´erio Nunes Wolff Mr. J. N. Wright
Dr. J. D. Wright Dr. D. Wright Dr. Chong-shi Wu Dr. D. Wu
Dr. Roahn Wynar Dr. Takashi Yanagisawa Dr. Michel Daoud Yacoub Dr. Yu S. Yakovlev Dr. H.-C. Yang Dr. J. J. Yang Dr. Z. J. Yang Dr. Mingwu Yao Dr. Yu-Min Yen
Mr. Chun Kin Au Yeung Dr. Steve Young
Dr. Kazuya Yuasa Dr. S. P. Yukon Dr. Zhuo-Quan Zeng Dr. B. Zhang
Dr. Peng Zhang Dr. Y. C. Zhang Dr. Y. Zhao Dr. Ralf Zimmer Dr. Chunhui Zhu Dr. Zeng Zhuo-Quan
The Order of Presentation of the Formulas
The question of the most expedient order in which to give the formulas, in particular, in what division to include particular formulas such as the definite integrals, turned out to be quite complicated. The thought naturally occurs to set up an order analogous to that of a dictionary. However, it is almost impossible to create such a system for the formulas of integral calculus. Indeed, in an arbitrary formula of the form
b
a f(x) dx=A
one may make a large number of substitutions of the form x = ϕ(t) and thus obtain a number of
“synonyms” of the given formula. We must point out that the table of definite integrals by Bierens de Haan and the earlier editions of the present reference both sin in the plethora of such “synonyms”
and formulas of complicated form. In the present edition, we have tried to keep only the simplest of the “synonym” formulas. Basically, we judged the simplicity of a formula from the standpoint of the simplicity of the arguments of the “outer” functions that appear in the integrand. Where possible, we have replaced a complicated formula with a simpler one. Sometimes, several complicated formulas were thereby reduced to a single simpler one. We then kept only the simplest formula. As a result of such substitutions, we sometimes obtained an integral that could be evaluated by use of the formulas of chapter two and the Newton–Leibniz formula, or to an integral of the form
a
−af(x) dx,
wheref(x) is an odd function. In such cases the complicated integrals have been omitted.
Let us give an example using the expression π/4
0
(cotx−1)p−1
sin2x ln tanxdx=−π
pcosecpπ. (0.1)
By making the natural substitutionu= cotx−1, we obtain
∞
0 up−1ln(1 +u) du= π
p cosecpπ. (0.2)
Integrals similar to formula (0.1) are omitted in this new edition. Instead, we have formula (0.2).
xxv
xxvi The Order of Presentation of the Formulas
As a second example, let us take I=
π/2
0
ln (tanpx+ cotpx) ln tanxdx= 0. The substitutionu= tanxyields
I=
∞
0
ln (up+u−p) lnu 1 +u2 du.
If we now setυ= lnu, we obtain I=
∞
−∞
υeυ 1 +e2υln
epυ+e−pυ dυ=
∞
−∞υln (2 coshpυ) 2 coshυ dυ.
The integrand is odd and, consequently, the integral is equal to 0.
Thus, before looking for an integral in the tables, the user should simplify as much as possible the arguments (the “inner” functions) of the functions in the integrand.
The functions are ordered as follows: First we have the elementary functions:
1. The functionf(x) =x. 2. The exponential function.
3. The hyperbolic functions.
4. The trigonometric functions.
5. The logarithmic function.
6. The inverse hyperbolic functions. (These are replaced with the corresponding logarithms in the formulas containing definite integrals.)
7. The inverse trigonometric functions.
Then follow the special functions:
8. Elliptic integrals.
9. Elliptic functions.
10. The logarithm integral, the exponential integral, the sine integral, and the cosine integral functions.
11. Probability integrals and Fresnel’s integrals.
12. The gamma function and related functions.
13. Bessel functions.
14. Mathieu functions.
15. Legendre functions.
16. Orthogonal polynomials.
17. Hypergeometric functions.
18. Degenerate hypergeometric functions.
19. Parabolic cylinder functions.
20. Meijer’s and MacRobert’s functions.
21. Riemann’s zeta function.
The integrals are arranged in order of outer function according to the above scheme: the farther down in the list a function occurs, (i.e., the more complex it is) the later will the corresponding formula appear
The Order of Presentation of the Formulas xxvii
in the tables. Suppose that several expressions have the same outer function. For example, consider sinex, sinx, sin lnx. Here, the outer function is the sine function in all three cases. Such expressions are then arranged in order of the inner function. In the present work, these functions are therefore arranged in the following order: sinx, sinex, sin lnx.
