Titel: Complex Analysis: In the Spirit of Lipman Bers
Autor/en: Irwin Kra, Rubi E. Rodriguez, Jane P. Gilman
Dezember 2007 - gebunden - 235 Seiten
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
From the reviews:"This is a fairly conventional text for a first course in complex analysis. It is an interesting mix of the concrete and the abstract, and of the formulaic and the geometric. It has good exercises ... . It is nominally a graduate text (it is in Springer's series of Graduate Texts in Mathematics) ... . The book covers all the usual topics for a first course and includes a lot of advanced topics ... ." (Allen Stenger, MathDL, February, 2008)"This book is based on the original courses of Complex Analysis that was delivered by the well-known American specialist and lecturer in Analysis, Professor Lipman Bers. ... the book is written in a clear and easily readable manner. ... this book is useful for all specialists and lecturers in Complex Analysis ... and also all specialists who deal with applications of Complex Analysis. I also recommend this book to beginners who want to study Complex Analysis ... ." (Peter Zabreiko, Zentralblatt MATH, Vol. 1139 (17), 2008)"An introduction to complex analysis based on original lectures by Lipman Bers ... . the whole text is clearly suitable as a first course in complex analysis. ... In addition, it contains many exercises and a brief guide to the literature." (G. Teschl, Monatshefte für Mathematik, Vol. 156 (4), April, 2009)