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Table of Integrals,Series,and ProductsEighth EditionTable ofIntegrals,Series,and ProductsEighth EditionI.S.Gradshteyn and I.M.RyzhikDaniel Zwillinger,EditorRensselaer Polytechnic Institute,USAVictor Moll(Scientific Editor)Tulane University,Department of MathematicsTranslated from Russian by Scripta Technica,IncAcademic Press is an imprint of Elsevier225 Wyman Street,Waltham,MA 02451,USA525 B Street,Suite 1800,San Diego,CA 92101-4495,USA32 Jamestown Road,London NW1 7BY,UKThe Boulevard,Langford Lane,Kidlington,Oxford OX5 1GB,UKEighth Edition:Copyright c?2015,2007,2000,1994 Elsevier Inc.All rights reserved.No part of this publication may be reproduced or transmitted in any form or by any means,electronic or mechanical,including photocopying,recording,or any information storage and retrieval system,without permission in writing fromthe publisher.Details on how to seek permission,further information about the Publishers permissions policies and ourarrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency,can be foundat our website: book and the individual contributions contained in it are protected under copyright by the Publisher(other than asmay be noted herein).NoticesKnowledge and best practice in this field are constantly changing.As new research and experience broaden our understand-ing,changes in research methods,professional practices,or medical treatment may become necessary.Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any informa-tion,methods,compounds,or experiments described herein.In using such information or methods they should be mindfulof their own safety and the safety of others,including parties for whom they have a professional responsibility.To the fullest extent of the law,neither the Publisher nor the authors,contributors,or editors,assume any liability for anyinjury and/or damage to persons or property as a matter of products liability,negligence or otherwise,or from any use oroperation of any methods,products,instructions,or ideas contained in the material herein.ISBN:978-0-12-384933-5Library of Congress Cataloging-in-Publication DataGradshtein,I.S.(Izrail?Solomonovich)Tablitsy integralov,summ,riadov i proizvedenii.EnglishTable of integrals,series,and products.Eighth edition/Daniel Zwillinger.pages cmIncludes bibliographical references and index.ISBN 978-0-12-384933-51.MathematicsTables.I.Zwillinger,Daniel,1957-II.Title.QA55.G6613 2014510.2?1dc232014010276British Library Cataloguing-in-Publication DataA catalogue record for this book is available from the British Library.For information on all Academic Press publicationsvisit our web site at Preface to the Eighth EditionGradshteyn 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 revisedseventh edition,users have continued