Examples and theorems in analysis在分析中的例子和定理

內(nèi)容概要

“This book is a unique and very practical contribution to the teaching of calculus … 。The aim of this book is to try to give the subject concreteness and immediacy by giving the well-chosen examples equal status with the theorems。this excellent book is written primarily for first- and second-year undergraduates in mathematics； but it will also be of interest to students of statistics，computer science and engineering 。 We warmly recommend it as an entertaining and stimulating companion ”（Ferenc Móricz，Acta Scientiarum Mathematicarum，71，2005）　“This book takes a unique and very practical approach to mathematical analysis。 It makes the subject more accessible by giving the examples equal status with the theorems。 … A number of applications show what the subject is about，and what can be done with it。 Exercises at the end of each chapter，of varying levels of difficulty，develop new ideas and present open problems。”（L’enseignement mathematique，50：1-2，2004）　　“The author presents a book on analysis in which theorems and examples are equally important。It is a good textbook for students to obtain a more complete picture of the material and to master basic methods of work in mathematical analysis。”

書籍目錄

1  Sequences  1.1 Examples，F(xiàn)ormulae and Recuion  1.2 Monotone and Bounded Sequences  1.3 Convergence  1.4 Subsequenees   1.5 Cauchy Sequences  Exercises2.Functions and Continuity  2.1 Examples  2.2 Monotone and Bounded Functions  2.3 Limits and Continuity  2.4 Bounds and Intermediate Values  2.5 Inverse Functions  2.6 Recursive Limits and Iteration  2.7 Ohe-Sided and Infinite Limits Regulated Fu  2.8 Countability  Exercises3.Differentlation  3.1 Differentiable Functions  3.2 The Significance of the Derivative  3.3 Rules for Differentiation  3.4 Mean Value Theorems and Estimation  3.5 More on Iteration  3.6 Optimisation   Exercises4.Constructive Integration.  4.1 Step Functions  4.2 The Integral of a Regulated Function  4.3 Integration and Differentiation  4.4 Applications  4.5 Further Mean Value Theorems  Exercises5.Improper Integrals  5.1 Improper Integrals on an Interval  5.2 Improper Integrals at Infinity  5.3 The Gamma Function  Exercises6.  Series  6.1 Convergence  6.2 Series with Positive Terms  6.3 Series with Arbitrary Terms  6.4 Power Series  6.5 Exponential and Trigonometric F unctions  6.6 Sequences and Series of Functions  6.7 Infinite Products  Exercises7.  Applications  7.1 F0urier Series  7.2 Fourier Integrals  7.3 Distributions  7.4 Asymptotics  ExercisesA.Fubini’S TheoremB.Hints and Solutions for ExercisesBibliographyIndex

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