Geometric Function Theory: Explorations in Complex Analysis / Edition 1

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Complex variables is a precise, elegant, and captivating subject. Presented from the point of view of modern work in the field, this new book addresses advanced topics in complex analysis that verge on current areas of research, including invariant geometry, the Bergman metric, the automorphism groups of domains, harmonic measure, boundary regularity of conformal maps, the Poisson kernel, the Hilbert transform, the boundary behavior of harmonic and holomorphic functions, the inhomogeneous Cauchy–Riemann equations, and the corona problem.

The author adroitly weaves these varied topics to reveal a number of delightful interactions. Perhaps more importantly, the topics are presented with an understanding and explanation of their interrelations with other important parts of mathematics: harmonic analysis, differential geometry, partial differential equations, potential theory, abstract algebra, and invariant theory. Although the book examines complex analysis from many different points of view, it uses geometric analysis as its unifying theme.

This methodically designed book contains a rich collection of exercises, examples, and illustrations within each individual chapter, concluding with an extensive bibliography of monographs, research papers, and a thorough index. Seeking to capture the imagination of advanced undergraduate and graduate students with a basic background in complex analysis—and also to spark the interest of seasoned workers in the field—the book imparts a solid education both in complex analysis and in how modern mathematics works.

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Editorial Reviews

From the Publisher
"The geometric point of view is the unifying theme in this fine textbook in complex function theory. But the author also studies byways that come from analysis and algebra.... Altogether, the author treats advanced topics that lead the reader to modern areas of research. And what is important, the topics are presented with an explanation of their interaction with other important parts of mathematics. The presentations of the topics are clear and the text makes [for] very good reading; basic ideas of many concepts and proofs are carefully described, non-formal introductions to each chapter are very helpful, a rich collection of exercises is well composed and helps the student to understand the subject. The book under review leads the student to see what complex function theory has to offer and thereby gives him or her a taste of some of the areas of current research. As such it is a welcome addition to the existing literature in complex function theory.... In this reviewer's opinion, the book can warmly be recommended both to experts and to a new generation of mathematicians." —Zentralblatt MATH

"This book provides a very good and deep point of view of modern and advanced topics in complex analysis. … Each chapter contains a rich collection of exercises of different level, examples and illustrations. … The book is very clearly written, with rigorous proofs, in a pleasant and accessible style. It is warmly recommended to advanced undergraduate and graduate students with a basic background in complex analysis, as well as to all researchers that are interested in modern and advanced topics in complex analysis." —Studia Universitatis Babes-Bolyai Mathematica

"This book is an exploration in Complex Analysis as a synthesis of many different areas; the prejudice in the subject is geometric, but the reader may [need] basic information from analysis…, partial differential equations, algebra, and other parts of mathematics. This synthesis is addressed to the students; it gives them the possibility of writing their thesis on the subject and introduces them to some research problems…This captivat[ing] book also contains a collection of exercises, examples and illustrations, as well as an extensive bibliography and a thorough index." —Analele Stiintifice ale Universitatii “Al. I. Cuza” din Iasi

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Product Details

  • ISBN-13: 9780817643393
  • Publisher: Birkhauser Verlag
  • Publication date: 9/20/2005
  • Series: Cornerstones Series
  • Edition description: 2006
  • Edition number: 1
  • Pages: 314
  • Product dimensions: 9.21 (w) x 6.14 (h) x 0.75 (d)

Table of Contents

* Preface Part I: Classical Function Theory
• Invariant Geometry
• Variations on the Theme of the Schwarz Lemma
• Normal Families
• The Riemann Mapping Theorem and its Generalizations
• Boundary Regularity of Conformal Maps
• The Boundary Behavior of Holomorphic Functions Part II: Real and Harmonic Analysis
• The Cauchy–Riemann Equations
• The Green's Function and the Poisson Kernel
• Harmonic Measure
• Conjugate Functions and the Hilbert Transform
• Wolff's Proof of the Corona Theorem Part III: Algebraic Topics
• Automorphism Groups of Domains in the Plane
• Cousin Problems, Cohomology, and Sheaves
• Bibliography
• Index

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