04602nam 22007575 450 991079948530332120240313210643.03-031-30832-810.1007/978-3-031-30832-1(MiAaPQ)EBC31054873(Au-PeEL)EBL31054873(DE-He213)978-3-031-30832-1(EXLCZ)992951617360004120240104d2023 u| 0engurcnu||||||||txtrdacontentcrdamediacrrdacarrierThe Big Book of Real Analysis[electronic resource] From Numbers to Measures /by Syafiq Johar1st ed. 2023.Cham :Springer Nature Switzerland :Imprint: Springer,2023.1 online resource (950 pages)Print version: Johar, Syafiq The Big Book of Real Analysis Cham : Springer,c2024 9783031308314 Preface -- 1. Logic and Sets -- 2. Integers -- 3. Construction of the Real Numbers -- 4. The Real Numbers -- 5. Real Sequences -- 6. Some Applications of Real Sequences -- 7. Real Series -- 8. Additional Topics in Real Series -- 9. Functions and Limits -- 10. Continuity -- 11. Function Sequences and Series -- 12. Power Series -- 13. Differentiation -- 14. Some Applications of Differentiation -- 15. Riemann and Darboux Integration -- 16. The Fundamental Theorem of Calculus -- 17. Taylor and MacLaurin Series -- 18. Introduction to Measure Theory -- 19. Lebesgue Integration -- 20. Double Integrals -- Solutions to the Exercises -- Bibliography -- Index.This book provides an introduction to real analysis, a fundamental topic that is an essential requirement in the study of mathematics. It deals with the concepts of infinity and limits, which are the cornerstones in the development of calculus. Beginning with some basic proof techniques and the notions of sets and functions, the book rigorously constructs the real numbers and their related structures from the natural numbers. During this construction, the readers will encounter the notions of infinity, limits, real sequences, and real series. These concepts are then formalised and focused on as stand-alone objects. Finally, they are expanded to limits, sequences, and series of more general objects such as real-valued functions. Once the fundamental tools of the trade have been established, the readers are led into the classical study of calculus (continuity, differentiation, and Riemann integration) from first principles. The book concludes with an introduction to the study of measures and how one can construct the Lebesgue integral as an extension of the Riemann integral. This textbook is aimed at undergraduate students in mathematics. As its title suggests, it covers a large amount of material, which can be taught in around three semesters. Many remarks and examples help to motivate and provide intuition for the abstract theoretical concepts discussed. In addition, more than 600 exercises are included in the book, some of which will lead the readers to more advanced topics and could be suitable for independent study projects. Since the book is fully self-contained, it is also ideal for self-study.MathematicsMathematical analysisSequences (Mathematics)Differential equationsMeasure theoryFunctions of real variablesCàlculthubAnàlisi matemàticathubSuccessions (Matemàtica)thubMathematicsAnalysisSequences, Series, SummabilityDifferential EquationsMeasure and IntegrationReal FunctionsLlibres electrònicsthubMathematics.Mathematical analysis.Sequences (Mathematics).Differential equations.Measure theory.Functions of real variables.CàlculAnàlisi matemàticaSuccessions (Matemàtica)Mathematics.Analysis.Sequences, Series, Summability.Differential Equations.Measure and Integration.Real Functions.510Johar Syafiq1586365MiAaPQMiAaPQMiAaPQBOOK9910799485303321The Big Book of Real Analysis3872753UNINA