This book organizes the basic material of complex analysis in a unique manner.The first part of the book is a study of the many equivalent ways of understanding the concept of analyticity. The many ways of formulating the concept of an analytic function are summarized in what we term the Fundamental Theorem for functions of a complex variable. The organization of these conditions into a single unifying theorem is a hallmark of Bers's mathematical style with an emphasis on clarity and elegance. Here it provides a conceptual framework for results that are highly technical and often computational. The framework comes from an insight that, once articulated, will drive the subsequent mathematics and lead to new results.In the second part, the text proceeds to a leisurely exploration of interesting ramifications of the main concepts.The book covers most, if not all, of the material contained in Bers s courses on first year complex analysis. In addition, topics of current interest such as zeros of holomorphic functions and the connection between hyperbolic geometry and complex analysis are explored.The organization of material in this book allows for an elegant and economical treatment of many important topics.
From the contents.
Preface.- Standard notation and commonly used symbols.- The fundamental theorem in complex function theory.- Foundations.- Power series.- The cauchy theory-a fundamental theorem.- The cauchy theory-key consequences.- Cauchy theory: local behavior and singularities of holomorphic functions.- Sequences and series of holomorphic functions.- Conformal equivalence.- Harmonic functions.- Zeros of holomorphic functions.- Bibliographical notes.- Bibliography.- Index
The Fundamental Theorem in Complex Function Theory.- Foundations.- Power Series.- The Cauchy Theory-A Fundamental Theorem.- The Cauchy Theory-Key Consequences.- Cauchy Theory: Local Behavior and Singularities of Holomorphic Functions.- Sequences and Series of Holomorphic Functions.- Conformal Equivalence.- Harmonic Functions.- Zeros of Holomorphic Functions.
The authors' aim here is to present a precise and concise treatment of those parts of complex analysis that should be familiar to every research mathematician. They follow a path in the tradition of Ahlfors and Bers by dedicating the book to a very precise goal: the statement and proof of the Fundamental Theorem for functions of one complex variable. They discuss the many equivalent ways of understanding the concept of analyticity, and offer a leisure exploration of interesting consequences and applications. Readers should have had undergraduate courses in advanced calculus, linear algebra, and some abstract algebra. No background in complex analysis is required.