Table Of ContentA COURSE OF MODERN ANALYSIS
Thisclassicworkhasbeenauniqueresourceforthousandsofmathematicians,scientists
andengineerssinceitsfirstappearancein1902.Neveroutofprint,itscontinuingvalue
liesinitsthoroughandexhaustivetreatmentofspecialfunctionsofmathematicalphysics
and the analysis of differential equations from which they emerge. The book also is of
historicalvalueasitwasthefirstbookinEnglishtointroducethethenmodernmethods
ofcomplexanalysis.
This fifth edition preserves the style and content of the original, but it has been
supplemented with more recent results and references where appropriate. All the
formulas have been checked and many corrections made. A complete bibliographical
search has been conducted to present the references in modern form for ease of
use. A new foreword by Professor S. J. Patterson sketches the circumstances of the
book’s genesis and explains the reasons for its longevity. A welcome addition to any
mathematician’s bookshelf, this will allow a whole new generation to experience the
beautycontainedinthistext.
e.t. whittaker was Professor of Mathematics at the University of Edinburgh. He
wasawardedtheCopleyMedalin1954,‘forhisdistinguishedcontributionstobothpure
andappliedmathematicsandtotheoreticalphysics’.
g.n. watson was Professor of Pure Mathematics at the University of Birmingham.
He is known, amongst other things, for the 1918 result now known as Watson’s lemma
andwasawardedtheDeMorganMedalin1947.
victor h. mollisProfessorintheDepartmentofMathematicsatTulaneUniversity.
Heco-authoredEllipticCurves(Cambridge,1997)andwasawardedtheWeissPresiden-
tialAwardin2017forhisGraduateTeaching.HefirstreceivedacopyofWhittakerand
WatsonduringhisownundergraduatestudiesattheUniversidadSantaMariainChile.
(Left):EdmundTaylorWhittaker(1873–1956);(Right):GeorgeNevilleWatson
(1886–1965):UniversalHistoryArchive/Contributor/GettyImages.
A COURSE OF MODERN ANALYSIS
Fifth Edition
An introduction to the general theory of infinite
processes and of analytic functions with an account
of the principal transcendental functions
E.T. WHITTAKER AND G.N. WATSON
Fiftheditioneditedandpreparedforpublicationby
VictorH.Moll
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DOI:10.1017/9781009004091
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Firstedition1902
Secondedition1915
Thirdedition1920
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Reprinted1935,1940,1946,1950,1952,1958,1962,1963
ReissuedintheCambridgeMathematicalLibrarySeries1996
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Contents
ForewordbyS.J.Patterson xvii
PrefacetotheFifthEdition xxi
PrefacetotheFourthEdition xxiii
PrefacetotheThirdEdition xxiv
PrefacetotheSecondEdition xxv
PrefacetotheFirstEdition xxvi
Introduction xxvii
PartI TheProcessofAnalysis 1
1 ComplexNumbers 3
1.1 Rationalnumbers 3
1.2 Dedekind’stheoryofirrationalnumbers 4
1.3 Complexnumbers 6
1.4 Themodulusofacomplexnumber 7
1.5 TheArganddiagram 8
1.6 Miscellaneousexamples 9
2 TheTheoryofConvergence 10
2.1 Thedefinitionofthelimitofasequence 10
2.11 Definitionofthephrase‘oftheorderof’ 10
2.2 Thelimitofanincreasingsequence 10
2.21 Limit-pointsandtheBolzano–Weierstrasstheorem 11
2.22 Cauchy’stheoremonthenecessaryandsufficientconditionfortheexistenceofa
limit 12
2.3 Convergenceofaninfiniteseries 13
2.31 Dirichlet’stestforconvergence 16
2.32 Absoluteandconditionalconvergence 17
2.33 Thegeometricseries,andtheseries(cid:205)∞ 1 17
n=1 ns
2.34 Thecomparisontheorem 18
v
vi Contents
2.35 Cauchy’stestforabsoluteconvergence 20
2.36 D’Alembert’sratiotestforabsoluteconvergence 20
2.37 Ageneraltheoremonseriesforwhich lim |un+1/un|=1 21
n→∞
2.38 Convergenceofthehypergeometricseries 22
2.4 Effectofchangingtheorderofthetermsinaseries 23
2.41 Thefundamentalpropertyofabsolutelyconvergentseries 24
2.5 Doubleseries 24
2.51 Methodsofsummingadoubleseries 25
2.52 Absolutelyconvergentdoubleseries 26
