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 -- 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 -- Algebraic Topics -- Automorphism Groups of Domains in the Plane -- Cousin Problems, Cohomology, and Sheaves.

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.