分析Ⅰ（影印版）

图书标准信息

• 作者 [法]戈德门特 著
• 出版社 高等教育出版社
• 出版时间 2009-12
• 版次 1
• ISBN 9787040279559
• 定价 34.20元
• 装帧 平装
• 开本 16开
• 纸张 胶版纸
• 页数 431页
• 字数 450千字
• 正文语种 简体中文

【内容简介】

　　《分析1（影印版）》第一卷的内容包括集合与函数、离散变量的收敛性、连续变量的收敛性、幂函数、指数函数与三角函数；第二卷的内容包括Fourier级数和Fourier积分以及可以通过Fourier级数解释的Weierstrass的解析函数理论。
　　《分析1（影印版）》是作者在巴黎第七大学讲授分析课程数十年的结晶，其目的是阐明分析是什么，它是如何发展的。《分析1（影印版）》非常巧妙地将严格的数学与教学实际、历史背景结合在一起，对主要结论常常给出各种可能的探索途径，以使读者理解基本概念、方法和推演过程。作者在《分析1（影印版）》中较早地引入了一些较深的内容，如在第一卷中介绍了拓扑空间的概念，在第二卷中介绍了Lebesgue理论的基本定理和Weierstrass椭圆函数的构造。

【目录】

Preface
I-SetsandFunctions
§1.SetTheory
1-Membership，equality，emptyset
2-Thesetdefinedbyarelation.Intersectionsandunions
3-Wholenumbers.Infinitesets
4-Orderedpairs，Cartesianproducts，setsofsubsets
5-Functions，maps，correspondences
6-Injections，surjections，bijections
7-Equipotentsets.Countablesets
8-Thedifferenttypesofinfinity
9-Ordinalsandcardinals
§2.Thelogicoflogicians

II-Convergence：Discretevariables
§1.Convergentsequencesandseries
0-Introduction：whatisarealnumber?
1-Algebraicoperationsandtheorderrelation：axiomsofR
2-Inequalitiesandintervals
3-Localorasymptoticproperties
4-Theconceptoflimit.Continuityanddifferentiability
5-Convergentsequences：definitionandexamples
6-Thelanguageofseries
7-Themarvelsoftheharmonicseries
8-Algebraicoperationsonlimits
§2.Absolutelyconvergentseries
9-Increasingsequences.Upperboundofasetofrealnumber
10-Thefunctionlogx.Rootsofapositivenumber
11-Whatisanintegral?
12-Serieswithpositiveterms
13-Alternatingseries
14-Classicalabsolutelyconvergentseries
15-Unconditionalconvergence：generalcase
16-Comparisonrelations.CriteriaofCauchyanddAlembert
17-Infinitelimits
18-Unconditionalconvergence：associativity
§3.Firstconceptsofanalyticfunctions
19-TheTaylorseries
20-Theprincipleofanalyticcontinuation
21-Thefunctioncotxandtheseries∑1/n2k
22-Multiplicationofseries.Compositionofanalyticfunctions.Formalseries
23-TheellipticfunctionsofWeierstrass

III-Convergence：Continuousvariables
§1.Theintermediatevaluetheorem
1-Limitvaluesofafunction.Openandclosedsets
2-Continuousfunctions
3-Rightandleftlimitsofamonotonefunction
4-Theintermediatevaluetheorem
§2.Uniformconvergence
5-Limitsofcontinuousfunctions
6-AslipupofCauchys
7-Theuniformmetric
8-Seriesofcontinuousfunctions.Normalconvergence
§3.Bolzano-WeierstrassandCauchyscriterion
9-Nestedintervals，Bolzano-Weierstrass，compactsets
10-Cauchysgeneralconvergencecriterion
11-Cauchyscriterionforseries：examples
12-Limitsoflimits
13-Passingtothelimitinaseriesoffunctions
§4.Differentiablefunctions
14-Derivativesofafunction
15-Rulesforcalculatingderivatives
16-Themeanvaluetheorem
17-Sequencesandseriesofdifferentiablefunctions
18-Extensionstounconditionalconvergence
§5.Differentiablefunctionsofseveralvariables
19-Partialderivativesanddifferentials
20-DifferentiabilityoffunctionsofclassC1
21-Differentiationofcompositefunctions
22-Limitsofdifferentiablefunctions
23-Interchangingtheorderofdifferentiation
24-Implicitfunctions
AppendixtoChapterIII
1-Cartesianspacesandgeneralmetricspaces
2-Openandclosedsets
3-LimitsandCauchyscriterioninametricspace;completespaces
4-Continuousfunctions
5-AbsolutelyconvergentseriesinaBanachspace
6-Continuouslinearmaps
7-Compactspaces
8-Topologicalspaces

IV-Powers，Exponentials，Logarithms，TrigonometricFunctions
§1.Directconstruction
1-Rationalexponents
2-Definitionofrealpowers
3-Thecalculusofrealexponents
4-Logarithmstobasea.Powerfunctions
5-Asymptoticbehaviour
6-Characterisationsoftheexponential，powerandlogarithmicfunctions
7-Derivativesoftheexponentialfunctions：directmethod
8-Derivativesofexponentialfunctions，powersandlogarithms
§2.Seriesexpansions
9-Thenumbere.Napierianlogarithms
10-Exponentialandlogarithmicseries：directmethod
11-Newtonsbinomialseries
12-Thepowerseriesforthelogarithm
13-Theexponentialfunctionasalimit
14-Imaginaryexponentialsandtrigonometricfunctions
15-EulersrelationchezEuler
16-Hyperbolicfunctions
§3.Infiniteproducts
17-Absolutelyconvergentinfiniteproducts
18-Theinfiniteproductforthesinefunction
19-Expansionofaninfiniteproductinseries
20-Strangeidentities
§4.ThetopologyofthefunctionsArg(z)andLogz
Index