2.53 Cauchy’stheoremonthemultiplicationofabsolutelyconvergentseries 27
2.6 Powerseries 28
2.61 Convergenceofseriesderivedfromapowerseries 29
2.7 Infiniteproducts 30
2.71 Someexamplesofinfiniteproducts 31
2.8 Infinitedeterminants 34
2.81 Convergenceofaninfinitedeterminant 34
2.82 Therearrangementtheoremforconvergentinfinitedeterminants 35
2.9 Miscellaneousexamples 36
3 ContinuousFunctionsandUniformConvergence 40
3.1 Thedependenceofonecomplexnumberonanother 40
3.2 Continuityoffunctionsofrealvariables 40
3.21 Simplecurves.Continua 41
3.22 Continuousfunctionsofcomplexvariables 42
3.3 Seriesofvariableterms.Uniformityofconvergence 43
3.31 Ontheconditionforuniformityofconvergence 44
3.32 Connexionofdiscontinuitywithnon-uniformconvergence 45
3.33 Thedistinctionbetweenabsoluteanduniformconvergence 46
3.34 Acondition,duetoWeierstrass,foruniformconvergence 47
3.35 Hardy’stestsforuniformconvergence 48
3.4 Discussionofaparticulardoubleseries 49
3.5 Theconceptofuniformity 51
3.6 ThemodifiedHeine–Boreltheorem 51
3.61 Uniformityofcontinuity 52
3.62 Arealfunction,ofarealvariable,continuousinaclosedinterval,attainsitsupper
bound 53
3.63 Arealfunction,ofarealvariable,continuousinaclosedinterval,attainsallvalues
betweenitsupperandlowerbounds 54
3.64 Thefluctuationofafunctionofarealvariable 54
3.7 Uniformityofconvergenceofpowerseries 55
3.71 Abel’stheorem 55
3.72 Abel’stheoremonmultiplicationofconvergentseries 55
3.73 Powerserieswhichvanishidentically 56
3.8 Miscellaneousexamples 56
4 TheTheoryofRiemannIntegration 58
4.1 Theconceptofintegration 58
4.11 Upperandlowerintegrals 58
4.12 Riemann’sconditionofintegrability 59
Contents vii
4.13 Ageneraltheoremonintegration 60
4.14 Mean-valuetheorems 62
4.2 Differentiationofintegralscontainingaparameter 64
4.3 Doubleintegralsandrepeatedintegrals 65
4.4 Infiniteintegrals 67
4.41 Infiniteintegralsofcontinuousfunctions.Conditionsforconvergence 67
4.42 Uniformityofconvergenceofaninfiniteintegral 68
4.43 Testsfortheconvergenceofaninfiniteintegral 68
4.44 Theoremsconcerninguniformlyconvergentinfiniteintegrals 71
4.5 Improperintegrals.Principalvalues 72
4.51 Theinversionoftheorderofintegrationofacertainrepeatedintegral 73
4.6 Complexintegration 75
4.61 Thefundamentaltheoremofcomplexintegration 76
4.62 Anupperlimittothevalueofacomplexintegral 76
4.7 Integrationofinfiniteseries 77
4.8 Miscellaneousexamples 79
5 The Fundamental Properties of Analytic Functions; Taylor’s, Laurent’s and
Liouville’sTheorems 81
5.1 Propertyoftheelementaryfunctions 81
5.11 Occasionalfailureoftheproperty 82
5.12 Cauchy’sdefinitionofananalyticfunctionofacomplexvariable 82
5.13 AnapplicationofthemodifiedHeine–Boreltheorem 83
5.2 Cauchy’stheoremontheintegralofafunctionroundacontour 83
5.21 Thevalueofananalyticfunctionatapoint,expressedasanintegraltakenrounda
contourenclosingthepoint 86
5.22 Thederivativesofananalyticfunction f(z) 88
5.23 Cauchy’sinequalityfor f(n)(a) 89
5.3 Analyticfunctionsrepresentedbyuniformlyconvergentseries 89
5.31 Analyticfunctionsrepresentedbyintegrals 90
5.32 Analyticfunctionsrepresentedbyinfiniteintegrals 91
5.4 Taylor’stheorem 91
5.41 FormsoftheremainderinTaylor’sseries 94
5.5 Theprocessofcontinuation 95
5.51 Theidentityoftwofunctions 97
5.6 Laurent’stheorem 98
5.61 Thenatureofthesingularitiesofone-valuedfunctions 100
5.62 The‘pointatinfinity’ 101
5.63 Liouvillle’stheorem 103
5.64 Functionswithnoessentialsingularities 104
5.7 Many-valuedfunctions 105
5.8 Miscellaneousexamples 106
6 TheTheoryofResidues;ApplicationtotheEvaluationofDefiniteIntegrals 110
6.1 Residues 110
6.2 Theevaluationofdefiniteintegrals 111
6.21 Theevaluationoftheintegralsofcertainperiodicfunctionstakenbetweenthe
limits0and2π 111
6.22 Theevaluationofcertaintypesofintegralstakenbetweenthelimits−∞and+∞ 112
viii Contents
6.23 Principalvaluesofintegrals 116
6.24 Evaluationofintegralsoftheform∫∞xa−1Q(x)dx 117
0
6.3 Cauchy’sintegral 118
6.31 Thenumberofrootsofanequationcontainedwithinacontour 119