Our list does not include polynomials, rational functions, powers, or other algebraic functions. An algebraic function that is included in tables of definite integrals can usually be reduced to a finite com- bination of roots of rational power. Therefore, for classifying our formulas, we can conditionally treat a power function as a generalization of an algebraic and, consequently, of a rational function.∗ We shall distinguish between all these functions and those listed above and we shall treat them as operators.
Thus, in the expression sin2ex, we shall think of the squaring operator as applied to the outer function, namely, the sine. In the expression sinsinx+cosx−cosxx, we shall think of the rational operator as applied to the trigonometric functions sine and cosine. We shall arrange the operators according to the following order:
1. Polynomials (listed in order of their degree).
2. Rational operators.
3. Algebraic operators (expressions of the formAp/q, whereqandpare rational, andq >0; these are listed according to the size ofq).
4. Power operators.
Expressions with the same outer and inner functions are arranged in the order of complexity of the operators. For example, the following functions (whose outer functions are all trigonometric, and whose inner functions are allf(x) =x) are arranged in the order shown:
sinx, sinxcosx, 1
sinx = cosecx, sinx
cosx= tanx, sinx+ cosx
sinx−cosx, sinmx, sinmxcosx.
Furthermore, if two outer functionsϕ1(x) andϕ2(x), whereϕ1(x) is more complex thanϕ2(x), appear in an integrand and if any of the operations mentioned are performed on them, the corresponding integral will appear (in the order determined by the position ofϕ2(x) in the list) after all integrals containing only the function ϕ1(x). Thus, following the trigonometric functions are the trigonometric and power functions (that is,ϕ2(x) =x). Then come
• combinations of trigonometric and exponential functions,
• combinations of trigonometric functions, exponential functions, and powers, etc.,
• combinations of trigonometric and hyperbolic functions, etc.
Integrals containing two functionsϕ1(x) andϕ2(x) are located in the division and order corresponding to the more complicated function of the two. However, if the positions of several integrals coincide because they contain the same complicated function, these integrals are put in the position defined by the complexity of the second function.
To these rules of a general nature, we need to add certain particular considerations that will be easily understood from the tables. For example, according to the above remarks, the functione1x comes after ex as regards complexity, but lnxand ln1
x are equally complex since ln1
x =−lnx. In the section on
“powers and algebraic functions”, polynomials, rational functions, and powers of powers are formed from power functions of the form (a+bx)n and (α+βx)ν.
∗For any natural numbern, the involution (a+bx)nof the binomiala+bxis a polynomial. Ifnis a negative integer, (a+bx)nis a rational function. Ifnis irrational, the function (a+bx)nis not even an algebraic function.
Use of the Tables ∗
For the effective use of the tables contained in this book it is necessary that the user should first become familiar with the classification system for integrals devised by the authors Gradshteyn and Ryzhik. This classification is described in detail in the section entitledThe Order of Presentation of the Formulas (see page xxv) and essentially involves the separation of the integrand intoinner and outer functions. The principal function involved in the integrand is called theouter function and its argument, which is itself usually another function, is called theinner function. Thus, if the integrand comprised the expression ln sinx, the outer function would be the logarithmic function while its argument, the inner function, would be the trigonometric function sinx. The desired integral would then be found in the section dealing with logarithmic functions, its position within that section being determined by the position of theinner function (here a trigonometric function) in Gradshteyn and Ryzhik’s list of functional forms.
It is inevitable that some duplication of symbols will occur within such a large collection of integrals and this happens most frequently in the first part of the book dealing with algebraic and trigonometric integrands. The symbols most frequently involved areα, β, γ, δ, t, u, z, zk, and Δ. The expressions associated with these symbols are used consistently within each section and are defined at the start of each new section in which they occur. Consequently, reference should be made to the beginning of the section being used in order to verify the meaning of the substitutions involved.
Integrals of algebraic functions are expressed as combinations of roots with rational power indices, and definite integrals of such functions are frequently expressed in terms of the Legendre elliptic integrals F(φ, k),E(φ, k) and Π(φ, n, k), respectively, of the first, second and third kinds.
