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. 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.

Presented from a geometric analytical viewpoint, this work addresses advanced topics in complex analysis that verge on modern areas of research Contains an extensive bibliography of both monographs and research papers and a thorough index Methodically designed with individual chapters containing a rich collection of exercises, examples, and illustrations Seeks to capture the imagination of advanced undergraduate and graduate students with a basic background in complex analysis

Autorentext

Peter V. Dovbush is an Associate Professor and Leading Researcher at the Institute of Mathematics and Computer Science of Moldova State University, Moldova. His research interests lie on the geometric theory of functions of several complex variables. Steven G. Krantz earned his B.A. from the University of California at Santa Cruz (1971) and his PhD from Princeton University, USA (1974). With teaching periods at UCLA, Princeton, Penn State, and Washington University in St. Louis, he chaired the latter's mathematics department for five years. Dr. Krantz has authored or co-authored over 160 books and 350 scholarly papers, and he has edited numerous journals. His contributions to mathematics have earned him awards such as the Chauvenet Prize, the Beckenbach Book Award, and the Kemper Prize.

Inhalt
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.