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This book introduces complex analysis in several variables. Applications focus on global extension problems for CR functions, such as the Hartogs-Bochner phenomenon. Includes appendices on differential manifolds, sheaf theory and functional analysis.

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This book provides an introduction to complex analysis in several variables. The viewpoint of integral representation theory together with Grauert´s bumping method offers a natural extension of single variable techniques to several variables analysis and leads rapidly to important global results. Applications focus on global extension problems for CR functions, such as the Hartogs-Bochner phenomenon and removable singularities for CR functions. Three appendices on differential manifolds, sheaf theory and functional analysis make the book self-contained.
Each chapter begins with a detailed abstract, clearly demonstrating the structure and relations of following chapters. New concepts are clearly defined and theorems and propositions are proved in detail. Historical notes are also provided at the end of each chapter.
Clear and succinct, this book will appeal to post-graduate students, young researchers seeking an introduction to holomorphic function theory in several variables and lecturers seeking a concise book on the subject.

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Elementary local properties of holomorphic functions of several complex variables.- Currents and complex structures.- The Bochner-Martinelli-Koppelman kernel and formula applications.- Extensions of CR functions.- Extensions of holomorphic and CR functions on manifolds.- Domains of holomorphy and pseudoconvexity.- The Levi problem and the resolution of in strictly pseudoconvex domains.- Characterisation of removable singularities of CR functions on a strictly pseudoconvex boundary.- Appendices.

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From the reviews:
"This introduction to complex analysis in several variables uses integral representation theory together with Grauert´s bumping method to characterize domains of holomorphy, in other words to solve the Levi problem. ... Each part is provided with a detailed abstract and interesting historical notes. This concise and self-contained book is a welcome enrichment to the existing literature for both graduate students as well as researchers." (F. Haslinger, Monatshefte für Mathematik, Vol. 165 (1), January, 2012)

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