Introduction to Real Analysis: An Educational Approach
July 2009, ©2009
This price is valid for United States. Change location to view local pricing and availability.
Other Available Formats: E-book
1 Elementary Calculus.
1.1 Preliminary Concepts.
1.2 Limits and Continuity.
1.5 Sequences and Series of Constants.
1.6 Power Series and Taylor Series.
Interlude: Fermat, Descartes, and theTangent Problem.
2 Introduction to Real Analysis.
2.1 Basic Topology of the Real Numbers.
2.2 Limits and Continuity.
2.4 Riemann and Riemann-Stieltjes Integration.
2.5 Sequences, Series, and Convergence Tests.
2.6 Pointwise and Uniform Convergence.
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.
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.
Appendix A: Definitions and Theorems of Elementary Real Analysis.
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.