The Elementary Mathematical Works of Euler

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1、TheElementaryMathematicalWorksofLeonhardEuler(17071783)PaulYiuDepartmentofMathematicsFloridaAtlanticUniversitySummer19991IA.IntroductiontoEulersOperaOmnia1IB.Solutionofcubicequations4IC.Solutionofquarticequations5IIA.Factorizationofaquarticasaproductoftworealquadratics7IIB.E

2、ulersproofoftheFTAforquarticpolynomials10IIC.Relationsbetweenrootsandcoefficients11IIIA.Partialfractiondecomposition17IIIB.IntroductiontoIntroductio,I20IIIC.Exponentialsandlogarithms21IVA.Trigonometricfunctions28IVB.Logarithmsofcomplexnumbers31VA.Eulersworksoninfiniteseries,in

3、troduction35VB.AnexamplefromEulersearliestpaperoninfiniteseries38VC.EulersPaper72:Variaeobservationescircaseriesinfinitas39VIA.Eulerfirstcalculationof1+1+1+1+···4322k32k42kVIB.ChapterXofIntroductio,I48VIC.EulersforgottenpaperDemonstrationdelasommedecettsuite1+1+1+1+1+···4949253

4、6VIIA.JacobBernoullissummationofthepowersofnaturalnumbers54VIIB.Eulerssummationformula57VIIC.Eulerssummationformula(continued)59VIIIA.Theseriesexpansionsofcotangent,tangent,andsecant62VIIIB.Eulersconstantγ66VIIIC.SummaryonBernoulliandEulernumbers681DiagramsredrawninAugust,20

5、07.IIXA.Continuedfractions69IXB.Continuedfractionsandinfiniteseries73IXC.Continuedfractionexpansionoffunctions77XA.EulersproofofHeronsformula80XB.Areaofacyclicquadrilateral81XC.TheexcirclesandHeronsformula82XIA.Trianglecenters86XIB.Distancesbetweenvariouscenters88XIC.Thenine-

6、pointcircleandFeuerbachstheorem90XIIA.Eulersconstructionofatrianglewithgivencircumcenter,incenter,andorthocenter93XIIB.Specialcases95XIIC.Constructions97XIIIA.EulerandtheFermatnumbers100XIIIB.PerfectnumbersandMersenneprimes103XIIIC.EulersproofofFermatslittletheorem107XIVA.Eu

7、lersTractatusdenumerorumdoctrina109XIVB.Eulersextraordinaryrelationinvolvingsumsofdivisors110XIVC.Eulerspentagonalnumbertheorem114XVA.ThePellequation118XVB.EulersproofofFermatslasttheoremforn=4121XVC.Sumsoftwosquares122XVIA.EulersproofofFermatslasttheoremforn=3124XVIB.Sumsof

8、fourthpowers125XVIC.OnEulersfearsomefoursome129FurtherReading134Exercisesan