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2017 | Book

Very Naive Set Theory, Functions, and Proofs Read chapter Read first chapter

Amazing and Aesthetic Aspects of Analysis

Author: Paul Loya

Publisher: Springer New York

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

Lively prose and imaginative exercises draw the reader into this unique introductory real analysis textbook. Motivating the fundamental ideas and theorems that underpin real analysis with historical remarks and well-chosen quotes, the author shares his enthusiasm for the subject throughout. A student reading this book is invited not only to acquire proficiency in the fundamentals of analysis, but to develop an appreciation for abstraction and the language of its expression.
In studying this book, students will encounter:the interconnections between set theory and mathematical statements and proofs;
the fundamental axioms of the natural, integer, and real numbers;
rigorous ε-N and ε-δ definitions;
convergence and properties of an infinite series, product, or continued fraction;
series, product, and continued fraction formulæ for the various elementary functions and constants.
Instructors will appreciate this engaging perspective, showcasing the beauty of these fundamental results.

Table of Contents

Frontmatter

Some Standard Curriculum

Frontmatter
Chapter 1. Very Naive Set Theory, Functions, and Proofs
Abstract
One of the goals of this text is to get you proving mathematical statements in real analysis. Set theory provides a safe environment in which to learn about math statements, “if ... then,” “if and only if,” etc., and to learn the logic behind proofs. Since this is an introductory book on analysis, our treatment of sets is “very naive,” in the sense that we actually don’t define sets rigorously, only informally; we are mostly interested in how “they work,” not really what they are.
Paul Loya
Chapter 2. Numbers, Numbers, and More Numbers
Abstract
This chapter is on the study of numbers. Of course, we all have a working understanding of the real numbers, and we use many aspects of these numbers in everyday life: tallying up tuition and fees, figuring out how much we have left on our food cards, etc. We have accepted from our childhood all the properties of numbers that we use every day.
Paul Loya
Chapter 3. Infinite Sequences of Real and Complex Numbers
Abstract
Notable enough, however, are the controversies over the series \(1 - 1 + 1 - 1 + 1 -\).
Paul Loya
Chapter 4. Limits, Continuity, and Elementary Functions
Abstract
In this chapter we study what are without doubt the most important functions in all of analysis and topology, the continuous functions.
Paul Loya
Chapter 5. Some of the Most Beautiful Formulas in the World I–III
Abstract
In this chapter we present a small sample of some of the most beautiful formulas in the world.
Paul Loya

Extracurricular Activities

Frontmatter
Chapter 6. Advanced Theory of Infinite Series

This chapter is about going in depth into the theory and application of infinite series. One infinite series that will come up again and again in this chapter and the next chapter as well is the Riemann zeta function.

Paul Loya
Chapter 7. More on the Infinite: Products and Partial Fractions
Abstract
This chapter is devoted entirely to the theory and application of infinite products, and as a consolation prize, we also talk about partial fractions.
Paul Loya
Chapter 8. Infinite Continued Fractions
Abstract
We dabbled a little into the theory of continued fractions (that is, fractions that continue on and on and on ...) way back on the exercises of Section 3.​4. In this chapter we concentrate on this fascinating subject.
Paul Loya
Backmatter
Metadata
Title
Amazing and Aesthetic Aspects of Analysis
Author
Paul Loya
Copyright Year
2017
Publisher
Springer New York
Electronic ISBN
978-1-4939-6795-7
Print ISBN
978-1-4939-6793-3
DOI
https://doi.org/10.1007/978-1-4939-6795-7

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