دانلود کتاب Introduction to Real Analysis: An Educational Approach

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Cover
Title
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 the Tangent 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 Is Very Large and Very 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 & 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 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