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eBook Real Analysis and Foundations, 4th Edition By Steven G. Krantz

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[eBook] [PDF] Real Analysis and Foundations, 4th Edition By Steven G. Krantz Half Title Title Page Copyright Page Dedication Table of Contents Preface to the Fourth Edition Preface to the Third Edition Preface to the Second Edition Preface to the First Edition 1: Number Systems 1.1 The Real Numbers 1.1 Appendix: Construction of the Real Numbers 1.2 The Complex Numbers 2: Sequences 2.1 Convergence of Sequences 2.2 Subsequences 2.3 Lim sup and Lim inf 2.4 Some Special Sequences 3: Series of Numbers 3.1 Convergence of Series 3.2 Elementary Convergence Tests 3.3 Advanced Convergence Tests 3.4 Some Special Series 3.5 Operations on Series 4: Basic Topology 4.1 Open and Closed Sets 4.2 Further Properties of Open and Closed Sets 4.3 Compact Sets 4.4 The Cantor Set 4.5 Connected and Disconnected Sets 4.6 Perfect Sets 5: Limits and Continuity of Functions 5.1 Basic Properties of the Limit of a Function 5.2 Continuous Functions 5.3 Topological Properties and Continuity 5.4 Classifying Discontinuities and Monotonicity 6: Differentiation of Functions 6.1 The Concept of Derivative 6.2 The Mean Value Theorem and Applications 6.3 More on the Theory of Differentiation 7: The Integral 7.1 Partitions and the Concept of Integral 7.2 Properties of the Riemann Integral 7.3 Change of Variable and Related Ideas 7.4 Another Look at the Integral 7.5 Advanced Results on Integration Theory 8: Sequences and Series of Functions 8.1 Partial Sums and Pointwise Convergence 8.2 More on Uniform Convergence 8.3 Series of Functions 8.4 The Weierstrass Approximation Theorem 9: Elementary Transcendental Functions 9.1 Power Series 9.2 More on Power Series: Convergence Issues 9.3 The Exponential and Trigonometric Functions 9.4 Logarithms and Powers of Real Numbers 10: Differential Equations 10.1 Picard’s Existence and Uniqueness Theorem 10.1.1 The Form of a Differential Equation 10.1.2 Picard’s Iteration Technique 10.1.3 Some Illustrative Examples 10.1.4 Estimation of the Picard Iterates 10.2 Power Series Methods 11: Introduction to Harmonic Analysis 11.1 The Idea of Harmonic Analysis 11.2 The Elements of Fourier Series 11.3 An Introduction to the Fourier Transform 11.3 Appendix: Approximation by Smooth Functions 11.4 Fourier Methods and Differential Equations 11.4.1 Remarks on Different Fourier Notations 11.4.2 The Dirichlet Problem on the Disc 11.4.3 Introduction to the Heat and Wave Equations 11.4.4 Boundary Value Problems 11.4.5 Derivation of the Wave Equation 11.4.6 Solution of the Wave Equation 11.5 The Heat Equation 12: Functions of Several Variables 12.1 A New Look at the Basic Concepts of Analysis 12.2 Properties of the Derivative 12.3 The Inverse and Implicit Function Theorems Appendix I: Elementary Number Systems Appendix II: Logic and Set Theory Appendix III: Review of Linear Algebra Table of Notation Glossary Bibliography Index

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