Andrei Bourchtein's Books

Elementary Functions

By: Andrei Bourchtein,Ludmila Bourchtein

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Elementary Functions

By: Andrei Bourchtein,Ludmila Bourchtein

This textbook focuses on the study of different kinds of elementary functions ubiquitous both in high school Algebra and Calculus. To analyze the functions ranging from polynomial to trigonometric ones, it uses rudimentary techniques available to high school students, and at the same time follows the mathematical rigor appropriate for university level courses. Contrary to other books of Pre-Calculus, this textbook emphasizes the study of elementary functions with rigor appropriate for university level courses in mathematics, although the exposition is confined to the pre-limit topics and techniques. This makes the book useful, on the one hand, as an introduction to mathematical reasoning and methods of proofs in mathematical analysis, and on the other hand, as a preparatory course on the properties of different kinds of elementary functions. The textbook is aimed at university freshmen and high-school students interested in learning strict mathematical reasoning and in preparing a solid base for subsequent study of elementary functions at advanced level of Calculus and Analysis. The required prerequisites correspond to the level of the high school Algebra. All the preliminary concepts and results related to the elementary functions are covered in the initial part of the text. This makes the textbook suitable for both classroom use and self-study.

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Page Count

488

Publisher

Springer Nature

Published Date

ISBN 10

3031290755

ISBN 13

9783031290756

Complex Analysis

By: Andrei Bourchtein,Ludmila Bourchtein

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Complex Analysis

By: Andrei Bourchtein,Ludmila Bourchtein

This book discusses all the major topics of complex analysis, beginning with the properties of complex numbers and ending with the proofs of the fundamental principles of conformal mappings. Topics covered in the book include the study of holomorphic and analytic functions, classification of singular points and the Laurent series expansion, theory of residues and their application to evaluation of integrals, systematic study of elementary functions, analysis of conformal mappings and their applications—making this book self-sufficient and the reader independent of any other texts on complex variables. The book is aimed at the advanced undergraduate students of mathematics and engineering, as well as those interested in studying complex analysis with a good working knowledge of advanced calculus. The mathematical level of the exposition corresponds to advanced undergraduate courses of mathematical analysis and first graduate introduction to the discipline. The book contains a large number of problems and exercises, making it suitable for both classroom use and self-study. Many standard exercises are included in each section to develop basic skills and test the understanding of concepts. Other problems are more theoretically oriented and illustrate intricate points of the theory. Many additional problems are proposed as homework tasks whose level ranges from straightforward, but not overly simple, exercises to problems of considerable difficulty but of comparable interest.

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Page Count

357

Publisher

Springer Nature

Published Date

ISBN 10

9811592195

ISBN 13

9789811592195

CounterExamples

From Elementary Calculus to the Beginnings of Analysis

By: Andrei Bourchtein,Ludmila Bourchtein

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CounterExamples

From Elementary Calculus to the Beginnings of Analysis

By: Andrei Bourchtein,Ludmila Bourchtein

This book provides a one-semester undergraduate introduction to counterexamples in calculus and analysis. It helps engineering, natural sciences, and mathematics students tackle commonly made erroneous conjectures. The book encourages students to think critically and analytically, and helps to reveal common errors in many examples. In this book, the authors present an overview of important concepts and results in calculus and real analysis by considering false statements, which may appear to be true at first glance. The book covers topics concerning the functions of real variables, starting with elementary properties, moving to limits and continuity, and then to differentiation and integration. The first part of the book describes single-variable functions, while the second part covers the functions of two variables. The many examples presented throughout the book typically start at a very basic level and become more complex during the development of exposition. At the end of each chapter, supplementary exercises of different levels of complexity are provided, the most difficult of them with a hint to the solution. This book is intended for students who are interested in developing a deeper understanding of the topics of calculus. The gathered counterexamples may also be used by calculus instructors in their classes.

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363

Publisher

CRC Press

Published Date

ISBN 10

1482246678

ISBN 13

9781482246674

Theory of Infinite Sequences and Series

By: Ludmila Bourchtein,Andrei Bourchtein

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Theory of Infinite Sequences and Series

By: Ludmila Bourchtein,Andrei Bourchtein

This textbook covers the majority of traditional topics of infinite sequences and series, starting from the very beginning – the definition and elementary properties of sequences of numbers, and ending with advanced results of uniform convergence and power series. The text is aimed at university students specializing in mathematics and natural sciences, and at all the readers interested in infinite sequences and series. It is designed for the reader who has a good working knowledge of calculus. No additional prior knowledge is required. The text is divided into five chapters, which can be grouped into two parts: the first two chapters are concerned with the sequences and series of numbers, while the remaining three chapters are devoted to the sequences and series of functions, including the power series. Within each major topic, the exposition is inductive and starts with rather simple definitions and/or examples, becoming more compressed and sophisticated as the course progresses. Each key notion and result is illustrated with examples explained in detail. Some more complicated topics and results are marked as complements and can be omitted on a first reading. The text includes a large number of problems and exercises, making it suitable for both classroom use and self-study. Many standard exercises are included in each section to develop basic techniques and test the understanding of key concepts. Other problems are more theoretically oriented and illustrate more intricate points of the theory, or provide counterexamples to false propositions which seem to be natural at first glance. Solutions to additional problems proposed at the end of each chapter are provided as an electronic supplement to this book.

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Page Count

388

Publisher

Springer Nature

Published Date

ISBN 10

3030794318

ISBN 13

9783030794316

Counterexamples on Uniform Convergence

Sequences, Series, Functions, and Integrals

By: Andrei Bourchtein,Ludmila Bourchtein

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Counterexamples on Uniform Convergence

Sequences, Series, Functions, and Integrals

By: Andrei Bourchtein,Ludmila Bourchtein

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.

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Page Count

270

Publisher

John Wiley & Sons

Published Date

ISBN 10

1119303389

ISBN 13

9781119303381

Theory of Infinite Sequences and Series

By: Ludmila Bourchtein,Andrei Bourchtein

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Theory of Infinite Sequences and Series

By: Ludmila Bourchtein,Andrei Bourchtein

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Page Count

388

Publisher

Springer Nature

Published Date

ISBN 10

3030794318

ISBN 13

9783030794316

Counterexamples on Uniform Convergence

Sequences, Series, Functions, and Integrals

By: Andrei Bourchtein,Ludmila Bourchtein

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Counterexamples on Uniform Convergence

Sequences, Series, Functions, and Integrals

By: Andrei Bourchtein,Ludmila Bourchtein

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Solve jigsaw puzzle

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Page Count

270

Publisher

John Wiley & Sons

Published Date

ISBN 10

1119303389

ISBN 13

9781119303381

CounterExamples

From Elementary Calculus to the Beginnings of Analysis

By: Andrei Bourchtein,Ludmila Bourchtein

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CounterExamples

From Elementary Calculus to the Beginnings of Analysis

By: Andrei Bourchtein,Ludmila Bourchtein

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Solve jigsaw puzzle

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Page Count

363

Publisher

CRC Press

Published Date

ISBN 10

1482246678

ISBN 13

9781482246674

Book's by Andrei Bourchtein

Elementary Functions

Elementary Functions

Complex Analysis

Complex Analysis

CounterExamples

CounterExamples

Theory of Infinite Sequences and Series

Theory of Infinite Sequences and Series

Counterexamples on Uniform Convergence

Counterexamples on Uniform Convergence

Theory of Infinite Sequences and Series

Theory of Infinite Sequences and Series

Counterexamples on Uniform Convergence

Counterexamples on Uniform Convergence

CounterExamples

CounterExamples

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