Special Topics in Mathematical Analysis

Author: Gao Hong
Publisher:
Publish Date: 2003-08-01
Features: This textbook is written for the "Special Topics in Mathematical Analysis" course of the Mathematics and Applied Mathematics program at the Open University of China Central Radio and Television University. Based on the fact that students have already studied mathematical analysis in their associate degree and are currently teaching mathematics in secondary schools, this textbook selects secondary school mathematics as the research subject, using the knowledge of mathematical analysis as a research tool, and systematically studies secondary school mathematics from the perspective and methods of higher mathematics. Through the study of this course, students can not only review the mathematical analysis knowledge they have already learned but also deepen their understanding of secondary school mathematics. The book is divided into six chapters: Chapter 1 is on sets and relations, which serve as the foundation of the entire book. In Chapter 2, we introduce the extension of number systems, with a focus on the theory of real numbers. In Chapter 3, we review the analytical theory of functions, introduce transcendental theory, and introduce linear functions in a broad sense. Chapters 4 and 5 use axiomatic methods to study logarithmic functions, exponential functions, and trigonometric functions. Chapter 6 introduces the theory of convex functions and discusses more broadly the extreme value problems of functions, which can be read by students interested in this topic. During the writing process, we fully considered the self-study nature of the Open University of China's education and strived to make the textbook as easy to understand as possible for self-study. The written textbook adopts a "combined" format, integrating teaching content and supplementary content to facilitate student learning. Exercises marked with an asterisk are generally more challenging and are provided for interested students to attempt. At the end of the book, answers to computational exercises are provided, and hints are given for some proof exercises for reference.