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Most every mathematics Ph.D. student must take a qualifying exam in complex variables. The task is a bit daunting. Complex variables is one of the oldest areas in mathematics, it is beautiful and compelling, and there is a plethora of material. Its literature 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 discipline. The book can also be considered as a refresher for those who have not seen this lovely subject for a while. 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. 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 (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.
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