分析Ⅰ(影印版)
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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
装帧平装
货号4.31
上书时间2024-09-25
商品详情
- 品相描述:八五品
图书标准信息
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作者
[法]戈德门特 著
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出版社
高等教育出版社
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出版时间
2009-12
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版次
1
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ISBN
9787040279559
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定价
34.20元
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装帧
平装
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开本
16开
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纸张
胶版纸
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页数
431页
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字数
450千字
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正文语种
简体中文
- 【内容简介】
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《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】
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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
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