The four inverse hyperbolic functions arcsinhz, arccoshz, arctanhz and arccothz are introduced through the definitions
arcsinz=1
i arcsinh(iz) arccosz=1
i arccosh(z) arctanz=1
i arctanh(iz) arccotz=iarccoth(iz)
∗Prepared by Alan Jeffrey for the English language edition.
xxix
xxx Use of the Tables
or
arcsinhz= 1
iarcsin(iz) arccoshz=iarccosz arctanhz= 1
iarctan(iz) arccothz= 1
iarccot(−iz)
The numerical constantsCandGwhich often appear in the definite integrals denote Euler’s constant and Catalan’s constant, respectively. Euler’s constantC is defined by the limit
C= lim
s→∞
s
m=1
1 m−lns
= 0.577215. . . .
On occasions other writers denote Euler’s constant by the symbolγ, but this is also often used instead to denote the constant
γ=eC= 1.781072. . . . Catalan’s constantGis related to the complete elliptic integral
K≡K(k)≡ π/2
0
dx 1−k2sin2x by the expression
G= 1 2
1
0 Kdk= ∞ m=0
(−1)m
(2m+ 1)2 = 0.915965. . ..
Since the notations and definitions for higher transcendental functions that are used by different authors are by no means uniform, it is advisable to check the definitions of the functions that occur in these tables. This can be done by identifying the required function by symbol and name in theIndex of Special Functions and Notation on page xxxvii, and by then referring to the defining formula or section number listed there. We now present a brief discussion of some of the most commonly used alternative notations and definitions for higher transcendental functions.
Bernoulli and Euler Polynomials and Numbers
Extensive use is made throughout the book of the Bernoulli and Euler numbers Bn and En that are defined in terms of the Bernoulli and Euler polynomials of ordern,Bn(x) andEn(x), respectively. These polynomials are defined by the generating functions
text et−1 =
∞ n=0
Bn(x)tn
n! for|t|<2π and
2ext et+ 1 =
∞ n=0
En(x)tn
n! for|t|< π.
The Bernoulli numbers are always denoted byBn and are defined by the relation Bn=Bn(0) forn= 0,1, . . . ,
when
B0= 1, B1=−1
2, B2=1
6, B4=−1 30, . . . .
Use of the Tables xxxi
The Euler numbersEn are defined by setting En = 2nEn
1 2
forn= 0,1, . . . TheEn are all integral andE0= 1,E2=−1,E4= 5,E6=−61,. . ..
An alternative definition of Bernoulli numbers, which we shall denote by the symbol B∗n, uses the same generating function but identifies theBn∗ differently in the following manner:
t
et−1 = 1−1
2t+B1∗t2
2!−B2∗t4 4! +. . . . This definition then gives rise to the alternative set of Bernoulli numbers
B1∗= 1/6, B2∗= 1/30, B∗3= 1/42, B4∗= 1/30, B5∗= 5/66, B6∗= 691/2730, B∗7= 7/6, B8∗= 3617/510, . . . .
These differences in notation must also be taken into account when using the following relationships that exist between the Bernoulli and Euler polynomials:
Bn(x) = 1 2n
n k=0
n k
Bn−kEk(2x) n= 0,1, . . .
En−1(x) = 2n n
Bn
x+ 1 2
−Bn x 2
or
En−1(x) = 2 n
Bn(x)−2nBn x 2
n= 1,2, . . . and
En−2(x) = 2 n 2
−1n−2
k=0
n
k 2n−k−1
Bn−kBk(x) n= 2,3, . . .
There are also alternative definitions of the Euler polynomial of ordern, and it should be noted that some authors, using a modification of the third expression above, call
2
n+ 1 Bn(x)−2nBn x 2
the Euler polynomial of ordern.
