Real analysis and foundations. (2022)

• Record Type: Book
• Further Information: Note: Steven G. Krantz.
• Authors: Krantz, Steven G (Steven George), 1951-
• Contents: Preface 0 Background Material; 0.1 Number Systems; 0.1.1 The Natural Numbers; 0.1.2 The Integers; 0.1.3 The Rational Numbers 02 Logic and Set; 0.2.1 And” and “Or”; 0.2.2 “not” and “if then”; 0.2.3 Contrapositive, Converse, and “Iff”; 0.2.4 Quantifiers; 0.2.5 Set Theory and Venn Diagrams; 0.2.6 Relations and Functions; 0.2.7 Countable and Uncountable Sets 1 Real and Complex Numbers; 1.1 The Real Numbers; Appendix: Construction of the Real Numbers; 1.2 The Complex Numbers; 2 Sequences 71; 2.1 Convergence of Sequences; 2.2 Subsequences; 2.3 Limsup and Liminf; 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 Functions of Several Variables; 10.1 A New Look at the Basic Concepts of Analysis; 10.2 Properties of the Derivative; 10.3 The Inverse and Implicit Function Theorems 11 Advanced Topics; 11.1 Metric Spaces; 11.2 Topology in a Metric Space; 11.3 The Baire Category Theorem; 11.4 The Ascoli-Arzela Theorem 12 Differential Equations; 12.1 Picard’s Existence and Uniqueness Theorem; 12.1.1 The Form of a Differential Equation; 12.1.2 Picard’s Iteration Technique; 12.1.3 Some Illustrative Examples; 12.1.4 Estimation of the Picard Iterates; 12.2 Power Series Methods 13 Introduction to Harmonic Analysis; 13.1 The Idea of Harmonic Analysis; 13.2 The Elements of Fourier Series; 13.3 An Introduction to the Fourier Transform; Appendix: Approximation by Smooth Functions; 13.4 Fourier Methods and Differential Equations; 13.4.1 Remarks on Different Fourier Notations; 13.4.2 The Dirichlet Problem on the Disc; 13.4.3 Introduction to the Heat and Wave Equations; 13.4.4 Boundary Value Problems; 13.4.5 Derivation of the Wave Equation; 13.4.6 Solution of the Wave Equation; 13.5 The Heat Equation Appendix: Review of Linear Algebra; Table of Notation; Glossary; Bibliography; Index;
• Edition: Fifth edition
• Publisher Details: Boca Raton : CRC Press
• Publication Date: 2022
• Extent: 1 online resource (491 pages), illustrations (black and white)
• Subjects: 515.8; Functions of real variables; Mathematical analysis
• Languages: English
• ISBNs: 9781000593242; 9781000593228; 9781003222682
• Related ISBNs: 9781032102726
• Notes: Note: Includes bibliographical references and index.; Note: Description based on online resource; title from PDF title page (viewed on June 10, 2022).
• Access Rights: Legal Deposit; Only available on premises controlled by the deposit library and to one user at any one time; The Legal Deposit Libraries (Non-Print Works) Regulations (UK).
• Access Usage: Restricted: Printing from this resource is governed by The Legal Deposit Libraries (Non-Print Works) Regulations (UK) and UK copyright law currently in force.