Ebook: Geometric Function Theory: Explorations in Complex Analysis

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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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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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.

Content:
Front Matter....Pages i-xiii
Front Matter....Pages 1-3
Invariant Geometry....Pages 5-27
Variations on the Theme of the Schwarz Lemma....Pages 29-63
Normal Families....Pages 65-81
The Riemann Mapping Theorem and Its Generalizations....Pages 83-107
Boundary Regularity of Conformal Maps....Pages 109-133
The Boundary Behavior of Holomorphic Functions....Pages 135-154
Front Matter....Pages 155-157
The Cauchy-Riemann Equations....Pages 159-167
The Green’s Function and the Poisson Kernel....Pages 169-183
Harmonic Measure....Pages 185-204
Conjugate Functions and the Hilbert Transform....Pages 205-223
Wolff’s Proof of the Corona Theorem....Pages 225-251
Front Matter....Pages 253-255
Automorphism Groups of Domains in the Plane....Pages 257-279
Cousin Problems, Cohomology, and Sheaves....Pages 281-302
Back Matter....Pages 303-314

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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.

Content:
Front Matter....Pages i-xiii
Front Matter....Pages 1-3
Invariant Geometry....Pages 5-27
Variations on the Theme of the Schwarz Lemma....Pages 29-63
Normal Families....Pages 65-81
The Riemann Mapping Theorem and Its Generalizations....Pages 83-107
Boundary Regularity of Conformal Maps....Pages 109-133
The Boundary Behavior of Holomorphic Functions....Pages 135-154
Front Matter....Pages 155-157
The Cauchy-Riemann Equations....Pages 159-167
The Green’s Function and the Poisson Kernel....Pages 169-183
Harmonic Measure....Pages 185-204
Conjugate Functions and the Hilbert Transform....Pages 205-223
Wolff’s Proof of the Corona Theorem....Pages 225-251
Front Matter....Pages 253-255
Automorphism Groups of Domains in the Plane....Pages 257-279
Cousin Problems, Cohomology, and Sheaves....Pages 281-302
Back Matter....Pages 303-314
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