Elliptic Functions and Elliptic Integrals
The following notations are often used in connection with the inverse elliptic functions snu, cnu, and dnu:
nsu= 1
snu ncu= 1
cnu ndu= 1
dnu scu= snu
cnu csu= cnu
snu dsu= dnu
snu sdu= snu
dnu cdu= cnu
dnu dcu= dnu
cnu
xxxii Use of the Tables
The elliptic integral of the third kind is defined by Gradshteyn and Ryzhik to be Π
ϕ, n2, k
= ϕ
0
da
1−n2sin2a 1−k2sin2a
= sinϕ
0
dx (1−n2x2)
(1−x2) (1−k2x2)
−∞< n2<∞
The Jacobi Zeta Function and Theta Functions
The Jacobi zeta function zn(u, k), frequently writtenZ(u), is defined by the relation zn(u, k) =Z(u) =
u
0
dn2υ− E K
dυ=E(u)− E Ku.
This is related to the theta functions by the relationship zn(u, k) = ∂
∂uln Θ(u) giving
(i). zn(u, k) = π 2K
ϑ1 πu 2K ϑ1 πu
2K
−cnudnu snu
(ii). zn(u, k) = π 2K
ϑ2 πu 2K ϑ2 πu
2K
+dnusnu cnu
(iii). zn(u, k) = π 2K
ϑ3 πu 2K ϑ3 πu
2K
−k2snucnu dnu
(iv). zn(u, k) = π 2K
ϑ4 πu 2K ϑ4 πu
2K
Many different notations for the theta function are in current use. The most common variants are the replacement of the argumentuby the argumentu/πand, occasionally, a permutation of the identification of the functionsϑ1 toϑ4with the function ϑ4 replaced byϑ.
The Factorial (Gamma) Function
In older reference texts the gamma function Γ(z), defined by the Euler integral Γ(z) =
∞
0 tz−1e−tdt, is sometimes expressed in the alternative notation
Γ(1 +z) =z! = Π(z).
On occasions the related derivative of the logarithmic factorial function Ψ(z) is used where d(lnz!)
dz = (z!)
z! = Ψ(z+ 1).
Use of the Tables xxxiii
This function satisfies the recurrence relation
Ψ(z) = Ψ(z−1) + 1 z−1 and is defined by the series
Ψ(z) =−C+ ∞ n=0
1
n+ 1 − 1 z+n
. The derivative Ψ(z) satisfies the recurrence relation
Ψ(z+ 1) = Ψ(z)− 1 z2 and is defined by the series
Ψ(z) = ∞ n=0
1 (z+n)2. Exponential and Related Integrals
The exponential integralsEn(z) have been defined by Schloemilch using the integral En(z) =
∞
1 e−ztt−ndt (n= 0,1, . . . , Rez >0)
They should not be confused with the Euler polynomials already mentioned. The function E1(z) is related to the exponential integral Ei(z) through the expressions
E1(z) =−Ei(−z) = ∞
z e−tt−1dt and
li(z) = z
0
dt
lnt = Ei (lnz) [z >1]
The functionsEn(z) satisfy the recurrence relations En(z) = 1
n−1
e−z−zEn−1(z)
[n >1]
and
En(z) =−En−1(z) with
E0(z) =e−z/z.
The function En(z) has the asymptotic expansion En(z)∼e−z
z
1−n
z +n(n+ 1)
z2 −n(n+ 1)(n+ 2) z3 +· · ·
|argz|< 3π 2
while for largen,
En(x) = e−x x+n
1 + n
(x+n)2 +n(n−2x) (x+n)4 +n
6x2−8nx+n2
(x+n)6 +R(n, x)
, where
−0.36n−4≤R(n, x)≤
1 + 1
x+n−1
n−4 [x >0]
The sine and cosine integrals si(x) and ci(x) are related to the functions Si(x) and Ci(x) by the integrals
Si(x) = x
0
sint
t dt= si(x) +π 2 and
xxxiv Use of the Tables
Ci(x) =C+ lnx+ x
0
(cost−1) t dt.
The hyperbolic sine and cosine integrals shi(x) and chi(x) are defined by the relations shi(x) =
x
0
sinht t dt and
chi(x) =C+ lnx+ x
0
(cosht−1) t dt.
Some authors write
Cin(x) = x
0
(1−cost) t dt so that
Cin(x) =−Ci(x) + lnx+C.
The error function erf(x) is defined by the relation erf(x) = Φ(x) = 2
√π x
0 e−t2dt
and the complementary error function erfc(x) is related to the error function erfc(x) and to Φ(x) by the expression
erfc(x) = 1−erf(x).