to submit corrections and new results that improve the book,andmake suggestions for changes that improve the presentation of the material.We regret that the structureof 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,togetherwith 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 entryreference 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 entryin a previous edition has been removed,the entry numbers will jump.This preserves the continuity ofnumbering between the new and older editions.This edition has also removed chapters present in the7th edition that were not aligned with integrals,series,and products(e.g.,the chapters on matrices andnorms).We wish to express our gratitude to all who have been in contact with us with the object of improvingand extending the book.Special thanks are extended to both Dr.Francis J.O Brien,Jr.of the NavalStation in Newport,Rhode Island and Dr.Andrej Tenne-Sens of Ottowa,Ontario,Canada.They haveeach 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 generatesmany suggestions for new entries and new errata.Hence,we do not expect this new edition to be free fromerrors.All users who identify errors,or who wish to propose new entries,are invited to contact the authorswhose email addresses are listed below.Corrections will be posted on the web site mourn the passing of Professor Alan Jeffrey who,as Editor,guided this book from its4th edition in 1965 to the 7th edition in 2007.Alan s dedication to this book was evident both by hislongevitywell over 40 years of improvementsand in the consideration he gave to all of our correspon-dents for whom he evaluated a huge number of integrals.Daniel ZwillingerZwillingerBVictor Mollvhmtulane.eduxviiAcknowledgmentsThe publisher and editors would like to take this opportunity to express their gratitude to the followingusers of the Table of Integrals,Series,and Products who either directly or through errata publishedin Mathematics of Computation have generously contributed corrections and addenda to the originalprinting.AnonymousDr.Artem G.AbanovDr.A.AbbasDr.S.M.AbrarovDr.P.B.AbrahamDr.Ari AbramsonDr.Jose AdachiDr.R.J.AdlerDr.N.AgmonDr.M.AhmadDr.S.A.AhmadDr.Rajai S.AlassarDr.Luis Alvarez-RusoDr.Maarten H P AmbaumDr.R.K.AmietDr.L.U.AncaraniDr.M.AntoineDr.Marian ApostolDr.C.R.AppledornDr.D.R.AppletonDr.Mitsuhiro ArikawaDr.Ir.Luk R.ArnautDr.Peter ArnoldDr.P.AshoshauvatiDr.C.L.AxnessDr.Scott BaalrudDr.E.BadralexeDr.S.B.BagchiDr.L.J.BakerDr.R.BallDr.Ingo BarthDr.M.P.BarnettDr.Fabio BernardoniDr.Florian BaumannDr.Norman C.BeaulieuDr.Jerome BenoitMr.V.BentleyDr.Laurent BergerDr.M.van den BergDr.N.F.BerkDr.C.A.BertulaniDr.J.Betancort-RijoDr.P.BickerstaffDr.Iwo Bialynicki-BirulaDr.Chris BidinostiDr.G.R.BiggDr.Ian BindlossDr.L.BlanchetDr.Mike BlaskiewiczDr.R.D.BlevinsDr.Anders