Hadamard expansions and hyperasymptotic evaluation: an extension of the method of steepest descents

253 Pages·2011·3.593 MB·English

by  Paris, R. B

HADAMARD EXPANSIONS AND
HYPERASYMPTOTIC EVALUATION
AnExtensionoftheMethodofSteepestDescents
TheauthordescribestherecentlydevelopedtheoryofHadamardexpansions
appliedtothehigh-precision(hyperasymptotic)evaluationofLaplaceand
Laplace-typeintegrals.Thisnewmethodbuildsonthewell-knownasymptotic
methodofsteepestdescents,ofwhichtheopeningchaptergivesadetailedaccount
illustratedbyaseriesofexamplesofincreasingcomplexity.Adiscussionof
uniformityproblemsassociatedwithvariouscoalescencephenomena,theStokes
phenomenonandhyperasymptoticsofLaplace-typeintegralsfollows.The
remainingchaptersdealwiththeHadamardexpansionofLaplaceintegrals,with
andwithoutsaddlepoints.Problemsofdifferenttypesofsaddlecoalescenceare
alsodiscussed.Thetextisillustratedwithmanynumericalexamples,whichhelp
thereadertounderstandthelevelofaccuracyachievable.Theauthoralsoconsiders
applicationstosomeimportantspecialfunctions.
Thisbookisidealforgraduatestudentsandresearchersworkinginasymptotics.
R. B. PARIS isaReaderinMathematicsattheUniversityofAbertay,Dundee.
EncyclopediaofMathematicsandItsApplications
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BooksintheEncyclopediaofMathematicsandItsApplicationscovertheir
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86 R.J.McElieceTheTheoryofInformationandCoding,2ndedn
87 B.A.MagurnAnAlgebraicIntroductiontoK-Theory
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ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS
Hadamard Expansions and
Hyperasymptotic Evaluation
An Extension of the Method of Steepest
Descents
R. B. PARIS
UniversityofAbertay,Dundee
CAMBRIDGE UNIVERSITY PRESS
Cambridge,NewYork,Melbourne,Madrid,CapeTown,
Singapore,SãoPaulo,Delhi,Tokyo,MexicoCity
CambridgeUniversityPress
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PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYork
www.cambridge.org
Informationonthistitle:www.cambridge.org/9781107002586
(cid:2)c R.B.Paris2011
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
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permissionofCambridgeUniversityPress.
Firstpublished2011
PrintedintheUnitedKingdomattheUniversityPress,Cambridge
AcataloguerecordforthepublicationisavailablefromtheBritishLibrary
LibraryofCongressCataloguinginPublicationdata
Paris,R.B.(RichardBruce),1946–
HadamardExpansionsandHyperasymptoticEvaluation:AnExtensionoftheMethod
ofSteepestDescents/R.B.Paris.
p. cm.–(EncyclopediaofMathematicsanditsApplications;141)
Includesbibliographicalreferencesandindex.
ISBN978-1-107-00258-6(hardback)
1. Integralequations–Asymptotictheory. 2. Asymptoticexpansions. I. Title.
QA431.P287 2011
(cid:3)
515.45–dc22
2010051563
ISBN978-1-107-00258-6Hardback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceor
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Contents
Preface pagevii
1 AsymptoticsofLaplace-typeintegrals 1
1.1 Historicalintroduction 1
1.2 Themethodofsteepestdescents 5
1.3 Examples 20
1.4 Furtherexamples 37
1.5 Uniformexpansions 56
1.6 Optimaltruncationandsuperasymptotics 73
1.7 TheStokesphenomenon 78
1.8 Hyperasymptotics 83
2 HadamardexpansionofLaplaceintegrals 100
2.1 Introduction 100
2.2 TheHadamardseriesfor Iν(x) 101
2.3 RapidlyconvergentHadamardseries 122
2.4 Hadamardseriesonaninfiniteinterval 126
2.5 Examples 134
2.6 BoundsonthetailsofHadamardseries 140
3 HadamardexpansionofLaplace-typeintegrals 144
3.1 Introduction 144
3.2 Expansionschemes 145
3.3 Examples 151
3.4 Coalescenceproblems 175
3.5 Examplesofcoalescence 178
4 Applications 197
4.1 Introduction 197
4.2 TheBesselfunction Jν(νz) 198
4.3 ThePearceyintegral 207
v
