analysis-数学分析习题课讲义

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1、数学分析习题课讲义

2、分析十段天元手筋张毅byDr.YiZhangInstituteofMathematicalSciencesFudanUniversityShanghai200433P.R.Chinazhangyimath@fudan.edu.cnLecturegivenatFudanUniversity,2006.http://homepage.fudan.edu.cn/~yizhang/teaching/analysis-exercise.pdfTheauthorholdsthecopyrightofthislecturenotes.An

3、yperson(s)intendingtocopyapartorwholeofthematerialsinthenotesinaproposedpublicationmustseekcopyrightreleasefromtheauthor.1Analysisi}DedicatetoKatie创造的神秘,有如夜间的黑暗,是伟大的。而知识的幻影,不过如晨间之雾。iiLectureNotesatFudanContentsReferencesii1.ProblemsonSets,SequencesandLimits11.1.ElementaryTec

4、hnique11.2.ApplicationsoftheStolztheorem112.Inequality143.OrdersEstimateofIn¯nitesimal203.1.NotationsandExamples203.2.Exercisesandhomework234.ApplicationofIn¯nitesimal:WallisFormulaandStirlingFormula275.TopicontheGammafunctions305.1.¡functions305.2.Exercises346.WorkingTechni

5、queinfunctiontheory366.1.Iterationtechnique366.2.ExercisesandHomework397.ApplicationsofDi®erential418.TreasuresinCalculus428.1.Euler'sidenti¯cation³(2)=1+1¢¢¢+1+¢¢¢´¼24222n268.2.Irrationalityof¼;log2;³(2);³(3).438.3.ThepropertiesofTchebychev'sfunctions468.4.Mertens'Theoreman

6、dSelberg'sInequality538.5.AnElementaryProofofThePrimeNumberTheorembySelbergandErdÄos588.6.Gauss'sProofofTheFundamentalTheoremofAlgebra599.AdvancedTechniquesinAnalysis609.1.PreliminaryResultsinAnalysis609.2.ScalingTechniqueandSchauderEstimate729.3.Campanato'scharacterizationo

7、fL2functionstobeHÄoldercontinuous7910.AdvancedTopicsinAnalysis8210.1.TopiconRiemannZetaFunction(TobeContinuous)8210.2.ElementaryonNevanlinnaTheory(TobeContinuous)8310.3.Elementaryonp-adicSeries(TobeContinuous)84References[F]¡:M.菲赫金哥尔茨:微积分学教程,VOLI.,II.,III.HigherEducationPres

8、s.[H]G.H.Hardy:ACourseofPureMathematics(TenthEdition)CombridgeUniversityPress2002[HUA]华罗庚:数论导引,科学出版社1979.[J]J.Jost:PartialDi®erentialEquations(GTM214),Spinger2002.[P]ProblemsSelectioninTheWilliamLowellPutnamMathematicalCompetitions.[P-Y]潘承洞,于秀源:阶的估计.Analysisiii}曾子曰：大学之道，在明明德

9、，在亲民，在止于至善。知止而后定，定而后能静，静而后能安，安而后能虑，虑而后能得。物有本末，事有始终，知所先后，则近道矣。Analysis11.Pro