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Graduate Studies in Mathematics
2008; 489 pp; hardcover
List Price: US$81
Member Price: US$64.80
Order Code: GSM/97
Function Theory of One Complex Variable: Third Edition - Robert E Greene and Steven G Krantz
Complex Variables - Joseph L Taylor
An Introduction to Complex Analysis and Geometry - John P D'Angelo
Perhaps uniquely among mathematical topics, complex analysis presents the student with the opportunity to learn a thoroughly developed subject that is rich in both theory and applications. Even in an introductory course, the theorems and techniques can have elegant formulations. But for any of these profound results, the student is often left asking: What does it really mean? Where does it come from?
In Complex Made Simple, David Ullrich shows the student how to think like an analyst. In many cases, results are discovered or derived, with an explanation of how the students might have found the theorem on their own. Ullrich explains why a proof works. He will also, sometimes, explain why a tempting idea does not work.
Complex Made Simple looks at the Dirichlet problem for harmonic functions twice: once using the Poisson integral for the unit disk and again in an informal section on Brownian motion, where the reader can understand intuitively how the Dirichlet problem works for general domains. Ullrich also takes considerable care to discuss the modular group, modular function, and covering maps, which become important ingredients in his modern treatment of the often-overlooked original proof of the Big Picard Theorem.
This book is suitable for a first-year course in complex analysis. The exposition is aimed directly at the students, with plenty of details included. The prerequisite is a good course in advanced calculus or undergraduate analysis.
Graduate students interested in complex analysis.
"This is an excellent book for a first-year graduate student doing a course in complex analysis. ...students will enjoy and profit from Ullrichs careful explanation of why the theorems work the way they do and also sometimes why seemingly nice ideas that promised to work do not (but often can be patched so that they do). ... In short, Ullrich has managed to write a book about a classical subject that is unusual because its exposition is aimed directly at students, not instructors. I strongly recommend this book to everyone."
-- MAA Reviews
"In general, the entire exposition stands out by its particular didactic features, by its expository mastery, and by its lucid style helping students grasp both the matter and the beauty of complex function theory profoundly. The prerequisites are kept to minimum, or recalled in the appendices, whereas the scope of the book is remarkably wide. Altogether, the current book offers a nearly irresistible invitation to the fascinating subject of complex analysis."
-- Zentralblatt MATH
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