Complex variables is a precise, elegant, and captivating subject. Presented from a geometric analytical viewpoint, this work addresses advanced topics in complex analysis that verge on modern areas of research, with an understanding and explanation of their interrelations with other important parts of mathematics. 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.

* Presented from a geometric analytical viewpoint, this work addresses advanced topics in complex analysis that verge on modern areas of research

* Methodically designed with individual chapters containing a rich collection of exercises, examples, and illustrations

* 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