Counterexamples on Uniform Convergence (eBook)

Sequences, Series, Functions, and Integrals

Andrei Bourchtein, Ludmila Bourchtein (Autoren)

eBook Download: PDF

2017
John Wiley & Sons (Verlag)
978-1-119-30340-4 (ISBN)

Infos

A comprehensive and thorough analysis of concepts and results on uniform convergence

Counterexamples on Uniform Convergence: Sequences, Series, Functions, and Integrals presents counterexamples to false statements typically found within the study of mathematical analysis and calculus, all of which are related to uniform convergence. The book includes the convergence of sequences, series and families of functions, and proper and improper integrals depending on a parameter. The exposition is restricted to the main definitions and theorems in order to explore different versions (wrong and correct) of the fundamental concepts and results.

The goal of the book is threefold. First, the authors provide a brief survey and discussion of principal results of the theory of uniform convergence in real analysis. Second, the book aims to help readers master the presented concepts and theorems, which are traditionally challenging and are sources of misunderstanding and confusion. Finally, this book illustrates how important mathematical tools such as counterexamples can be used in different situations.

The features of the book include:

• An overview of important concepts and theorems on uniform convergence
• Well-organized coverage of the majority of the topics on uniform convergence studied in analysis courses
• An original approach to the analysis of important results on uniform convergence based/ on counterexamples
• Additional exercises at varying levels of complexity for each topic covered in the book
• A supplementary Instructor's Solutions Manual containing complete solutions to all exercises, which is available via a companion website

Counterexamples on Uniform Convergence: Sequences, Series, Functions, and Integrals is an appropriate reference and/or supplementary reading for upper-undergraduate and graduate-level courses in mathematical analysis and advanced calculus for students majoring in mathematics, engineering, and other sciences. The book is also a valuable resource for instructors teaching mathematical analysis and calculus.

ANDREI BOURCHTEIN, PhD, is Professor in the Department of Mathematics at Pelotas State University in Brazil. The author of more than 100 referred articles and five books, his research interests include numerical analysis, computational fluid dynamics, numerical weather prediction, and real analysis. Dr. Andrei Bourchtein received his PhD in Mathematics and Physics from the Hydrometeorological Center of Russia.

LUDMILA BOURCHTEIN, PhD, is Senior Research Scientist at the Institute of Physics and Mathematics at Pelotas State University in Brazil. The author of more than 80 referred articles and three books, her research interests include real and complex analysis, conformal mappings, and numerical analysis. Dr. Ludmila Bourchtein received her PhD in Mathematics from Saint Petersburg State University in Russia.

ANDREI BOURCHTEIN, PhD, is Professor in the Department of Mathematics at Pelotas State University in Brazil. The author of more than 100 referred articles and five books, his research interests include numerical analysis, computational fluid dynamics, numerical weather prediction, and real analysis. Dr. Andrei Bourchtein received his PhD in Mathematics and Physics from the Hydrometeorological Center of Russia.

LUDMILA BOURCHTEIN, PhD, is Senior Research Scientist at the Institute of Physics and Mathematics at Pelotas State University in Brazil. The author of more than 80 referred articles and three books, her research interests include real and complex analysis, conformal mappings, and numerical analysis. Dr. Ludmila Bourchtein received her PhD in Mathematics from Saint Petersburg State University in Russia.

Inhalt

Title Page 5
Copyright Page 6
Contents 9
Preface 11
List of Examples 13
List of Figures 31
About the CompanionWebsite 35
Introduction 37
I.1 Comments 37
I.1.1 On the Structure of This Book 37
I.1.2 OnMathematical Language and Notation 39
I.2 Background (Elements of Theory) 40
I.2.1 Sequences of Functions 40
I.2.2 Series of Functions 43
I.2.3 Families of Functions 46
Chapter1 Conditions of Uniform Convergence 53
1.1 Pointwise, Absolute, and Uniform Convergence. Convergence on a Set and Subset 53
1.2 Uniform Convergence of Sequences and Series of Squares and Products 67
1.3 Dirichlet’s and Abel’s Theorems 83
Exercises 91
Further Reading 94
Chapter 2 Properties of the Limit Function: Boundedness, Limits, Continuity 97
2.1 Convergence and Boundedness 97
2.2 Limits and Continuity of Limit Functions 103
2.3 Conditions of Uniform Convergence. Dini’s Theorem 120
2.4 Convergence and Uniform Continuity 131
Exercises 140
Further Reading 145
Chapter3 Properties of the Limit Function: Differentiability and Integrability 147
3.1 Differentiability of the Limit Function 147
3.2 Integrability of the Limit Function 169
Exercises 180
Further Reading 183
Chapter 4 Integrals Depending on a Parameter 185
4.1 Existence of the Limit and Continuity 185
4.2 Differentiability 196
4.3 Integrability 206
Exercises 214
Further Reading 218
Chapter 5 Improper Integrals Depending on a Parameter 219
5.1 Pointwise, Absolute, and Uniform Convergence 219
5.2 Convergence of the Sum and Product 228
5.3 Dirichlet’s and Abel’s Theorems 237
5.4 Existence of the Limit and Continuity 244
5.5 Differentiability 250
5.6 Integrability 254
Exercises 262
Further Reading 266
Bibliography 267
Index 269

Rezensionen

"The features of the book include An overview of important concepts and theorems on uniform convergence, Well-organized coverage of the majority of the topics on uniform convergence studied in analysis courses, An original approach to the analysis of important results on uniform convergence
based on counterexamples, Additional exercises at varying levels of complexity for each topic covered in the book & A supplementary Instructor's Solutions Manual containing complete solutions to
all exercises, which is available via a companion website" Mathematical Reviews, Sept 2017

Produkt-Details

• Erscheint lt. Verlag: 17.1.2017
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Technik
• Schlagworte: Analysis • Calculus • counterexamples, mathematical analysis, calculus, proper and improper integrals, real analysis, uniform convergence • gegenbeispiel • Mathematical Analysis • Mathematics • Mathematik • Mathematische Analyse • Real analysis • reelle Analysis • Reelle Zahl
• ISBN-10: 1-119-30340-0 / 1119303400
• ISBN-13: 978-1-119-30340-4 / 9781119303404