The Fresnel integralsS(x) andC(x) are defined by Gradshteyn and Ryzhik as S(x) = 2
√2π x
0
sint2dt and
C(x) = 2
√2π x
0
cost2dt.
Other definitions that are in use are S1(x) =
x
0
sinπt2
2 dt, C1(x) = x
0
cosπt2 2 dt and
S2(x) = 1
√2π x
0
sin√t
t dt, C2(x) = 1
√2π x
0
cos√t t dt These are related by the expressions
S(x) =S1
x
2 π
=S2 x2 and
C(x) =C1
x
2 π
=C2 x2
Hermite and Chebyshev Orthogonal Polynomials
The Hermite polynomialsHn(x) are related to the Hermite polynomialsHen(x) by the relations Hen(x) = 2−n/2Hn
√x 2
and
Hn(x) = 2n/2Hen x√ 2
.
Use of the Tables xxxv
These functions satisfy the differential equations d2Hn
dx2 −2xdHn
dx + 2n Hn= 0 and
d2Hen
dx2 −xdHen
dx +n Hen= 0. They obey the recurrence relations
Hn+1= 2xHn−2nHn−1
and
Hen+1=xHen−nHen−1
The first six orthogonal polynomialsHen are
He0= 1, He1=x, He2=x2−1, He3=x3−3x, He4=x4−6x2+ 3, He5=x5−10x3+ 15x.
Sometimes the Chebyshev polynomialUn(x) of the second kind is defined as a solution of the equation (1−x2)d2y
dx2 −3xdy
dx+n(n+ 2)y= 0. Bessel Functions
A variety of different notations for Bessel functions are in use. Some common ones involve the replacement ofYn(z) by Nn(z) and the introduction of the symbol
Λn(z) = 1
2z −n
Γ(n+ 1)Jn(z).
In the book by Gray, Mathews and MacRobert the symbol Yn(z) is used to denote 1
2πYn(z) + (ln 2−C)Jn(z) while Neumann uses the symbolY(n)(z) for the identical quantity.
The Hankel functions H(1)ν (z) and H(2)ν (z) are sometimes denoted by Hsν(z) and Hiν(z) and some authors writeGν(z) =
1 2
πiH(1)ν (z).
The Neumann polynomial On(t) is a polynomial of degree n+ 1 in 1/t, with O0(t) = 1/t. The polynomialsOn(t) are defined by the generating function
1
t−z =J0(z)O0(t) + 2 ∞ k=1
Jk(z)Ok(t), giving
On(t) =1 4
[n/2]
k=0
n(n−k−1)!
k!
2 t
n−2k+1
forn= 1,2, . . . , where 1
2n
signifies the integral part of 12n. The following relationship holds between three successive polynomials:
(n−1)On+1(t) + (n+ 1)On−1(t)−2 n2−1
t On(t) =2n
t sin2 nπ 2 .
xxxvi Use of the Tables
The Airy functions Ai(z) and Bi(z) are independent solutions of the equation d2u
dz2 −zu= 0.
The solutions can be represented in terms of Bessel functions by the expressions Ai(z) = 1
3
√z
I−1/3
2 3z3/2
−I1/3
2 3z3/2
= 1 π
z 3K1/3
2 3z3/2
Ai(−z) = 1 3
√z
J1/3
2 3z3/2
+J−1/3
2 3z3/2
and by
Bi(z) = z
3
I−1/3
2 3z3/2
+I1/3
2 3z3/2
, Bi(−z) =
z 3
J−1/3
2 3z3/2
−J1/3
2 3z3/2
.
Parabolic Cylinder Functions and Whittaker Functions The differential equation
d2y
dz2 + (az2+bz+c)y= 0 has associated with it the two equations
d2y dz2+
1 4z2+a
y= 0 and d2y dz2−
1 4z2+a
y= 0
the solutions of which are parabolic cylinder functions. The first equation can be derived from the second by replacingz byzeiπ/4 andaby−ia.
The solutions of the equation
d2y dz2−
1 4z2+a
y= 0
are sometimes writtenU(a, z) andV(a, z). These solutions are related to Whittaker’s functionDp(z) by the expressions
U(a, z) =D−a−12(z) and
V(a, z) = 1 πΓ
1
2 +a D−a−12(−z) + (sinπa)D−a−12(z) .