BlomDr.L.M.BlumbergDr.R.BlumelDr.S.E.BodnerDr.Simone BoiDr.M.BonsagerDr.George BorosDr.S.BosanacDr.Ruben Van BoxemDr.Christoph BruderDr.Patrick BrunoDr.B.Van den BosscheDr.A.Bostr omDr.J.E.BowcockDr.T.H.BoyerDr.K.M.BriggsDr.D.J.BroadhurstDr.Chris Van Den BroeckDr.W.B.BrowerDr.H.N.BrowneDr.Christoph BrueggerDr.William J.BrunoDr.Vladimir BubanjaDr.D.J.BuchDr.D.J.BukmanDr.F.M.BurrowsDr.R.CabozDr.T.CallowayDr.F.CalogeroDr.D.Dal CappelloDr.David CardonDr.J.A.Carlson GallosDr.B.CarrascalDr.A.R.CarrDr.Neal CarronDr.Florian CartariusDr.S.CarterDr.Miguel CarvajalDr.G.CavalleriMr.W.H.L.CawthorneDr.Alexandre Caz eDr.A.CecchiniDr.B.ChanxixxxAcknowledgmentsDr.M.A.ChaudhryDr.Sabino Chavez-CerdaDr.Julian ChengDr.H.W.ChewDr.D.ChinDr.Young-seek ChungDr.S.CiccarielloDr.N.S.ClarkeDr.R.W.ClearyDr.A.ClementDr.Alain CochardDr.P.CochraneDr.D.K.CohoonDr.Howard S.CohlDr.L.ColeDr.Filippo ColomoDonal ConnonDr.J.R.D.CopleyHenry Corback,Esq.Dr.Daniel Benevides da CostaDr.Barry J.CoxDr.D.CoxDr.J.CoxDr.J.W.CrissDr.A.E.CurzonDr.D.DadyburjorDr.D.DajaputraDr.C.Dal CappelloDr.P.DalyDr.S.DasguptaDr.Charles E.DavidsonDr.John DaviesDr.C.L.DavisDr.A.DegasperisDr.Gustav DeliusDr.B.C.DenardoDr.R.W.DentDr.E.DeutschDr.D.deVriesDr.Eran DganiDr.Enno DiekemaDr.P.DitaDr.P.J.de DoelderDr.Mischa DohlerDr.G.D omeDr.Shi-Hai DongDr.Balazs DoraDr.M.R.D OrsognaDr.Forrest DossDr.Adrian A.DragulescuDr.Zvi DreznerDr.Eduardo DuenezMr.Tommi J.DufvaDr.Duc V.DuongDr.E.B.Dussan,VDr.Percy DusekDr.C.A.EbnerDr.M.van der EndeDr.Jonathan EngleDr.G.EngDr.E.S.ErckDr.Grant ErdmannDr.Jan ErkelensDr.Olivier EspinosaDr.G.A.Est evezDr.K.EvansDr.G.EvendonDr.Valery I.FabrikantDr.L.A.FalkovskyDr.Kambiz FarahmandDr.Richard J.FatemanDr.G.FedeleDr.A.R.FerchminDr.P.FerrantDr.Andr e FerrariDr.H.E.FettisDr.W.B.FichterDr.George FikiorisMr.J.C.S.S.FilhoDr.L.FordDr.Nicolao FornengoDr.J.FranceDr.B.FrankDr.S.FrasierDr.Stefan FredenhagenDr.A.J.FreemanDr.A.FrinkDr.Jason M.GallaspyDr.J.A.C.GallasDr.J.A.Carlson GallasDr.G.R.GamertsfelderDr.Jianliang GaoDr.T.GaravagliaDr.Jaime Zaratiegui GarciaDr.C.G.GardnerDr.D.GarfinkleDr.P.N.GarnerDr.F.GasserDr.E.GathDr.P.GattDr.D.GayDr.M.P.GelfandDr.M.R.GellerDr.Ali I.GencDr.Vincent GenotDr.M.F.GeorgeDr.Teschl GeraldDr.P.GermainDr.Ing.Christoph GierullDr.S.P.GillDr.Federico GirosiDr.E.A.GislasonDr.M.I.GlasserDr.P.A.GlendinningDr.L.I.GoldfischerDr.Denis GolosovDr.I.J.GoodDr.J.GoodMr.L.GorinDr.Martin G otzDr.R.GovindarajDr.M.De GraufDr.Gabriele GradoniDr.L.GreenMr.Leslie O.GreenDr.R.GreenwellDr.K.D.GrimsleyDr.Albert GroenenboomDr.V.GudmundssonDr.J.GuilleraDr.K.GunnDr.D.L.GunterDr.Julio C.Guti errez-VegaDr.Roger HaagmansDr.Howard HaberDr.H.van HaeringenDr.B.HafiziDr.Bahman HafiziDr.T.HagforsDr.M.J.HaggertyDr.Timo HakulinenAcknowledgmentsxxiDr.S.E.HammelDr.E.HansenDr.Wes HarkerDr.T.HarrettDr.D.O.HarrisDr.Frank HarrisMr.Mazen