Complex Made Simple (Graduate Studies in Mathematics) by David C. Ullrich (PDF)

Description

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.

User’s Reviews

Editorial Reviews: Review Altogether, the current book offers a nearly irresistible invitation to the fascinating subject of complex analysis. –Zentralblatt

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐I searched for a complex variables textbook that explained and proved the Schwarz Reflection Theorem for harmonic functions in a way that I could understand. I found that Ullrich accomplished this goal in only two pages (pp.215-216). This is an important result that allows you to calculate image forces/fields in electrostatics and dislocation theory.

⭐I have no fault with his book which is much betterthan any of his papers which are nearly unreadable.In point of fact his modular group/ modular form section is betterthan I have seen in many texts.I thought a lot before posting a review of this book.

⭐I give this book high marks as a text for a first course in complexanalysis. Ullrich’s presentation is both rigorous and well motivated,a combination that’s very hard to achieve.Motivation is where this book really shines. For example, in anumber of places, Ullrich pauses in the presentation to remark anddemonstrate how surprising the world of complex-differentiablefunctions is, when compared to the world of real-differentiablefunctions. In other cases, e.g. in his derivation of Euler’s formulafor sin(z), Ullrich shows how an intuitive idea can lead, throughseveral false starts, to a true result. This is an invaluableinsight, seldom found in textbooks.The reader should be comfortable with concepts from “AdvancedCalculus” (the real numbers, continuity, convergence, integration,and so on). Ullrich constructs everything else that’s needed. Whenthe formalism gets heavier, e.g. when he’s covering analyticcontinuation and the complete analytic function, he introduces itcarefully, explains why it’s used, and advises the reader on howto make sense of it. This makes the book valuable for anotherreason — it works as a fairly gentle introduction to modern,abstract mathematical techniques.The book also seems well suited to self-study: it’s self-containedand very clear.

⭐It is really a very standard book on Complex Analysis and is written in keeping a view of modern complex analysis

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