Mathieu Functions
There are several accepted notations for Mathieu functions and for their associated parameters. The defining equation used by Gradshteyn and Ryzhik is
d2y dz2 +
a−2k2cos 2z
y= 0 withk2=q.
Different notations involve the replacement of a and q in this equation by h and θ, λ and h2 and b and c = 2√q, respectively. The periodic solutions sen(z, q) and cen(z, q) and the modified periodic solutions Sen(z, q) and Cen(z, q) are suitably altered and, sometimes, re-normalized. A description of these relationships together with the normalizing factors is contained in: Tables relating to Mathieu functions. National Bureau of Standards, Columbia University Press, New York, 1951.
Index of Special Functions
Notation Name of the function and the number of
the formula containing its definition
β(x) 8.37
Γ(z) Gamma function 8.31–8.33
γ(a, x), Γ(a, x) Incomplete gamma functions 8.35
Δ(n−k) Unit integer pulse function 18.1
ξ(s) 9.56
λ(x, y) 9.640
μ(x, β), μ(x, β, α) 9.640
ν(x), ν(x, α) 9.640
Π(x) Lobachevskiy angle of parallelism 1.48
Π(ϕ, n, k) Elliptic integral of the third kind 8.11
ζ(u) Weierstrass zeta function 8.17
ζ(z, q), ζ(z) Riemann zeta functions 9.51–9.54
Θ(u) =ϑ4πu
2K
, Θ1(u) =ϑ3πu
2K
Jacobi theta function 8.191–8.196
⎧⎪
⎨
⎪⎩
ϑ0(υ|τ) =ϑ4(υ|τ), ϑ1(υ|τ), ϑ2(υ|τ),
ϑ3(υ|τ)
⎫⎪
⎬
⎪⎭ Elliptic theta functions 8.18, 8.19
σ(u) Weierstrass sigma function 8.17
Φ(x) Probability integral 8.25
Φ(z, s, υ) Lerch function 9.55
Φ(α, γ;⎧ x) = 1F1(α; γ; x) Confluent hypergeometric function 9.21
⎪⎨
⎪⎩
Φ1(α, β, γ, x, y) Φ2(β, β, γ, x, y) Φ3(β, γ, x, y)
⎫⎪
⎬
⎪⎭
Degenerate hypergeometric series in two
variables 9.26
ψ(x) Euler psi function 8.36
℘(u) Weierstrass elliptic function 8.16
Ai(x) Airy function page xxxvi
am(u, k) Amplitude (of an elliptic function) 8.141
Bi(x) Bairy function page xxxvi
Bn Bernoulli numbers 9.61, 9.71
Bn(x) Bernoulli polynomials 9.620
B(x, y) Beta functions 8.38
continued on next page xxxvii
xxxviii Index of Special Functions
continued from previous page
Notation Name of the function and the number of
the formula containing its definition
Bx(p, q) Incomplete beta functions 8.39
bei(z), ber(z) Thomson functions 8.56
C Euler constant 9.73, 8.367
C(x) Fresnel cosine integral 8.25
Cν(a) Young functions 3.76
Cλn(t) Gegenbauer polynomials 8.93
Cλn(x) Gegenbauer functions 8.932 1
ce2n(z, q), ce2n+1(z, q) Periodic Mathieu functions (Mathieu
functions of the first kind) 8.61 Ce2n(z, q), Ce2n+1(z, q) Associated (modified) Mathieu functions of
the first kind 8.63
chi(x) Hyperbolic cosine integral function 8.22
ci(x) Cosine integral 8.23
cn(u) Cosine amplitude 8.14
D(k)≡D Elliptic integral 8.112
D(ϕ, k) Elliptic integral 8.111
Dn(z), Dp(z) Parabolic cylinder functions 9.24–9.25
dnu Delta amplitude 8.14
e1, e2, e3 (used with the Weierstrass function) 8.162
En Euler numbers 9.63, 9.72
E(ϕ, k) Elliptic integral of the second kind 8.11–8.12 E(k) =E
E(k) =E
Complete elliptic integral of the second
kind 8.11-8.12
E(p;αr:q, ϕs:x) MacRobert function 9.4
Eν(z) Weber function 8.58
Ei(z) Exponential integral function 8.21
erf(x) Error function 8.25
erfc(x) = 1−erf(x) Complementary error function 8.25
F(ϕ, k) Elliptic integral of the first kind 8.11–8.12
pFq(α1, . . . , αp;β1, . . . , βq;z) Generalized hypergeometric series 9.14
2F1(α, β;γ;z) =F(α, β;γ;z) Gauss hypergeometric function 9.10–9.13
1F1(α;γ;z) = Φ(α, γ;z) Degenerate hypergeometric function 9.21 FA(α:β1, . . . , βn;
γ1, . . . , γn:z1, . . . , zn)
Hypergeometric function of several
variables 9.19
F1, F2, F3, F4 Hypergeometric functions of two variables 9.18 fen(z, q),Fen(z, q). . .
