Real Analysis (eBook)

A Constructive Approach

Mark Bridger (Autor)

eBook Download: PDF

2011 | 1. Auflage
320 Seiten
John Wiley & Sons (Verlag)
9781118031568 (ISBN)

Infos

A unique approach to analysis that lets you apply mathematicsacross a range of subjects

This innovative text sets forth a thoroughly rigorous modernaccount of the theoretical underpinnings of calculus: continuity,differentiability, and convergence. Using a constructive approach,every proof of every result is direct and ultimatelycomputationally verifiable. In particular, existence is neverestablished by showing that the assumption of non-existence leadsto a contradiction. The ultimate consequence of this method is thatit makes sense--not just to math majors but also to studentsfrom all branches of the sciences.

The text begins with a construction of the real numbersbeginning with the rationals, using interval arithmetic. Thisintroduces readers to the reasoning and proof-writing skillsnecessary for doing and communicating mathematics, and it sets thefoundation for the rest of the text, which includes:
* Early use of the Completeness Theorem to prove a helpfulInverse Function Theorem
* Sequences, limits and series, and the careful derivation offormulas and estimates for important functions
* Emphasis on uniform continuity and its consequences, such asboundedness and the extension of uniformly continuous functionsfrom dense subsets
* Construction of the Riemann integral for functions uniformlycontinuous on an interval, and its extension to improperintegrals
* Differentiation, emphasizing the derivative as a functionrather than a pointwise limit
* Properties of sequences and series of continuous anddifferentiable functions
* Fourier series and an introduction to more advanced ideas infunctional analysis

Examples throughout the text demonstrate the application of newconcepts. Readers can test their own skills with problems andprojects ranging in difficulty from basic to challenging.

This book is designed mainly for an undergraduate course, andthe author understands that many readers will not go on to moreadvanced pure mathematics. He therefore emphasizes an approach tomathematical analysis that can be applied across a range ofsubjects in engineering and the sciences.

Autor

MARK BRIDGER, PHD, is Associate Professor of Mathematics at Northeastern University in Boston, Massachusetts. The author of numerous journal articles, Dr. Bridger's research focuses on constructive analysis, the philosophy of science, and the use of technology in mathematics education.

Inhalt

Preface.

Acknowledgements.

Introduction.

0 Preliminaries.

0.1 The Natural Numbers.

0.2 The Rationals.

1 The Real Numbers and Completeness.

1.0 Introduction.

1.1 Interval Arithmetic.

1.2 Families of Intersecting Intervals.

1.3 Fine Families.

1.4 Definition of the Reals.

1.5 Real Number Arithmetic.

1.6 Rational Approximations.

1.7 Real Intervals and Completeness.

1.8 Limits and Limiting Families.

Appendix: The Goldbach Number and Trichotomy.

2 An Inverse Function Theorem and its Application.

2.0 Introduction.

2.1 Functions and Inverses.

2.2 An Inverse Function Theorem.

2.3 The Exponential Function.

2.4 Natural Logs and the Euler Number.

3 Limits. Sequences and Series.

3.1 Sequences and Convergence.

3.2 Limits of Functions.

3.3 Series of Numbers.

Appendix I: Some Properties of Exp and Log.

Appendix 11: Rearrangements of Series.

4 Uniform Continuity.

4.1 Definitions and Elementary Properties.

4.2 Limits and Extensions.

Appendix I: Are there Non-Continuous Functions?

Appendix XI: Continuity of Double-Sided Inverses.

Appendix III: The Goldbach Function.

5 The Riemann Integral.

5.1 Definition and Existence.

5.2 Elementary Properties.

5.3 Extensions and Improper Integrals.

6 Differentiation.

6.1 Definitions and Basic Properties.

6.2 The Arithmetic of Differentiability.

6.3 Two Important Theorems.

6.4 Derivative Tools.

6.5 Integral Tools.

7 Sequences and Series of Functions.

7.1 Sequences of Functions.

7.2 Integrals and Derivatives of Sequences.

7.3 Power Series.

7.4 Taylor Series.

7.5 The Periodic Functions.

Appendix: Binomial Issues.

8 The Complex Numbers and Fourier Series.

8.0 Introduction.

8.1 The Complex Numbers C.

8.2 Complex Functions and Vectors.

8.3 Fourier Series Theory.

References.

Index.

Rezensionen

"The first chapters are presented at a very nice leisurelypace, which makes reading and learning enjoyable."(Zentralblatt MATH, 2007)

"Very suitable for self-study by undergraduates at alllevels..." (CHOICE, August 2007)

"...deserves to be read. Even if you do not subscribe to theconstructive viewpoint, you'll learn something and find plenty ofmaterial to exploit in your classical analysis courses." (MAAReviews, December 23, 2006)

Produkt-Details

• Erscheint lt. Verlag: 14.10.2011
• Reihe/Serie: Wiley Series in Pure and Applied Mathematics
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Technik
• Schlagworte: Analysis • Mathematical Analysis • Mathematics • Mathematik • Mathematische Analyse • Real analysis • reelle Analysis
• ISBN-13: 9781118031568 / 9781118031568