Elementary Real and Complex Analysis / Edition 2

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Overview

In this book the renowned Russian mathematician Georgi E. Shilov brings his unique perspective to real and complex analysis, an area of perennial interest in mathematics. Although there are many books available on the topic, the present work is specially designed for undergraduates in mathematics, science and engineering. A high level of mathematical sophistication is not required.
The book begins with a systematic study of real numbers, understood to be a set of objects satisfying certain definite axioms. The concepts of a mathematical structure and an isomorphism are introduced in Chapter 2, after a brief digression on set theory, and a proof of the uniqueness of the structure of real numbers is given as an illustration. Two other structures are then introduced, namely n-dimensional space and the field of complex numbers.
After a detailed treatment of metric spaces in Chapter 3, a general theory of limits is developed in Chapter 4. Chapter 5 treats some theorems on continuous numerical functions on the real line, and then considers the use of functional equations to introduce the logarithm and the trigonometric functions. Chapter 6 is on infinite series, dealing not only with numerical series but also with series whose terms are vectors and functions (including power series). Chapters 7 and 8 treat differential calculus proper, with Taylor's series leading to a natural extension of real analysis into the complex domain. Chapter 9 presents the general theory of Riemann integration, together with a number of its applications. Analytic functions are covered in Chapter 10, while Chapter 11 is devoted to improper integrals, and makes full use of the technique of analytic functions.
Each chapter includes a set of problems, with selected hints and answers at the end of the book. A wealth of examples and applications can be found throughout the text. Over 340 theorems are fully proved.

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Editorial Reviews

Booknews
**** Reprint of the MIT Press original of 1974 (which is honored by inclusion in BCL3. Annotation c. Book News, Inc., Portland, OR (booknews.com)
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Product Details

  • ISBN-13: 9780486689227
  • Publisher: Dover Publications
  • Publication date: 2/7/1996
  • Series: Dover Books on Mathematics Series
  • Edition description: REV
  • Edition number: 2
  • Pages: 528
  • Sales rank: 722,556
  • Product dimensions: 5.38 (w) x 8.48 (h) x 1.03 (d)

Table of Contents

Preface
1 Real Numbers
  1.1. Set-Theoretic Preliminaries
  1.2. Axioms for the Real Number System
  1.3. Consequences of the Addition Axioms
  1.4. Consequences of the Multiplication Axioms
  1.5. Consequences of the Order Axioms
  1.6. Consequences of the Least Upper Bound Axiom
  1.7. The Principle of Archimedes and Its Consequences
  1.8. The Principle of Nested Intervals
  1.9. The Extended Real Number System
  Problems
2 Sets
  2.1. Operations on Sets
  2.2. Equivalence of Sets
  2.3. Countable Sets
  2.4 Uncountable Sets
  2.5. Mathematical Structures
  2.6. n-Dimensional Space
  2.7. Complex Numbers
  2.8. Functions and Graphs
  Problems
3 Metric Spaces
  3.1. Definitions and Examples
  3.2. Open Sets
  3.3. Convergent Sequences and Homeomorphisms
  3.4. Limit Points
  3.5. Closed Sets
  3.6. Dense Sets and Closures
  3.7. Complete Metric Spaces
  3.8. Completion of a Metric Space
  3.9. Compactness
  Problems
4 Limits
  4.1. Basic Concepts
  4.2. Some General Theorems
  4.3. Limits of Numerical Functions
  4.4. Upper and Lower Limits
  4.5. Nondecreasing and Nonincreasing Functions
  4.6. Limits of Numerical Functions
  4.7. Limits of Vector Functions
  Problems
5 Continuous Functions
  5.1. Continuous Functions on a Metric Space
  5.2. Continuous Numerical Functions on the Real Line
  5.3. Monotonic Functions
  5.4. The Logarithm
  5.5. The Exponential
  5.6. Trignometric Functions
  5.7. Applications of Trigonometric Functions
  5.8. Continuous Vector Functions of a Vecor Variable
  5.9. Sequences of Functions
  Problems
6 Series
  6.1. Numerical Series
  6.2. Absolute and Conditional Convergences
  6.3. Operations on Series
  6.4. Series of Vectors
  6.5. Series of Functions
  6.6. Power Series
  Problems
7 The Derivative
  7.1. Definitions and Examples
  7.2. Properties of Differentiable Functions
  7.3. The Differential
  7.4. Mean Value Theorems
  7.5. Concavity and Inflection Points
  7.6. L'Hospital's Rules
  Problems
8 Higher Derivatives
  8.1. Definitions and Examples
  8.2. Taylor's Formula
  8.3. More on Concavity and Inflection Points
  8.4. Another Version of Taylor's Formula
  8.5. Taylor Series
  8.6. Complex Exponentials and Trigonometric Functions
  8.7. Hyperbolic Functions
  Problems
9 The Integral
  9.1. Definitions and Basic Properties
  9.2. Area and Arc Length
  9.3. Antiderivatives and Indefinite Integrals
  9.4. Technique of Indefinite Integrals
  9.5. Evaluation of Definite Integrals
  9.6. More on Area
  9.7. More on Arc Length
  9.8. Area of a Surface of Revolution
  9.9. Further Applications of Integration
  9.10. Integration of Sequences of Functions
  9.11. Parameter-Dependent Integrals
  9.12. Line Integrals
  Problems
10 Analytic Functions
  10.1. Basic Concepts
  10.2. Line Integrals of Complex Functions
  10.3. Cauchy's Theorem and Its Consequences
  10.4. Residues and Isolated Singular Points
  10.5. Mappings and Elementary Functions
  Problems
11 Improper Integrals
  11.1. Improper Integralsof the First Kind
  11.2. Convergence of Improper Integrals
  11.3. Improper Integrals of the Second and Third Kinds
  11.4 Evaluation of Improper Integrals by Residues
  11.5 Parameter-Dependent ImproperIntegrals
  11.6 The Gamma and Beta Functions
  Problems
Appendix A Elementary Symbolic Logic
Appendix B Measure and Integration on a Compact Metric Space
Selected Hints and Answers
Index
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  • Anonymous

    Posted January 8, 2005

    getting started in math analysis

    This book by Shilov covers the fundamentals in beginning analysis(both real and complex). It has in common with Walter Rudin's book (entitled 'Real and Complex Analysis') that it covers both real functions (integration theory and more), as well as Cauchy's theorems for analytic functions. Shilov's book is at an undergraduate level, and it can easily be used for self-study. The Dover edition is affordable. Rudin's book is for the beginning graduate level, and it is widely used in math departments around the world. Both books have stood the test of time. Comparison of Shilov with Rudin: Rudin's 'Real and Complex' has become an institution, and I have to admit I have loved it since I was a student myself, but conventional wisdom will have it that Shilov is a lot gentler on students, and much easier to get started with: It stresses motivation a bit more, the exercises are easier (some of Rudin's exercises are notorious, but I find the challenge charming--not all of my students do though!), and finally Shilov gets to touch upon a few applications; fashionable these days. But that part easily gets dated. I will expect that beginning students will enjoy Shilov's book. Personally, I find that with perseverance, students who keep at it with Rudin's book, will end up with a lot stronger foundation. They are more likely to have proofs in their blood. I guess Shilov can always serve as a leisurely supplementary reading to Rudin. There will never be another book like Rudin's 'Real and Complex', just like there will never be another van Gogh. But the fact that we love van Gogh doesn't prevent us from enjoying other paintings.

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