vi Contents
4.4 Theparaboliccylinderfunction 216
4.5 Theexpansionforlog(cid:3)(z) 219
AppendixA: Propertiesof P(a,z) 224
AppendixB: ConvergenceofHadamardseries 230
AppendixC: Connectionwiththeexp-arcintegrals 232
References 235
Index 241
Preface
The aims of this book are twofold. The first is to present a detailed account of
the classical method of steepest descents applied to the asymptotic evaluation of
Laplace-type integrals containing a large parameter, and the second is to give a
coherent account of the theory of Hadamard expansions. This latter topic, which
hasbeendevelopedduringthepastdecade,extendsthemethodofsteepestdescents
and effectively ‘exactifies’ the procedure since, in theory, the Hadamard expansion
ofaLaplaceorLaplace-typeintegralcanproduceunlimitedaccuracy.
Many texts deal with the method of steepest descents, some in more detail than
others. The well-known books by Copson Asymptotic Expansions (1965), Olver
Asymptotics and Special Functions (1997), Bleistein and Handelsman Asymptotic
Expansion of Integrals (1975), Wong Asymptotic Approximations of Integrals
(1989)andBenderandOrszagAdvancedMathematicalMethodsforScientistsand
Engineers (1978) are all good examples. It is our aim in the first chapter to give a
comprehensive account of the method of steepest descents accompanied by a set
of illustrative examples of increasing complexity. We also consider the common
causes of non-uniformity in the asymptotic expansions of Laplace-type integrals
and conclude the first chapter with a discussion of the Stokes phenomenon and
hyperasymptotics.
The next two chapters present the Hadamard expansion theory of Laplace and
of Laplace-type integrals possessing saddle points. A study of these chapters
makes it apparent how this theory builds upon and extends the method of steepest
descents. Considerable emphasis is devoted to explaining the problems associated
withcoalescencephenomena,suchasasaddlepointcoalescingeitherwithanother
saddle point or with an endpoint of the integration interval. Methods for dealing
withthesedifficultiesintheHadamardexpansionprocedurearecarefullydescribed.
Themonographcloseswithsophisticatedapplicationsoftheideasdevelopedinthe
earlier chapters to four particular special functions: the Bessel function Jν(νx) of
largeorderandargument,thePearceyintegral(atwo-variablegeneralisationofthe
classical Airy function), the parabolic cylinder function U(a,z) of large order and
argument,andthelogarithmofthegammafunction.
vii
viii Preface
Inkeepingwiththelast-mentionedtextabove,manyoftheexamplesinthelater
chapters are illustrated with numerical studies to better display the calibre of the
asymptoticapproximationsobtained,astrategythatgivesthenon-expertpractitioner
a good sense of the method being showcased. This book should be accessible to
anyone with a solid undergraduate background in functions of a single complex
variable.
Theauthoracknowledgesthesupportofhisinstitution,theUniversityofAbertay,
Dundee,whichfacilitatedthewritingofthisbook.Aconsiderabledebtofgratitudeis
owedtoseveralcolleagueswhogenerouslyundertookacarefulinspectionofvarious
sectionsandfortheircriticalcommentsthathavehelpedtoimprovethepresentation
of this text. The whole of Chapter 1 was read by N. M. Temme, with the first half
of this chapter and Chapter 2 being read by T. M. Dunster; Chapters 2 and 3 were
readbyD.Kaminski,andC.J.Howlsinspectedthesectiononhyperasymptoticsof
Laplace-typeintegralsinChapter1.Finally,thenon-specialistcommentsonthefirst
part of Chapter 1 by J. S. Dagpunar were helpful. It is almost inevitable, however,
thatinspiteofthiscarefulexaminationsomeerrorsormisprintswillhaveremained
undetected,andtheauthorrequeststhereader’sforebearanceforthosethatproveto
bevexatious.