Table Of ContentSeries and Products in the Development of Mathematics
Volume1
Thisisthefirstvolumeofatwo-volumeworkthattracesthedevelopmentofseries
and products from 1380 to 2000 by presenting and explaining the interconnected
concepts and results of hundreds of unsung as well as celebrated mathematicians.
Somechaptersdealwiththeworkofprimarilyonemathematicianonapivotaltopic,
and other chapters chronicle the progress over time of a given topic. This updated
secondeditionofSourcesintheDevelopmentofMathematicsaddsextensivecontext,
detail, and primary source material, with many sections rewritten to more clearly
reveal the significance of key developments and arguments. Volume 1, accessible
eventoadvancedundergraduatestudents,discussesthedevelopmentofthemethods
inseriesandproductsthatdonotemploycomplexanalyticmethodsorsophisticated
machinery. Volume 2 treats more recent work, including de Branges’s solution of
Bieberbach’sconjecture,andrequiresmoreadvancedmathematicalknowledge.
ranjan roy (1947–2020) was the Ralph C. Huffer Professor of Mathematics
and Astronomy at Beloit College, where he was a faculty member for 38 years.
Roy published papers and reviews on Riemann surfaces, differential equations,
fluid mechanics, Kleinian groups, and the development of mathematics. He was
an award-winning educator, having received the Allendoerfer Prize, the Wisconsin
MAAteachingaward,andtheMAAHaimoAwardforDistinguishedMathematics
TeachingandwastwicenamedTeacheroftheYearatBeloitCollege.Hecoauthored
SpecialFunctions(2001)withGeorgeAndrewsandRichardAskeyandcoauthored
chaptersintheNISTHandbookofMathematicalFunctions(2010);healsoauthored
Elliptic and Modular Functions from Gauss to Dedekind to Hecke (2017) and the
firsteditionofthisbook,SourcesintheDevelopmentofMathematics(2011).
RanjanRoy1948–2020
Series and Products in the
Development of Mathematics
Second Edition
Volume 1
RANJAN ROY
BeloitCollege
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www.cambridge.org
Informationonthistitle:www.cambridge.org/9781108709453
DOI:10.1017/9781108627702
Firstedition©RanjanRoy2011
Secondedition©RanjanRoy2021
Thispublicationisincopyright.Subjecttostatutoryexceptionandtotheprovisionsof
relevantcollectivelicensingagreements,noreproductionofanypartmaytakeplacewithoutthe
writtenpermissionofCambridgeUniversityPress.
FirstpublishedasSourcesintheDevelopmentofMathematics,2011
Secondedition2021
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ISBN 2-volumeSet978-1-108-70943-9Paperback
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ISBN Volume2978-1-108-70937-8Paperback
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Contents
ContentsofVolume2 pagexiii
Preface xvii
1 PowerSeriesinFifteenth-CenturyKerala 1
1.1 PreliminaryRemarks 1
1.2 TransformationofSeries 5
1.3 JyesthadevaonSumsofPowers 6
1.4 ArctangentSeriesintheYuktibhasa 8
1.5 DerivationoftheSineSeriesintheYuktibhasa 10
1.6 ContinuedFractions 15
1.7 Exercises 20
1.8 NotesontheLiterature 21
2 SumsofPowersofIntegers 23
2.1 PreliminaryRemarks 23
2.2 JohannFaulhaber 27
2.3 Fermat 28
2.4 Pascal 30
2.5 SekiandJakobBernoullionBernoulliNumbers 31
2.6 JakobBernoulli’sPolynomials 33
2.7 Euler 37
2.8 Lacroix’sProofofBernoulli’sFormula 40
2.9 JacobionFaulhaber 42
2.10 JacobiandRaabeonBernoulliPolynomials 43
2.11 Ramanujan’sRecurrenceRelationsforBernoulliNumbers 48
2.12 NotesontheLiterature 53
3 InfiniteProductofWallis 54
3.1 PreliminaryRemarks 54
3.2 Wallis’sInfiniteProductforπ 59
3.3 BrounckerandInfiniteContinuedFractions 61
v
vi Contents
3.4 Me´rayandStieltjes:TheProbabilityIntegral 64
3.5 Euler:SeriesandContinuedFractions 67
3.6 Euler:Riccati’sEquationandContinuedFractions 72
3.7 Exercises 75
3.8 NotesontheLiterature 76
4 TheBinomialTheorem 77
4.1 PreliminaryRemarks 77
4.2 Landen’sDerivationoftheBinomialTheorem 89
4.3 Euler:BinomialTheoremforRationalExponents 90
4.4 Cauchy:ProofoftheBinomialTheoremforRealExponents 94
4.5 Abel’sTheoremonContinuity 96
4.6 HarknessandMorley’sProofoftheBinomialTheorem 100
4.7 Exercises 101
4.8 NotesontheLiterature 103
5 TheRectificationofCurves 105
5.1 PreliminaryRemarks 105
