A Guide to Complex Variables (Steven G. Krantz 2007).pdf

高清PDF文件 Preface Most every mathematics Ph.D. student must take a qualifying exam in com- plex variables. The task is a bit daunting. This is one of the oldest areas in mathematics, it is beautiful and compelling, and there is a plethora of material. The literature in complex variables is vast and diverse. There are a great many textbooks in the subject, but each has a different point of view and places different emphases according to the tastes of the author. Thus it is a bit difficult for the student to focus on what are the essential parts of this subject. What must one absolutely know for the qualifying exam? What will be asked? What techniques will be stressed? What are the key facts? The purpose of this book is to answer these questions. This is definitely not a comprehensive textbook like [GRK]. It is rather an entree to the disci- pline. It will tell you the key ideas in a first-semester graduate course in the subject, map out the important theorems, and indicate most of the proofs. Here by “indicate” we mean that (i) if the proof is short then we include it, (ii) if the proof is of medium length then we outline it, and bf (iii) if the proof is long then we sketch it. This book has plenty of figures, plenty of examples, copious commentary, and even in-text exercises for the students. But, since it is not a formal textbook, it does not have exercise sets. It does not have a Glossary or a Table of Notation. This is meant to be a breezy book that you could read at one or two sittings, just to get the sense of what this subject is about and how it fits together. In that wise it is quite different from a typical mathematics text or monograph. After reading this book (or even while reading this book), you will want to pick up a more traditional and comprehensive tome and work your way through it. The present book will get you started on your journey.