Description
Book SynopsisAn accessible introduction to real analysis and its connection to elementary calculus Bridging the gap between the development and history of real analysis, Introduction to Real Analysis: An Educational Approach presents a comprehensive introduction to real analysis while also offering a survey of the field. With its balance of historical background, key calculus methods, and hands-on applications, this book provides readers with a solid foundation and fundamental understanding of real analysis.
The book begins with an outline of basic calculus, including a close examination of problems illustrating links and potential difficulties. Next, a fluid introduction to real analysis is presented, guiding readers through the basic topology of real numbers, limits, integration, and a series of functions in natural progression. The book moves on to analysis with more rigorous investigations, and the topology of the line is presented along with a discussion of limits and
Table of Contents
Preface.
Acknowledgments.
1 Elementary Calculus.
1.1 Preliminary Concepts.
1.2 Limits and Continuity.
1.3 Differentiation.
1.4 Integration.
1.5 Sequences and Series of Constants.
1.6 Power Series and Taylor Series.
Summary.
Exercises.
Interlude: Fermat, Descartes, and theTangent Problem.
2 Introduction to Real Analysis.
2.1 Basic Topology of the Real Numbers.
2.2 Limits and Continuity.
2.3 Differentiation.
2.4 Riemann and Riemann-Stieltjes Integration.
2.5 Sequences, Series, and Convergence Tests.
2.6 Pointwise and Uniform Convergence.
Summary.
Exercises.
Interlude: Euler and the "Basel Problem".
3 A Brief Introduction to Lebesgue Theory.
3.1 Lebesgue Measure and Measurable Sets.
3.2 The Lebesgue Integral.
3.3 Measure, Integral, and Convergence.
3.4 Littlewood’s Three Principles.
Summary.
Exercises.
Interlude: The Set of Rational Numbers isVery Large andVery Small.
4 Special Topics.
4.1 Modeling with Logistic Functions—Numerical Derivatives.
4.2 Numerical Quadrature.
4.3 Fourier Series.
4.4 Special Functions—The Gamma Function.
4.5 Calculus Without Limits: Differential Algebra.
Summary.
Exercises.
Appendix A: Definitions and Theorems of Elementary Real Analysis.
A.1 Limits.
A.2 Continuity.
A.3 The Derivative.
A.4 Riemann Integration.
A.5 Riemann-Stieltjes Integration.
A.6 Sequences and Series of Constants.
A.7 Sequences and Series of Functions.
Appendix B: A Very Brief Calculus Chronology.
Appendix C: Projects in Real Analysis.
C.1 Historical Writing Projects.
C.2 Induction Proofs: Summations, Inequalities, and Divisibility.
C.3 Series Rearrangements.
C.4 Newton and the Binomial Theorem.
C.5 Symmetric Sums of Logarithms.
C.6 Logical Equivalence: Completeness of the Real Numbers.
C.7 Vitali’s Nonmeasurable Set.
C.8 Sources for Real Analysis Projects.
C.9 Sources for Projects for Calculus Students.
Bibliography.
Index.