D.HasnaDr.Peter HawkinsDr.Joel G.HeinrichDr.Adam Dade HendersonDr.Franck HersantDr.Sten HerlitzDr.Chris HerzogDr.A.HiguchiDr.R.E.HiseDr.Henrik HolmDr.N.HolteDr.R.W.HopperDr.P.N.HouleDr.C.J.HowardJie HuDr.Ben Yu-Kuang HuDr.J.H.HubbellDr.Felix HuberDr.J.R.HullDr.W.HumphriesDr.Jean-Marc Hur eDr.Jamal A.HusseinDr.Y.IksbeDr.Philip IngenhovenMr.L.IossifDr.Sean A.IrvineDr.OttarIsbergDr.Kazuhiro IshidaDr.Cyril-Daniel IskanderDr.S.A.JacksonDr.John David JacksonDr.Francois JaclotDr.B.JacobsDr.Pierre JacobsDr.E.C.JamesDr.B.JancoviciDr.D.J.JeffreyDr.H.J.JensenDr.Bin JiangDr.Edwin F.JohnsonDr.I.R.JohnsonDr.Steven JohnsonDr.Joel T.JohnsonDr.Fredrik JohanssonDr.I.JohnstoneDr.Y.P.JoshiDr.Jae-Hun JungDr.Damir JuricDr.Florian KaempferDr.S.KanmaniDr.Z.KapalDr.Peter KaradimasDr.Dave KasperDr.M.KaufmanDr.Eduardo KauselDr.B.KayLouis KempeneersDr.Jack KerlinDr.Avinash KhareDr.Karen T.KohlDr.Ilki KimDr.Youngsun KimDr.S.KlamaDr.L.KlingenDr.C.KnesslDr.M.J.KnightDr.Yannis KohninosDr.D.KoksDr.L.P.KokDr.K.S.K olbigDr.Yannis KomninosDr.D.D.KonowalowDr.Z.KopalDr.I.KostyukovDr.R.A.KrajcikDr.Vincent KrakoviackDr.Stefan Kr amerDr.Tobias KramerDr.Hermann KrebsDr.Chethan KrishnanDr.J.W.KrozelDr.E.D.KrupnikovDr.Kun-Lin KuoDr.E.A.KuraevDr.Heinrich KuttlerDr.Konstantinos KyritsisDr.Velimir LabinacDr.Javier Navarro LaboulaisDr.A.D.J.LambertDr.A.LambertDr.A.LarrazaDr.K.D.LeeDr.M.Howard LeeDr.M.K.LeeDr.P.A.LeeDr.Todd LeeDr.J.LeggDr.Xianfu LeiDr.Yefim LeifmanDr.Remigijus LeipusDr.Armando LemusDr.Elvio LeonardoDr.Eugene LepelaarsDr.S.L.LevieDr.D.LeviDr.Michael LexaDr.Liang LiDr.Kuo Kan LiangDr.Sergey LiflandskyDr.B.LinetDr.Andrej LikarDr.M.A.LisaDr.Donald LivesayDr.H.LiDr.Xian-Fang LiDr.Georg LohoeferDr.I.M.LongmanDr.D.LongDr.Sylvie LorthoisDr.Wenzhou LuDr.Phil LuchtDr.Stephan LudwigDr.Arpad LukacsDr.Y.L.LukeDr.W.LukoszDr.T.LundgrenDr.E.A.LuraevDr.R.LynchDr.K.B.MaDr.Ilari MaasiltaDr.R.MahurinDr.R.MallierDr.G.A.MamonDr.A.MangiarottiDr.I.ManningxxiiAcknowledgmentsDr.Greg MarksDr.J.MarmurDr.A.MartinDr.Carmelo P.MartinSr.Yuzo MaruyamaDr.Richard MartharDr.David J.MasielloDr.Richard J.MatharDr.H.A.MavromatisDr.M.MazzoniDr.P.McCullaghDr.J.H.McDonnellDr.J.R.McGregorDr.Kim McInturffDr.N.McKinneyDr.David McA McKirdyDr.Andrew J.McHutchonDr.Rami MehremDr.W.N.MeiDr.Angelo MelinoMr.Jos e Ricardo MendesDr.Zvi MendlowitzDr.Andy MennimDr.Sarah MesserDr.J.P.MeunierDr.Haixing MiaoDr.Krys A.MichalskiDr.Gerard P.MichonDr.D.F.R.MildnerDr.D.L.MillerDr.Steve MillerDr.P.C.D.MillyDr.S.P.MitraDr.K.MiuraDr.N.MohankumarDr.M.MollDr.Victor H.MollDr.D.MonowalowMr.Tony MontagneseDr.Thierry MontaguDr.Jim MoreheadDr.J.MoriceDr.J.Guy MorganDr.W.MueckDr.C.MuhlhausenDr.S.MukherjeeDr.R.R.M ullerDr.Pablo Parmezani MunhozDr.Frank NaminDr.Paul NanningaDr.A.NatarajanDr.Sven Peter N asholmDr.Javier NavarroDr.Christian NetzelDr.Stefan