Function Theory in Several Complex Variables

Function Theory in Several Complex Variables examines advanced topics in complex analysis beyond a single variable. The text covers foundational concepts like domains of holomorphy, pseudoconvexity, and the Levi problem. The progression moves from basic principles through to modern developments in the field. Specific focus areas include complex manifolds, the ∂-equation, integral formulas, and applications to partial differential equations. Topics build systematically with rigorous proofs and exercises throughout the chapters. The material requires prior knowledge of basic complex analysis and some real analysis. This work serves as both a graduate textbook and reference for researchers, presenting key theoretical frameworks that connect complex analysis to other areas of mathematics. The treatment emphasizes geometric intuition while maintaining mathematical precision.

The book serves as a graduate-level text for students with prior exposure to complex analysis. Readers appreciate: - Clear explanations of advanced concepts - Progression from basic principles to research-level topics - Inclusion of exercises with varying difficulty levels - Thorough treatment of pluripotential theory Common criticisms: - Dense notation that can be hard to follow - Some proofs lack detail - Prerequisites not clearly stated - Limited discussion of applications Reviews across platforms: Amazon: 4.5/5 (7 reviews) - "Detailed but requires strong mathematical maturity" - "Good for self-study if you have the background" Goodreads: 4.0/5 (3 ratings, 0 written reviews) Mathematics Stack Exchange users recommend it for students who have completed a first course in complex analysis and want to study several complex variables, though they suggest supplementing with Hörmander's book for more complete proofs.

Several Complex Variables by Lars Hörmander This text develops the foundations of complex analysis in multiple variables with emphasis on partial differential equations and advanced analytic methods.

Complex Analysis in Several Variables by Volker Scheidemann The book connects complex analysis fundamentals to modern developments in complex geometry and sheaf theory.

Introduction to Complex Analysis in Several Variables by L.V. Ahlfors This work builds from basic complex analysis to advanced topics through a systematic treatment of plurisubharmonic functions and domains of holomorphy.

Complex Analysis of Several Variables by Anatole Katok and Hasselblatt The text presents complex analysis through geometric methods and includes applications to dynamical systems.

Complex Variables and Applications in Several Variables by Raghavan Narasimhan This book provides connections between complex analysis, algebraic geometry, and partial differential equations through concrete examples and applications.

🔹 Steven G. Krantz has authored more than 320 research papers and 130 books, making him one of the most prolific mathematics authors alive today. 🔹 The study of several complex variables, which this book covers, was revolutionized in the 1940s by Japanese mathematicians, particularly Kiyoshi Oka, who developed fundamental techniques still used today. 🔹 Unlike functions of a single complex variable, functions of several complex variables can't have isolated singularities - a property that leads to dramatically different behavior and makes the field especially challenging. 🔹 The book was first published in 1982 and has become a standard reference in graduate-level complex analysis courses, particularly for its comprehensive treatment of the Levi problem. 🔹 Krantz received the Chauvenet Prize (1992) and the Beckenbach Book Prize (1994) for his mathematical exposition, demonstrating his exceptional ability to explain complex mathematical concepts clearly.