5.2 Descartes’sMethodofFindingtheNormal 107
5.3 Hudde’sRuleforaDoubleRoot 109
5.4 VanHeuraet’sLetteronRectification 110
5.5 Newton’sRectificationofaCurve 112
5.6 Leibniz’sDerivationoftheArcLength 113
5.7 Exercises 113
5.8 NotesontheLiterature 114
6 Inequalities 116
6.1 PreliminaryRemarks 116
6.2 Harriot’sProofoftheArithmeticandGeometricMeansInequality 122
6.3 Maclaurin’sInequalities 124
6.4 CommentsonNewton’sandMaclaurin’sInequalities 125
6.5 Rogers 127
6.6 Ho¨lder 130
6.7 Jensen’sInequality 134
6.8 Riesz’sProofofMinkowski’sInequality 135
6.9 Exercises 137
6.10 NotesontheLiterature 142
7 TheCalculusofNewtonandLeibniz 143
7.1 PreliminaryRemarks 143
7.2 Newton’s1671CalculusText 147
7.3 Leibniz:DifferentialCalculus 150
7.4 LeibnizontheCatenary 153
7.5 JohannBernoulliontheCatenary 156
7.6 JohannBernoulli:TheBrachistochrone 157
7.7 Newton’sSolutiontotheBrachistochrone 158
7.8 NewtonontheRadiusofCurvature 161
Contents vii
7.9 JohannBernoulliontheRadiusofCurvature 162
7.10 Exercises 163
7.11 NotesontheLiterature 164
8 DeAnalysiperAequationesInfinitas 165
8.1 PreliminaryRemarks 165
8.2 AlgebraofInfiniteSeries 168
8.3 Newton’sPolygon 171
8.4 NewtononDifferentialEquations 172
8.5 Newton’sEarliestWorkonSeries 174
8.6 DeMoivreonNewton’sFormulaforsinnθ 176
8.7 Stirling’sProofofNewton’sFormula 177
8.8 Zolotarev:LagrangeInversionwithRemainder 179
8.9 Exercises 181
8.10 NotesontheLiterature 183
9 FiniteDifferences:InterpolationandQuadrature 186
9.1 PreliminaryRemarks 186
9.2 Newton:DividedDifferenceInterpolation 193
9.3 Gregory–NewtonInterpolationFormula 198
9.4 Waring,Lagrange:InterpolationFormula 199
9.5 EuleronInterpolation 201
9.6 Cauchy,Jacobi:Waring–LagrangeInterpolationFormula 202
9.7 NewtononApproximateQuadrature 204
9.8 Hermite:ApproximateIntegration 207
9.9 ChebyshevonNumericalIntegration 209
9.10 Exercises 211
9.11 NotesontheLiterature 212
10 SeriesTransformationbyFiniteDifferences 213
10.1 PreliminaryRemarks 213
10.2 Newton’sTransformation 219
10.3 Montmort’sTransformation 220
10.4 Euler’sTransformationFormula 222
10.5 Stirling’sTransformationFormulas 225
10.6 Nicole’sExamplesofSums 229
10.7 StirlingNumbers 233
10.8 Lagrange’sProofofWilson’sTheorem 241
10.9 Taylor’sSummationbyParts 242
10.10 Exercises 244
10.11 NotesontheLiterature 246
11 TheTaylorSeries 247
11.1 PreliminaryRemarks 247
11.2 Gregory’sDiscoveryoftheTaylorSeries 256
11.3 Newton:AnIteratedIntegralasaSingleIntegral 258
11.4 BernoulliandLeibniz:AFormoftheTaylorSeries 259
viii Contents
11.5 TaylorandEulerontheTaylorSeries 261
11.6 LacroixonD’Alembert’sDerivationoftheRemainder 262
11.7 Lagrange’sDerivationoftheRemainderTerm 264
11.8 Laplace’sDerivationoftheRemainderTerm 266
11.9 CauchyonTaylor’sFormulaandl’Hoˆpital’srule 267
11.10 Cauchy:TheIntermediateValueTheorem 270
11.11 Exercises 271
11.12 NotesontheLiterature 272
12 IntegrationofRationalFunctions 273
12.1 PreliminaryRemarks 273
12.2 Newton’s1666BasicIntegrals 280
12.3 Newton’sFactorizationofxn±1 282
12.4 CotesanddeMoivre’sFactorizations 284
12.5 Euler:IntegrationofRationalFunctions 286
12.6 Euler’s“InvestigatioValorisIntegralis” 293
12.7 Hermite’sRationalPartAlgorith√m 299
12.8 JohannBernoulli:Integrationof ax2+bx+c 301
12.9 Exercises 302
12.10 NotesontheLiterature 305
13 DifferenceEquations 306
13.1 PreliminaryRemarks 306
13.2 DeMoivreonRecurrentSeries 308
13.3 SimpsonandWaringonPartitioningSeries 311
13.4 Stirling’sMethodofUltimateRelations 317
13.5 DanielBernoullionDifferenceEquations 319
13.6 Lagrange:NonhomogeneousEquations 322
13.7 Laplace:NonhomogeneousEquations 325
13.8 Exercises 326
13.9 NotesontheLiterature 327
14 DifferentialEquations 328
14.1 PreliminaryRemarks 328
14.2 Leibniz:EquationsandSeries 338
14.3 NewtononSeparationofVariables 340
14.4 JohannBernoulli’sSolutionofaFirst-OrderEquation 341
14.5 EuleronGeneralLinearEquationswithConstantCoefficients 343
14.6 Euler:NonhomogeneousEquations 345
14.7 Lagrange’sUseoftheAdjoint 350
14.8 JakobBernoulliandRiccati’sEquation 352
14.9 Riccati’sEquation 353
14.10 SingularSolutions 354
14.11 MukhopadhyayonMonge’sEquation 358
14.12 Exercises 360
14.13 NotesontheLiterature 363
