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Sources in the Development of Mathematics
ThediscoveryofinfiniteproductsbyWallisandinfiniteseriesbyNewtonmarkedthe
beginningofthemodernmathematicalera.TheuseofseriesallowedNewtontofind
theareaunderacurvedefinedbyanyalgebraicequation,anachievementcompletely
beyondtheearliermethodsofTorricelli,Fermat,andPascal. TheworkofNewtonand
hiscontemporaries,includingLeibnizandtheBernoullis,wasconcentratedinmath-
ematical analysis and physics. Euler’s prodigious mathematical accomplishments
dramaticallyextendedthescopeofseriesandproductstoalgebra,combinatorics,and
numbertheory. SeriesandproductsprovedpivotalintheworkofGauss,Abel,and
Jacobiinellipticfunctions;inBooleandLagrange’soperatorcalculus;andinCayley,
Sylvester,andHilbert’sinvarianttheory. Seriesandproductsstillplayacriticalrole
inthemathematicsoftoday.ConsidertheconjecturesofLanglands,includingthatof
Shimura-Taniyama,leadingtoWiles’sproofofFermat’slasttheorem.
DrawingontheoriginalworkofmathematiciansfromEurope,Asia,andAmerica,
Ranjan Roy discusses many facets of the discovery and use of infinite series and
products. He gives context and motivation for these discoveries, including original
notation and diagrams when practical. He presents multiple derivations for many
importanttheoremsandformulasandprovidesinterestingexercises,supplementing
theresultsofeachchapter.
Roy deals with numerous results, theorems, and methods used by students,
mathematicians,engineers,andphysicists.Moreover,sincehepresentsoriginalmath-
ematical insights often omitted from textbooks, his work may be very helpful to
mathematicsteachersandresearchers.
ranjan roy istheRalphC.HufferProfessorofMathematicsandAstronomyat
BeloitCollege.Royhaspublishedpapersandreviewsindifferentialequations,fluid
mechanics, Kleinian groups, and the development of mathematics. He co-authored
Special Functions (2001) with GeorgeAndrews and RichardAskey, and authored
chaptersintheNISTHandbookofMathematicalFunctions(2010).Hehasreceived
theAllendoerfer prize, the Wisconsin MAAteaching award, and the MAAHaimo
awardfordistinguishedmathematicsteaching.
CoverimagebyNFNKalyan;CoverdesignbyDavidLevy.
Sources in the Development
of Mathematics
Infinite Series and Products from the
Fifteenth to the Twenty-first Century
RANJAN ROY
BeloitCollege
cambridge university press
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CambridgeUniversityPress
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Contents
Preface page xvii
1 PowerSeriesinFifteenth-CenturyKerala 1
1.1 PreliminaryRemarks 1
1.2 TransformationofSeries 4
1.3 JyesthadevaonSumsofPowers 5
1.4 ArctangentSeriesintheYuktibhasa 7
1.5 DerivationoftheSineSeriesintheYuktibhasa 8
1.6 ContinuedFractions 10
1.7 Exercises 12
1.8 NotesontheLiterature 14
2 SumsofPowersofIntegers 16
2.1 PreliminaryRemarks 16
2.2 JohannFaulhaberandSumsofPowers 19
2.3 JakobBernoulli’sPolynomials 20
2.4 ProofofBernoulli’sFormula 24
2.5 Exercises 25
2.6 NotesontheLiterature 26
3 InfiniteProductofWallis 28
3.1 PreliminaryRemarks 28
3.2 Wallis’sInfiniteProductforπ 32
3.3 BrounckerandInfiniteContinuedFractions 33
3.4 Stieltjes:ProbabilityIntegral 36
3.5 Euler:SeriesandContinuedFractions 38
3.6 Euler:ProductsandContinuedFractions 40
3.7 Euler:ContinuedFractionsandIntegrals 43
3.8 Sylvester:ADifferenceEquationandEuler’sContinuedFraction 45
3.9 Euler:Riccati’sEquationandContinuedFractions 46
3.10 Exercises 48
3.11 NotesontheLiterature 50
v
vi Contents
4 TheBinomialTheorem 51
4.1 PreliminaryRemarks 51
4.2 Landen’sDerivationoftheBinomialTheorem 57
4.3 Euler’sProofforRationalIndices 58
4.4 Cauchy:ProofoftheBinomialTheoremforRealExponents 60
4.5 Abel’sTheoremonContinuity 62
4.6 HarknessandMorley’sProofoftheBinomialTheorem 66
4.7 Exercises 67
4.8 NotesontheLiterature 69
5 TheRectificationofCurves 71
5.1 PreliminaryRemarks 71
5.2 Descartes’sMethodofFindingtheNormal 73
5.3 Hudde’sRuleforaDoubleRoot 74
5.4 VanHeuraet’sLetteronRectification 75
5.5 Newton’sRectificationofaCurve 76
5.6 Leibniz’sDerivationoftheArcLength 77
5.7 Exercises 78