6.4 Connexionbetweenthezerosofafunctionandthezerosofitsderivative 120
6.5 Miscellaneousexamples 121
7 TheExpansionofFunctionsinInfiniteSeries 125
7.1 AformuladuetoDarboux 125
7.2 TheBernoulliannumbersandtheBernoullianpolynomials 125
7.21 TheEuler–Maclaurinexpansion 127
7.3 Bürmann’stheorem 129
7.31 Teixeira’sextendedformofBürmann’stheorem 131
7.32 Lagrange’stheorem 133
7.4 Theexpansionofaclassoffunctionsinrationalfractions 134
7.5 Theexpansionofaclassoffunctionsasinfiniteproducts 137
7.6 ThefactortheoremofWeierstrass 138
7.7 Expansioninaseriesofcotangents 140
7.8 Borel’stheorem 141
7.81 Borel’sintegralandanalyticcontinuation 142
7.82 Expansionsinseriesofinversefactorials 143
7.9 Miscellaneousexamples 145
8 AsymptoticExpansionsandSummableSeries 153
8.1 Simpleexampleofanasymptoticexpansion 153
8.2 Definitionofanasymptoticexpansion 154
8.21 Anotherexampleofanasymptoticexpansion 154
8.3 Multiplicationofasymptoticexpansions 156
8.31 Integrationofasymptoticexpansions 156
8.32 Uniquenessofanasymptoticexpansion 157
8.4 Methodsofsummingseries 157
8.41 Borel’smethodofsummation 158
8.42 Euler’smethodofsummation 158
8.43 Cesàro’smethodofsummation 158
8.44 ThemethodofsummationofRiesz 159
8.5 Hardy’sconvergencetheorem 159
8.6 Miscellaneousexamples 161
9 FourierSeriesandTrigonometricSeries 163
9.1 DefinitionofFourierseries 163
9.11 Natureoftheregionwithinwhichatrigonometricalseriesconverges 164
9.12 Valuesofthecoefficientsintermsofthesumofatrigonometricalseries 167
9.2 OnDirichlet’sconditionsandFourier’stheorem 167
9.21 TherepresentationofafunctionbyFourierseriesforrangesotherthan(−π,π) 168
9.22 Thecosineseriesandthesineseries 169
9.3 ThenatureofthecoefficientsinaFourierseries 171
9.31 DifferentiationofFourierseries 172
9.32 Determinationofpointsofdiscontinuity 173
9.4 Fejér’stheorem 174
Contents ix
9.41 TheRiemann–Lebesguelemmas 177
9.42 TheproofofFourier’stheorem 179
9.43 TheDirichlet–BonnetproofofFourier’stheorem 181
9.44 TheuniformityoftheconvergenceofFourierseries 183
9.5 TheHurwitz–LiapounofftheoremconcerningFourierconstants 185
9.6 Riemann’stheoryoftrigonometricalseries 187
9.61 Riemann’sassociatedfunction 188
9.62 PropertiesofRiemann’sassociatedfunction;Riemann’sfirstlemma 189
9.63 Riemann’stheoremontrigonometricalseries 191
9.7 Fourier’srepresentationofafunctionbyanintegral 193
9.8 Miscellaneousexamples 195
10 LinearDifferentialEquations 201
10.1 Lineardifferentialequations 201
10.2 Solutionsinthevicinityofanordinarypoint 201
10.21 Uniquenessofthesolution 203
10.3 Pointswhichareregularforadifferentialequation 204
10.31 Convergenceoftheexpansionof§10.3 206
10.32 Derivationofasecondsolutioninthecasewhenthedifferenceoftheexponentsis
anintegerorzero 207
10.4 Solutionsvalidforlargevaluesof|z| 209
10.5 Irregularsingularitiesandconfluence 210
10.6 Thedifferentialequationsofmathematicalphysics 210
10.7 Lineardifferentialequationswiththreesingularities 214
10.71 TransformationsofRiemann’sP-equation 215
10.72 TheconnexionofRiemann’sP-equationwiththehypergeometricequation 215
10.8 Lineardifferentialequationswithtwosingularities 216
10.9 Miscellaneousexamples 216
11 IntegralEquations 219
11.1 Definitionofanintegralequation 219
11.11 Analgebraicallemma 220
11.2 Fredholm’sequationanditstentativesolution 221
11.21 InvestigationofFredholm’ssolution 223
11.22 Volterra’sreciprocalfunctions 226
11.23 Homogeneousintegralequations 228
11.3 Integralequationsofthefirstandsecondkinds 229
11.31 Volterra’sequation 229
11.4 TheLiouville–Neumannmethodofsuccessivesubstitutions 230
11.5 Symmetricnuclei 231
11.51 Schmidt’stheoremthat,ifthenucleusissymmetric,theequationD(λ)=0hasat
leastoneroot 232
11.6 Orthogonalfunctions 233
11.61 Theconnexionoforthogonalfunctionswithhomogeneousintegralequations 234
11.7 Thedevelopmentofasymmetricnucleus 236
11.71 ThesolutionofFredholm’sequationbyaseries 237
11.8 SolutionofAbel’sintegralequation 238
11.81 Schlömilch’sintegralequation 238
11.9 Miscellaneousexamples 239
