تحلیل (سری ریاضیات مدولار)

Building on the basic concepts through a careful discussion of covalence, (while adhering resolutely to sequences where possible), the main part of the book concerns the central topics of continuity, differentiation and integration of real functions. Throughout, the historical context in which the subject was developed is highlighted and particular attention is paid to showing how precision allows us to refine our geometric intuition. The intention is to stimulate the reader to reflect on the underlying concepts and ideas. Front Cover 1 Analysis 4 Copyright Page 5 Table of Contents 6 Series Preface 9 Preface 10 Acknowledgements 12 Chapter 1. Introduction: Why We Study Analysis 14 1.1 What the computer cannot see ... 14 1.2 From counting to complex numbers 15 1.3 From infinitesimals to limits 16 Chapter 2. Convergent Sequences and Series 18 2.1 Convergence and summation 18 2.2 Algebraic and order properties of limits 27 Summary 30 Further exercises 30 Chapter 3. Completeness and Convergence 32 3.1 Completeness and sequences 32 3.2 Completeness and series 41 3.3 Alternating series 45 3.4 Absolute and conditional convergence of series 48 Summary 51 Further exercises 51 Chapter 4. Functions Defined by Power Series 53 4.1 Polynomials – and what Euler did with them! 53 4.2 Multiplying power series: Cauchy products 56 4.3 The radius of convergence of a power series 59 4.4 The elementary transcendental functions 61 Summary 64 Further exercises 64 Chapter 5. Functions and Limits 65 5.1 Historical interlude: curves, graphs and functions 66 5.2 The modern concept of function: ordered pairs, domain and range 69 5.3 Combining real functions 72 5.4 Limits of real functions – what Cauchy meant! 75 Summary 80 Further exercises 80 Chapter 6. Continuous Functions 82 6.1 Limits that fit 82 6.2 Limits that do not fit: types of discontinuity 87 6.3 General power functions 90 6.4 Continuity of power series 95 Summary 98 Further exercises 98 Chapter 7. Continuity on Intervals 99 7.1 From interval to interval 99 7.2 Applications: fixed points, roots and iteration 103 7.3 Reaching the maximum: the Boundedness Theorem 107 7.4 Uniform continuity – what Cauchy meant? 109 Summary 112 Further exercises 112 Chapter 8. Differentiable Real Functions 113 8.1 Tangents: prime and ultimate ratios 113 8.2 The derivative as a limit 115 8.3 Differentiation and continuity 117 8.4 Combining derivatives 119 8.5 Extreme points and curve sketching 122 Summary 124 Further exercises 125 Chapter 9. Mean Values and Taylor Series 126 9.1 The Mean Value Theorem 126 9.2 Tests for extreme points 129 9.3 L'Hôpital's Rules and the calculation of limits 132 9.4 Differentiation of power series 134 9.5 Taylor's Theorem and series expansions 137 Summary 141 Further exercises 142 Chapter 10. The Riemann Integral 143 10.1 Primitives and the 'arbitrary constant' 144 10.2 Partitions and step functions: the Riemann Integral 146 10.3 Criteria for integrability 152 10.4 Classes of integrable functions 155 10.5 Properties of the integral 157 Summary 160 Further exercises 161 Chapter 11. Integration Techniques 162 11.1 The Fundamental Theorem of the Calculus 162 11.2 Integration by parts and change of variable 164 11.3 Improper integrals 167 11.4 Convergent integrals and convergent series 169 Summary 172 Further exercises 172 Chapter 12. What Next? Extensions and Developments 174 12.1 Generalizations of completeness 174 12.2 Approximation of functions 176 12.3 Integrals of real functions: yet more completeness 176 Appendix A: Program Listings 178 A.I Sequences program 178 A.2 Another sequence program 178 A.3 Taylor series 180 A.4 Newton's method in one dimension 182 Solutions to exercises 183 Index 198 Before you shut the book and file its contents under 'useless pedantry', let's reflect a little on what we know about the Calculus, what its operations are and on what sort of objects it operates.