Providing students with a clear and understandable introduction to the fundamentals of analysis, this book continues to present the fundamental concepts of analysis in as painless a manner as possible. To achieve this aim, the second edition has made many improvements in exposition.
Author(s): Rod Haggarty
Edition: Second Edition
Publisher: Addison-Wesley UK
Year: 1993
Language: English
Pages: C, VIII, 332, B
PROLOGUE: What is Analysis? Introduction Precakulus developments in Ancient Greece Precalculus developments in Western Europe The discovery of calculus From calculus to analysis Conclusion CHAPTER ONE Preliminaries 1.1 Logic 1.2 Sets 1.2.1 Laws of the algebra of sets 1.3 Functions CHAPTER TWO The Real Numbers 2.1 Numbers 2.1.1 Theorem 2.1.2 Theorem 2.1.3 Theorem 2.1.4 Theorem 2.2 Axioms for the real numbers 2.2.1 The axioms of arithmetic 2.2.2 The axioms of order 2.2.3 The completeness axiom 2.3 The completeness axiom 2.3.1 The Archimedean postulate 2.3.2 Density of the rationals 2.3.3 The principle of mathematical induction CHAPTER THREE Sequences 3.1 Convergent sequences 3.1.1 Definition 3.1.2 Rules 3.1.3 Sandwich rule 3.1.4 Composite rule 3.2 Null sequences 3.2.1 Power rule 3.2.2 Basic null sequences 3.3 Divergent sequences 3.3.1 Result 3.3.2 Definition 3.3.3 Rules 3.3.4 Reciprocal rule 3.3.5 Definition 3.3.6 Result 3.4 Monotone sequences 3.4.1 Principle of monotone sequences 3.4.2 Bolzano-Weierstrass theorem 3.4.3 Definition 3.4.4 Theorem CHAPTER FOUR Series 4.1 Infinite series 4.1.1 Result 4.1.2 The vanishing condition 4.1.3 Sum rule 4.1.4 Scalar product rule 4.1.5 Result 4.2 Series tests 4.2.1 First comparison test 4.2.2 Second comparison test 4.2.3 D’ Alembert’s ratio test 4.2.4 Altenating series test 4.2.5 Integral test 4.2.6 ’fheore111 4.2.7 l’heorern 4.2.8 The rearrangement rule 4.2. 9 Cauchy product of series 4.3 Power series 4.3.1 Theorem 4.3.2 Arithmetic of power series 4.3.3 Definition 4.3.4 Definition 4.3.5 Definition CHAPTER FIVE Continuous Functions 5.1 Limits 5.1.1 Definition 5.1.2 Definition 5.1.3 Rules 5.1.4 Sandwich rule 5.1..5 Composite rule 5.1.6 Theorem 5.2 Continuity 5.2.1 Definition 5.2.2 Definition 5.2.3 Theorem 5.2.4 Rules 5.2.5 Sandwich rule 5.2.6 Composite rule 5.2. 7 Inverse rule 5.3 Theorems 5.3.1 The boundedness property 5.3.2 The intermediate value property 5.3.3 The interval theorem 5.3.4 A fixed point theorem CHAPTER SIX Differentiation 6.1 Differentiable functions 6.1.1 Definition 6.1.2 Theorem 6.1.3 Rules 6.1.4 Quotient rule 6.1.5 Sandwich rule 6.1.6 Composite rule 6.1.7 Inverse rule 6.1.8 Definition 6.1.9 Local extremum theorem 6.2 Theorems 6.2.1 Rolle’s theorem 6.2.2 Mean value theorem 6.2.3 The increasing-decreasing theorem 6.2.4 Cauchy’s mean value theorem 6.2.5 L’Hopital’s rule (version A) 6.2.6 Leibniz’s formula 6.2.7 L’Hopital’s rule (version B) 6.3 Taylor polynomials 6.3.1 Definition 6.3.2 The first remainder theorem 6.3.3 Standard series 6.3.4 Definition 6.3.5 Taylor’s theorem 6.3.6 Result 6.3. 7 Classification theorem for local extrema 6.4 Alternative forms of Taylor’s theorem 6.4.1 Taylor’s theorem CHAPTER SEVEN Integration 7.1 The Riemann integral 7.1.1 Definition 7.1.2 Definition 7.1.3 Definition 7.1.4 Theorem 7.1.5 Riemann’s condition 7.1.6 Theorem 7.1.7 Definition 7.1.8 Uniform continuity theorem 7.1.9 Theorem 7.1.10 Properties of the Riemann integral 7.1.11 The fundamental theorem of calculus 7.1.12 The integral mean value theorem 7. l .13 Corollary 7.2 Techniques 7.2.1 Integration by parts 7.2.2 Change of variables 7.3 Improper integrals 7.3.1 Definition 7.3.2 Definition 7.3.3 Definition 7.3.4 Comparison test for integrals 7.3.5 Definition APPENDIX The Elementary Functions A.1 Theorem A.2 The calculus of power series A.3 Definition A.4 Definition A.5 Definition A.6 Definition A.7 Definition A.8 Definition A. 9 Definition A.10 Definition A.11 Definition Solutions to Exercises Answers to problems Index of Symbols Index
