Complex Analysis

Organizing the basic material of complex analysis in a unique manner, the authors of this versatile book aim is to present a precise and concise treatment of those parts of complex analysis that should be familiar to every research mathematician.

Here is a work that breaks with tradition and organizes the basic material of complex analysis in a unique manner. The authors' aim 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. The first part of the book is a study of the many equivalent ways of understanding the concept of analyticity, and move on to offer a leisurely exploration of interesting consequences and applications. 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. Readers should have had undergraduate courses in advanced calculus, linear algebra, and some abstract algebra.

New edition extensively revised and updated New topics include Bers's theorem on isomorphisms between rings of holomorphic functions on plane domains, necessary and sufficient conditions for the existence of a bounded analytic function on the disc, subharmonic functions, Perron's principle, the ring of holomorphic functions on a plane domain, solutions to the Dirichlet problem, Green's function and its relation to the Riemann Mapping Theorem Covers vast majority of the material needed for a beginning graduate level course on complex analysis Material may be of interest to mathematicians, computer scientists, physicists, and engineers Request lecturer material: sn.pub/lecturer-material

Autorentext
Rubí E. Rodríguez is a professor of mathematics at Pontificia Universidad Católica de Chile. Irwin Kra is an emeritus distinguished service professor of mathematics at SUNY, Stony Brook. Jane P. Gilman is a professor II of mathematics at Rutgers University.

Inhalt

Preface to Second Edition.- Preface to First Edition.- Standard Notation and Commonly Used Symbols.- 1 The Fundamental Theorem in Complex Function Theory.- 2 Foundations.- 3 Power Series.- 4 The Cauchy Theory - A Fundamental Theorem.- 5 The Cauchy Theory - Key Consequences.- 6 Cauchy Theory: Local Behavior and Singularities of Holomorphic Functions.- 7 Sequences and Series of Holomorphic Functions.- 8 Conformal Equivalence and Hyperbolic Geometry.- 9 Harmonic Functions.- 10 Zeros of Holomorphic Functions.- Bibliographical Notes.- Bibliography.- Index.