Feyn(z, q),Fekn(z, q). . .
Other nonperiodic solutions of Mathieu’s
equation 8.64, 8.663
G Catalan constant 9.73
g2, g3 Invariants of the℘(u)-function 8.161
gdx Gudermannian 1.49
continued on next page
Index of Special Functions xxxix
continued from previous page
Notation Name of the function and the number of
the formula containing its definition gen(z, q),Gen(z, q)
Geyn(z, q),Gekn(z, q)
Other nonperiodic solutions of Mathieu’s
equation 8.64, 8.663
Gm,np,q
xab11,... ,b,... ,aqp
Meijer functions 9.3
h(n) Unit integer function 18.1
heiν(z), herν(z) Thomson functions 8.56
H(1)ν (z), H(2)ν (z) Hankel functions of the first and second
kinds 8.405, 8.42
H(u) =ϑ1πu
2K
Theta function 8.192
H1(u) =ϑ2πu
2K
Theta function 8.192
Hn(z) Hermite polynomials 8.95
Hν(z) Struve functions 8.55
Iν(z) Bessel functions of an imaginary argument 8.406, 8.43 Ix(p, q) Normalized incomplete beta function 8.39
Jν(z) Bessel function 8.402, 8.41
Jν(z) Anger function 8.58
kν(x) Bateman function 9.210 3
K(k) =K, K(k) =K Complete elliptic integral of the first kind 8.11–8.12 Kν(z) Bessel functions of imaginary argument 8.407, 8.43
kei(z), ker(z) Thomson functions 8.56
L(x) Lobachevskiy function 8.26
Lν(z) Modified Struve function 8.55
Lαn(z) Laguerre polynomials 8.97
li(x) Logarithm integral 8.24
Mλ,μ(z) Whittaker functions 9.22, 9.23
On(x) Neumann polynomials 8.59
Pμν(z), Pμν(x) Associated Legendre functions of the first
kind 8.7, 8.8
Pν(z), Pν(x) Legendre functions and polynomials 8.82, 8.83, 8.91 P
⎧⎨
⎩
a b c
α β γ z
α β γ
⎫⎬
⎭ Riemann’s differential equation 9.160
P(α,β)n (x) Jacobi polynomials 8.96
Qμν(z), Qμν(x) Associated Legendre functions of the
second kind 8.7, 8.8
Qν(z), Qν(x) Legendre functions of the second kind 8.82, 8.83
RC(x, y) Elliptic Function 8.111
RD(x, y, z) Elliptic Function 8.111
RF(x, y, z) Elliptic Function 8.111
RJ(x, y, z, p) Elliptic Function 8.111
S(x) Fresnel sine integral 8.25
Sn(x) Schl¨afli polynomials 8.59
sμ,ν(z), Sμ,ν(z) Lommel functions 8.57
continued on next page
xl Index of Special Functions
continued from previous page
Notation Name of the function and the number of
the formula containing its definition se2n+1(z, q), se2n+2(z, q) Periodic Mathieu functions 8.61 Se2n+1(z, q), Se2n+2(z, q) Mathieu functions of an imaginary
argument 8.63
shi(x) Hyperbolic sine integral 8.22
si(x) Sine integral 8.23
snu Sine amplitude 8.14
Tn(x) Chebyshev polynomial of the 1st kind 8.94 Un(x) Chebyshev polynomials of the 2nd kind 8.94 Uν(w, z), Vν(w, z) Lommel functions of two variables 8.578
Wλ,μ(z) Whittaker functions 9.22, 9.23
Yν(z) Neumann functions 8.403, 8.41
Zν(z) Bessel functions 8.401
Zν(z) Bessel functions 5.5
