Based on courses given at Eotvos Lorand University (Hungary) over the past 30 years, this introductory textbook develops the central concepts of the analysis of functions of one variable -- systematically, with many examples and illustrations, and in a manner that builds upon, and sharpens, the student's mathematical intuition. The book provides a solid grounding in the basics of logic and proofs, sets, and real numbers, in preparation for a study of the main topics: limits, continuity, rational functions and transcendental functions, differentiation, and integration. Numerous applications to other areas of mathematics, and to physics, are given, thereby demonstrating the practical scope and power of the theoretical concepts treated.
In the spirit of learning-by-doing, Real Analysis includes more than 500 engaging exercises for the student keen on mastering the basics of analysis. The wealth of material, and modular organization, of the book make it adaptable as a textbook for courses of various levels; the hints and solutions provided for the more challenging exercises make it ideal for independent study.
Author(s): Miklós Laczkovich; Vera T Sos
Publisher: Springer
Year: 2015
Language: English
Pages: 483
Preface
Contents
1 A Brief Historical Introduction
2 Basic Concepts
2.1 A Few Words About Mathematics in General
2.2 Basic Concepts in Logic
2.3 Proof Techniques
2.4 Sets, Functions, Sequences
3 Real Numbers
3.1 Decimal Expansions: The Real Line
3.2 Bounded Sets
3.3 Exponentiation
3.4 First Appendix: Consequences of the Field Axioms
3.5 Second Appendix: Consequences of the Order Axioms
4 Infinite Sequences I
4.1 Convergent and Divergent Sequences
4.2 Sequences That Tend to Infinity
4.3 Uniqueness of Limit
4.4 Limits of Some Specific Sequences
5 Infinite Sequences II
5.1 Basic Properties of Limits
5.2 Limits and Inequalities
5.3 Limits and Operations
5.4 Applications
6 Infinite Sequences III
6.1 Monotone Sequences
6.2 The Bolzano–Weierstrass Theorem and Cauchy's Criterion
7 Rudiments of Infinite Series
8 Countable Sets
9 Real-Valued Functions of One Real Variable
9.1 Functions and Graphs
9.2 Global Properties of Real Functions
9.3 Appendix: Basics of Coordinate Geometry
10 Continuity and Limits of Functions
10.1 Limits of Functions
10.2 The Transference Principle
10.3 Limits and Operations
10.4 Continuous Functions in Closed and Bounded Intervals
10.5 Uniform Continuity
10.6 Monotonicity and Continuity
10.7 Convexity and Continuity
10.8 Arc Lengths of Graphs of Functions
10.9 Appendix: Proof of Theorem 10.81
11 Various Important Classes of Functions (Elementary Functions)
11.1 Polynomials and Rational Functions
11.2 Exponential and Power Functions
11.3 Logarithmic Functions
11.4 Trigonometric Functions
11.5 The Inverse Trigonometric Functions
11.6 Hyperbolic Functions and Their Inverses
11.7 First Appendix: Proof of the Addition Formulas
11.8 Second Appendix: A Few Words on Complex Numbers
12 Differentiation
12.1 The Definition of Differentiability
12.2 Differentiation Rules and Derivatives of the ElementaryFunctions
12.3 Higher-Order Derivatives
12.4 Linking the Derivative and Local Properties
12.5 Intermediate Value Theorems
12.6 Investigation of Differentiable Functions
13 Applications of Differentiation
13.1 L'Hôpital's Rule
13.2 Polynomial Approximation
13.3 The Indefinite Integral
13.4 Differential Equations
13.5 The Catenary
13.6 Properties of Derivative Functions
13.7 First Appendix: Proof of Theorem 13.20
13.8 Second Appendix: On the Definition of Trigonometric Functions Again
14 The Definite Integral
14.1 Problems Leading to the Definition of the Definite Integral
14.2 The Definition of the Definite Integral
14.3 Necessary and Sufficient Conditions for Integrability
14.4 Integrability of Continuous Functions and Monotone Functions
14.5 Integrability and Operations
14.6 Further Theorems Regarding the Integrability of Functions and the Value of the Integral
14.7 Inequalities for Values of Integrals
15 Integration
15.1 The Link Between Integration and Differentiation
15.2 Integration by Parts
15.3 Integration by Substitution
15.4 Integrals of Elementary Functions
15.4.1 Rational Functions
15.4.2 Integrals Containing Roots
15.4.3 Rational Functions of ex
15.4.4 Trigonometric Functions
15.5 Nonelementary Integrals of Elementary Functions
15.6 Appendix: Integration by Substitution for Definite Integrals (Proof of Theorem 15.22)
16 Applications of Integration
16.1 The General Concept of Area and Volume
16.2 Computing Area
16.3 Computing Volume
16.4 Computing Arc Length
16.5 Polar Coordinates
16.6 The Surface Area of a Surface of Revolution
16.7 Appendix: Proof of Theorem 16.20
17 Functions of Bounded Variation
18 The Stieltjes Integral
19 The Improper Integral
19.1 The Definition and Computation of Improper Integrals
19.2 The Convergence of Improper Integrals
19.3 Appendix: Proof of Theorem 17timphely
Erratum
Hints, Solutions
Notation
References
Index
