Complex Analysis: In the Spirit of Lipman Bers (Graduate Texts in Mathematics Book 245)

This book is intended for a graduate course in complex analysis, where the main focus is the theory of complex-valued functions of a single complex variable. This theory is a prerequisite for the study of many areas of mathematics, including the theory of several finitely and infinitely many complex variables, hyperbolic geometry, two- and three-manifolds, and number theory. Complex analysis has connections and applications to many other subjects in mathematics and to other sciences. Thus this material will also be of interest to computer scientists, physicists, and engineers.The book covers most, if not all, of the material contained in Lipman 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.In addition to many new exercises, this second edition introduces a variety of new and interesting topics. New features include a section on 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 with prescribed zeros; sections on subharmonic functions and Perron's principle; and a section on the ring of holomorphic functions on a plane domain. There are three new appendices: the first is a contribution by Ranjan Roy on the history of complex analysis, the second contains background material on exterior differential calculus, and the third appendix includes an alternate approach to the Cauchy theory. Read more... 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 and Hyperbolic Geometry -- Harmonic Functions -- Zeros of Holomorphic Functions This book is intended for a graduate course in complex analysis, where the main focus is the theory of complex-valued functions of a single complex variable. This theory is a prerequisite for the study of many areas of mathematics, including the theory of several finitely and infinitely many complex variables, hyperbolic geometry, two- and three-manifolds, and number theory. Complex analysis has connections and applications to many other subjects in mathematics and to other sciences. Thus this material will also be of interest to computer scientists, physicists, and engineers. The book covers most, if not all, of the material contained in Lipman 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. In addition to many new exercises, this second edition introduces a variety of new and interesting topics. New features include a section on 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 with prescribed zeros; sections on subharmonic functions and Perron's principle; and a section on the ring of holomorphic functions on a plane domain. There are three new appendices: the first is a contribution by Ranjan Roy on the history of complex analysis, the second contains background material on exterior differential calculus, and the third appendix includes an alternate approach to the Cauchy theory. This Book Is Intended For A Graduate Course On Complex Analysis, Also Known As Function Theory. The Main Focus Is The Theory Of Complex-valued Functions Of A Single Complex Variable. This Theory Is A Prerequisite For The Study Of Many Current And Rapidly Developing Areas Of Mathematics Including The Theory Of Several And Infinitely Many Complex Variables, The Theory Of Groups, Hyperbolic Geometry And Three-manifolds, And Number Theory. Complex Analysis Has Connections And Applications To Many Other Subjects In Mathematics And To Other Sciences. It Is An Area Where The Classic And The Modern Techniques Meet And Benefit From Each Other. This Material Should Be Part Of The Education Of Every Practicing Mathematician, And It Will Also Be Of Interest To Computer Scientists, Physicists, And Engineers. 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.--jacket. 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. Jane P. Gilman, Irwin Kra, Rubí E. Rodriguez. Includes Bibliographical References (p. 215-216) And Index. This bookpresents fundamental material that should be part of the education of every practicing mathematician. This material will also be of interest to computer scientists, physicists, and engineers. Complex analysis is also known as function theory. In this text we address the theory of complex-valued functions of a single complex variable. This is a prerequisite for the study of many current and rapidlydevelopingareasofmathematics, includingthetheoryofseveral andin?nitely many complex variables, thetheoryofgroups, hyperbolic geometry and three-manifolds, and number theory. Complex analysis has connections and applications to many other many other subjects in mathematics, and also to other sciences as an area where the classic and the modern techniques meet and bene?t from each other. We will try to illustrate this in the applications we give. Because function theory has been used by generations of practicing mathematicians working in a number of di?erent?elds, the basic - sults have been developed and redeveloped from a number of di?erent perspectives. We are not wedded to any one viewpoint. Rather we will try to exploit the richness of the subject and explain and interpret standard de?nitions and results using the most convenient tools from analysis, geometry, and algebra. 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. Front Matter....Pages i-xviii The Fundamental Theorem in Complex Function Theory....Pages 1-14 Foundations....Pages 15-38 Power Series....Pages 39-80 The Cauchy Theory: A Fundamental Theorem....Pages 81-117 The Cauchy Theory: Key Consequences....Pages 119-137 Cauchy Theory: Local Behavior and Singularities of Holomorphic Functions....Pages 139-169 Sequences and Series of Holomorphic Functions....Pages 171-197 Conformal Equivalence and Hyperbolic Geometry....Pages 199-228 Harmonic Functions....Pages 229-265 Zeros of Holomorphic Functions....Pages 267-295 Back Matter....Pages 297-306

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.

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