Mathematical Analysis: A Concise Introduction
- Length: 263 pages
- Edition: 1
- Language: English
- Publisher: World Scientific Pub Co Inc
- Publication Date: 2020-12-17
- ISBN-10: 9811221634
- ISBN-13: 9789811221637
- Sales Rank: #1400303 (See Top 100 Books)
Mathematical analysis serves as a common foundation for many research areas of pure and applied mathematics. It is also an important and powerful tool used in many other fields of science, including physics, chemistry, biology, engineering, finance, and economics. In this book, some basic theories of analysis are presented, including metric spaces and their properties, limit of sequences, continuous function, differentiation, Riemann integral, uniform convergence, and series.
After going through a sequence of courses on basic calculus and linear algebra, it is desirable for one to spend a reasonable length of time (ideally, say, one semester) to build an advanced base of analysis sufficient for getting into various research fields other than analysis itself, and/or stepping into more advanced levels of analysis courses (such as real analysis, complex analysis, differential equations, functional analysis, stochastic analysis, amongst others). This book is written to meet such a demand. Readers will find the treatment of the material is as concise as possible, but still maintaining all the necessary details.
Preface Contents Chapter 1 - Metric Spaces and Limits for Sequences 1.1 Metric Spaces 1.2 Sequences and Limits 1.3 Sets in Metric Spaces 1.4 Properties of Metric Spaces Chapter 2 - Functions on Metric Spaces 2.1 Continuity 2.2 Properties of Continuous Functions Chapter 3 - Differentiaion 3.1 Derivatives 3.2 Fréchet Differentiability 3.3 Inverse Function and Implicit Function Theorems 3.4 Higher Order Derivatives Chapter 4 - Riemann Integrals 4.1 Definition of Integrals 4.2 Properties of Integrals 4.3 Further Theorems Chapter 5 - Uniform Convergence 5.1 Observations and Examples 5.2 Uniform Convergence 5.3 The Metric of Uniform Convergence 5.4 Limit Theorems Chapter 6 - Series 6.1 Series of Numbers 6.2 Further Tests for Convergence 6.3 Series of Functions 6.4 More Convergence Criteria 6.5 Power Series 6.6 Fourier Series Appendix
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