NeumeierDr.C.T.NguyenDr.A.C.NicolDr.M.M.NietoDr.P.NoerdlingerDr.Andrew N.NorrisDr.K.H.NorwichDr.A.H.NuttallDr.F.O BrienDr.R.P.O KeeffeDr.A.OjoDr.O.OlendskiDr.Oleg OlendskiDr.P.OlssonDr.Gilad OrenDr.M.OrtnerDr.Matthew OrtonDr.S.OstlundDr.J.OverduinDr.J.PachnerMr.Robert A.PadgugDr.D.PapadopoulosDr.F.J.PappMr.Man Sik ParkDr.Jong-Do ParkDr.B.PattersonDr.R.F.PawulaDr.D.W.PeacemanDr.Vittorio PeanoDr.Vincent PegoraroDr.D.PelatDr.L.PelitiDr.Y.P.PellegriniDr.Thiago S.PereiraDr.G.J.PertDr.Nicola PessinaDr.J.B.PetersonDr.Rickard PeterssonDr.Arnaud PierensDr.Emilio PisantyDr.Ralph PixleyDr.Andrew PlumbDr.Dror PoratDr.E.A.PowerDr.E.PredazziDr.William S.PriceDr.Gunnar PruessnerDr.Paul RadmoreDr.Carl E.RasmussenDr.F.RaynalDr.X.R.ResendeDr.Guy A.ReynoldsDr.J.M.RiedlerDr.Sjoerd RienstraDr.Thomas RichardDr.E.RingelDr.T.M.RobertsDr.N.I.RobinsonDr.P.A.RobinsonDr.Elvira RomeraDr.D.M.RosenblumDr.R.A.RosthalDr.J.R.RothDr.Klaus RottbrandDr.Bastien RoucariesDr.D.RoyDr.E.RoyerDr.D.RudermannDr.Ahmad RushdiDr.Ali RushdiDr.Niall RyanDr.Sanjib SabhapanditDr.C.T.SachradjaDr.J.SadikuDr.A.SadiqDr.Motohiko SaitohDr.Naoki SaitoDr.A.SalimDr.Sherwood SamnDr.J.H.SamsonDr.Miguel A.Sanchis-LozanoDr.J.A.SandersDr.M.A.F.SanjunDr.P.SarquizDr.Avadh SaxenaDr.Vito ScarolaDr.O.Sch arpfAcknowledgmentsxxiiiDr.A.ScherzingerDr.B.SchizerDr.Martin SchmidDr.J.ScholesDr.Mel SchopperDr.H.J.SchulzDr.Andreas SchulzDr.Markus SchwarzDr.G.J.SearsDr.Kazuhiko SekiDr.B.SeshadriDr.Roger SewellDr.A.ShapiroDr.Sihong ShaoDr.Masaki ShigemoriDr.J.S.ShengDr.Kenneth Ing ShingDr.Tomohiro ShiraiDr.S.ShlomoDr.D.SiegelDr.Matthew StapletonDr.Leo SteinDr.Michael L.SteinDr.Steven H.SimonDr.Ashok Kumar SingalDr.Constantin SiriteanuDr.C.SmithDr.Richard SmithDr.G.C.C.SmithDr.Stefan Llewellyn SmithDr.S.SmithDr.Sasha SodinDr.G.SoltDr.J.SondowDr.A.SrenssenDr.Marcus SpradlinDr.Andrzej StaruszkiewiczDr.Philip C.L.StephensonDr.Edgardo StockmeyerDr.J.C.StratonMr.H.SuraweeraDr.N.F.SvaiterDr.V.SvaiterDr.R.SzmytkowskiDr.Sebastian S.SzyszkowiczDr.S.TabachnikDr.Erik TalvilaDr.G.TanakaDr.C.TanguyDr.G.K.TannahillDr.B.T.TanDr.C.TavardDr.Gon calo TavaresDr.Aba TelekiAndrej Tenne-SensDr.Gerald TeschlDr.Arash Dahi TaleghaniDr.D.TemperleyDr.A.J.TervoortDr.Theodoros TheodoulidisDr.D.J.ThomasDr.Michael ThorwartDr.S.T.ThynellDr.Tamer TlasDr.D.C.TorneyDr.R.ToughDr.Marwan ToutounjiDr.Dennis TredeDr.B.F.TreadwayDr.Ming TsaiDr.N.TurkkanDr.Sandeep TyagiDr.J.J.TysonDr.Takahiro UedaDr.S.UeharaDr.M.VadacchinoDr.Stathis VagenasDr.O.T.VallsDr.D.VandethDr.Klaas VantournhoutMr.Andras VanyolosDr.D.VeitchMr.Jose Lopez VicarioDr.Hari VishnuDr.K.VogelDr.J.M.M.J.VogelsDr.Alexis De VosDr.Emanuel VotoDr.Stuart WalshDr.Haiming WangDr.J.J.WangDr.Reinhold WannemacherDr.S.WanzuraDr.J.WardDr.S.I.WarshawDr.Alex WatsonDr.R.WeberDr.Steffen 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