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Complex Analysis [electronic resource] : In the Spirit of Lipman Bers / by Jane P. Gilman, Irwin Kra, Rubí E. Rodríguez.

By: Gilman, Jane P [author.].
Contributor(s): Kra, Irwin [author.] | Rodríguez, Rubí E [author.] | SpringerLink (Online service).
Material type: TextTextSeries: Graduate Texts in Mathematics: 245Publisher: New York, NY : Springer New York, 2007Description: XIV, 220 p. 20 illus. online resource.Content type: text Media type: computer Carrier type: online resourceISBN: 9780387747156.Subject(s): Functions of complex variables | Differential equations, partial | Functions of a Complex Variable | Several Complex Variables and Analytic SpacesAdditional physical formats: Printed edition:: No title; Printed edition:: No title; Printed edition:: No titleDDC classification: 515.9 Online resources: Click here to access online
Contents:
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.
In: Springer eBooksSummary: 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 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 is termed the Fundamental Theorem for functions of a complex variable. The organization of these conditions into a single unifying theorem with an emphasis on clarity and elegance is a hallmark of Lipman Bers's mathematical style. 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.
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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.

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 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 is termed the Fundamental Theorem for functions of a complex variable. The organization of these conditions into a single unifying theorem with an emphasis on clarity and elegance is a hallmark of Lipman Bers's mathematical style. 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.

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