5.8 NotesontheLiterature 79
6 Inequalities 81
6.1 PreliminaryRemarks 81
6.2 Harriot’sProofoftheArithmeticandGeometricMeansInequality 87
6.3 Maclaurin’sInequalities 88
6.4 Jensen’sInequality 89
6.5 Reisz’sProofofMinkowski’sInequality 90
6.6 Exercises 91
6.7 NotesontheLiterature 96
7 GeometricCalculus 97
7.1 PreliminaryRemarks (cid:1) 97
7.2 Pascal’sEvaluationof sinxdx 100
7.3 Gregory’sEvaluationof(cid:1)aBetaIntegral 101
7.4 Gregory’sEvaluationof(cid:1) secθdθ 104
7.5 Barrow’sEvaluationof (cid:1)s√ecθdθ 106
7.6 BarrowandtheIntegral x2+a2dx 108
7.7 Barrow’sProofof d tanθ =sec2θ 110
dθ
7.8 Barrow’sProductRuleforDerivatives 111
7.9 Barrow’sFundamentalTheoremofCalculus 114
7.10 Exercises 114
7.11 NotesontheLiterature 118
8 TheCalculusofNewtonandLeibniz 120
8.1 PreliminaryRemarks 120
8.2 Newton’s1671CalculusText 123
8.3 Leibniz:DifferentialCalculus 126
Contents vii
8.4 LeibnizontheCatenary 129
8.5 JohannBernoulliontheCatenary 131
8.6 JohannBernoulli:TheBrachistochrone 132
8.7 Newton’sSolutiontotheBrachistochrone 133
8.8 NewtonontheRadiusofCurvature 135
8.9 JohannBernoulliontheRadiusofCurvature 136
8.10 Exercises 137
8.11 NotesontheLiterature 138
9 DeAnalysiperAequationesInfinitas 140
9.1 PreliminaryRemarks 140
9.2 AlgebraofInfiniteSeries 142
9.3 Newton’sPolygon 145
9.4 NewtononDifferentialEquations 146
9.5 Newton’sEarliestWorkonSeries 147
9.6 DeMoivreonNewton’sFormulaforsinnθ 149
9.7 Stirling’sProofofNewton’sFormula 150
9.8 Zolotarev:LagrangeInversionwithRemainder 152
9.9 Exercises 153
9.10 NotesontheLiterature 156
10 FiniteDifferences:InterpolationandQuadrature 158
10.1 PreliminaryRemarks 158
10.2 Newton:DividedDifferenceInterpolation 163
10.3 Gregory–NewtonInterpolationFormula 165
10.4 Waring,Lagrange:InterpolationFormula 165
10.5 Cauchy,Jacobi:LagrangeInterpolationFormula 166
10.6 NewtononApproximateQuadrature 168
10.7 Hermite:ApproximateIntegration 170
10.8 ChebyshevonNumericalIntegration 172
10.9 Exercises 173
10.10 NotesontheLiterature 175
11 SeriesTransformationbyFiniteDifferences 176
11.1 PreliminaryRemarks 176
11.2 Newton’sTransformation 181
11.3 Montmort’sTransformation 182
11.4 Euler’sTransformationFormula 184
11.5 Stirling’sTransformationFormulas 187
11.6 Nicole’sExamplesofSums 190
11.7 StirlingNumbers 191
11.8 Lagrange’sProofofWilson’sTheorem 194
11.9 Taylor’sSummationbyParts 195
11.10 Exercises 196
11.11 NotesontheLiterature 199
viii Contents
12 TheTaylorSeries 200
12.1 PreliminaryRemarks 200
12.2 Gregory’sDiscoveryoftheTaylorSeries 206
12.3 Newton:AnIteratedIntegralasaSingleIntegral 209
12.4 BernoulliandLeibniz:AFormoftheTaylorSeries 210
12.5 TaylorandEulerontheTaylorSeries 211
12.6 Lacroixond’Alembert’sDerivationoftheRemainder 212
12.7 Lagrange’sDerivationoftheRemainderTerm 213
12.8 Laplace’sDerivationoftheRemainderTerm 215
12.9 CauchyonTaylor’sFormulaandl’Hôpital’sRule 216
12.10 Cauchy:TheIntermediateValueTheorem 218
12.11 Exercises 219
12.12 NotesontheLiterature 220
13 IntegrationofRationalFunctions 222
13.1 PreliminaryRemarks 222
13.2 Newton’s1666BasicIntegrals 228
13.3 Newton’sFactorizationofxn±1 230
13.4 CotesanddeMoivre’sFactorizations 231
13.5 Euler:IntegrationofRationalFunctions 233
13.6 Euler’sGeneralizationofHisEarlierWork 234
13.7 Hermite’sRationalPartAlgorithm√ 237
13.8 JohannBernoulli:Integrationof ax2+bx+c 238
13.9 Exercises 239
13.10 NotesontheLiterature 243
14 DifferenceEquations 245
14.1 PreliminaryRemarks 245
14.2 DeMoivreonRecurrentSeries 247
14.3 Stirling’sMethodofUltimateRelations 250
14.4 DanielBernoullionDifferenceEquations 252
14.5 Lagrange:NonhomogeneousEquations 254
14.6 Laplace:NonhomogeneousEquations 257
14.7 Exercises 258
14.8 NotesontheLiterature 259
15 DifferentialEquations 260
15.1 PreliminaryRemarks 260
15.2 Leibniz:EquationsandSeries 268
15.3 NewtononSeparationofVariables 270
15.4 JohannBernoulli’sSolutionofaFirst-OrderEquation 271
15.5 EuleronGeneralLinearEquationswithConstantCoefficients 272
15.6 Euler:NonhomogeneousEquations 274
15.7 Lagrange’sUseoftheAdjoint 276
15.8 JakobBernoulliandRiccati’sEquation 278
15.9 Riccati’sEquation 278
