Complex Analysis in Several Variables

Complex Analysis in Several Variables is a foundational mathematics text focused on holomorphic functions of multiple complex variables. The book covers key concepts including plurisubharmonic functions, domains of holomorphy, and pseudoconvex domains. The text progresses from basic principles through advanced topics like integral formulas, cohomology theory, and the ∂̄-Neumann problem. Each chapter builds systematically on previous material while introducing new theoretical frameworks and analytical techniques. Methods of complex analysis in multiple dimensions are presented with applications to partial differential equations and spectral theory. The book includes detailed proofs and exercises throughout to reinforce understanding of the concepts. This rigorous treatment serves as both an introduction to several complex variables and a reference work for researchers in complex analysis, differential geometry, and related fields. The material synthesizes fundamental developments in complex analysis from the mid-20th century while establishing connections to modern mathematical research.

Readers describe this as an advanced graduate-level text that requires significant mathematical maturity and prior knowledge of complex analysis in one variable. Likes: - Clear and precise presentation of proofs - Comprehensive coverage of analytic functions, plurisubharmonic functions, and cohomology theory - Mathematical rigor appreciated by experts and researchers - Useful as a reference for specific theorems and concepts Dislikes: - Too terse and abstract for self-study - Limited motivation and examples - Assumes familiarity with functional analysis concepts - Print quality issues noted in some editions One reader on Math Stack Exchange noted: "Hörmander writes for mathematicians who already understand the big picture and want the technical details filled in." Ratings: Goodreads: 4.33/5 (6 ratings) Amazon: 4.0/5 (2 ratings) Math Stack Exchange: Multiple positive mentions in discussions about complex analysis texts, though often with caveats about the advanced level

Several Complex Variables by Lars Hörmander and Bernard Malgrange This text builds upon introductory complex analysis to develop the theory of holomorphic functions in multiple variables with emphasis on functional analysis methods.

Functions of Several Complex Variables by B.A. Fuks The book presents fundamental concepts of complex manifolds, sheaf theory, and cohomology with applications to analytic continuation and domains of holomorphy.

Theory of Functions of Several Complex Variables by Salomon Bochner and William Ted Martin The text covers plurisubharmonic functions, pseudoconvex domains, and the solution of the Levi problem in complex analysis.

Complex Analysis in Several Variables by Volker Scheidemann The work provides a systematic treatment of power series methods, integral formulas, and the fundamental theorems of complex analysis in n-dimensional space.

Introduction to Complex Analysis in Several Variables by R. Michael Range The book develops the theory from basic principles through advanced topics including the edge-of-the-wedge theorem and applications to partial differential equations.

🔹 Lars Hörmander won the Fields Medal in 1962 for his work on partial differential operators, becoming the first Swedish mathematician to receive this prestigious award. 🔹 The book was first published in 1966 and became a foundational text in the field, bridging the gap between introductory complex analysis and advanced research topics. 🔹 Complex analysis in several variables, unlike its single-variable counterpart, requires sophisticated tools from algebra and topology, as the behavior of functions becomes dramatically more complicated in higher dimensions. 🔹 Hörmander wrote this book while at Stanford University, where he held a position after leaving his professorship at Stockholm University, and before returning to Sweden to take a position at Lund University. 🔹 The techniques developed in this book have significant applications in quantum field theory and string theory, making it relevant not only to mathematicians but also to theoretical physicists.