Problems and Solutions for Undergraduate Complex Analysis I

The aim of **Problems and Solutions for Undergraduate Complex Analysis I**, as the name reveals, is to assist undergraduate students in studying essential theories and techniques of complex analysis. The wide variety of problems, which are of varying difficulty, includes the following topics:* Complex Numbers * Geometry of Complex Numbers * __n__th Roots of a Complex Number * Complex Functions and the Analyticity * Power Series * Elementary Theory of Complex Integration * Properties of Analytic and Entire Functions * Further Properties of Analytic Functions * Isolated Singularities of Analytic Functions Furthermore, the main features of this book are listed as follows:1. The book contains 226 problems which cover the topics mentioned above. The solutions are __detailed__ and __complete__ in the sense that every step and every theorem that I applied will be presented. 2. Each chapter starts with a brief and concise note of introducing the notations, terminologies, basic mathematical concepts or important/famous/frequently used theorems (without proofs) relevant to the topic. 3. Three levels of difficulty have been assigned to problems so that you can sharpen your mathematics step-by-step. 4. Different colors are used in appropriate places so as to highlight or explain problems, examples, remarks, main points/formulas involved, or show the steps of manipulation in some complicated proofs. (ebook only) 5. Different authors may have different setup for the theory of complex integration. To reduce confusions and save your time of cross-checking, an appendix is made for recording some terminologies of some common complex analysis textbooks, such as Ahlfors, Asmar and Grafakos, Bak and Newmann, Conway, Gamelin, Rudin as well as Stein and Shakarchi. The aim of Problems and Solutions for Undergraduate Complex Analysis I , as the name reveals, is to assist undergraduate students in studying essential theories and techniques of complex analysis. The wide variety of problems, which are of varying difficulty, includes the following topics: Complex Numbers Geometry of Complex Numbers n th Roots of a Complex Number Complex Functions and the Analyticity Power Series Elementary Theory of Complex Integration Properties of Analytic and Entire Functions Further Properties of Analytic Functions Isolated Singularities of Analytic Functions Furthermore, the main features of this book are listed as follows: The book contains 226 problems which cover the topics mentioned above. The solutions are detailed and complete in the sense that every step and every theorem that I applied will be presented. Each chapter starts with a brief and concise note of introducing the notations, terminologies, basic mathematical concepts or important/famous/frequently used theorems (without proofs) relevant to the topic. Three levels of difficulty have been assigned to problems so that you can sharpen your mathematics step-by-step. Different colors are used in appropriate places so as to highlight or explain problems, examples, remarks, main points/formulas involved, or show the steps of manipulation in some complicated proofs. (ebook only) Different authors may have different setup for the theory of complex integration. To reduce confusions and save your time of cross-checking, an appendix is made for recording some terminologies of some common complex analysis textbooks, such as Ahlfors, Asmar and Grafakos, Bak and Newmann, Conway, Gamelin, Rudin as well as Stein and Shakarchi. About the author Preface List of Figures Contens Complex Numbers Geometry of Complex Numbers nth Roots of a Complex Number Complex Functions and the Analyticity Power Series Elementary Theory of Complex Integration Properties of Analytic and Entire Functions Further Properties of Analytic Functions Isolated Singularities of Analytic Functions Appendix Terminologies Index Bibliography