This work departs from earlier treatments of the subject by emphasizing integral formulas, the geometric theory of pseudoconvexity, estimates, partial differential equations, approximation theory, the boundary behavior of holomorphic functions, inner functions, invariant metrics, and mapping theory. While due homage is paid to the more traditional algebraic theory (sheaves, Cousin problems, etc.), the student with a background in real and complex variable theory, harmonic analysis, and differential equations will be most comfortable with this treatment.

Readership

Graduate students and research mathematicians interested in several complex variables and analytic spaces.

Reviews

"This book makes available a comprehensive, detailed and carefully organized treatment of the foundations for multidimensional complex analysis. ...provides all sorts of topics and exercises which stretch the imagination and show how useful the subject really is."

-- Zentralblatt MATH

"One of the remarkable features of this book is that it contains a large number of exercises and problems. ...the student needs a book like Krantz's, which is written as a text with explainations and exercises."

-- Bulletin of the American Mathematical Society

Table of Contents

• An introduction to the subject
• Some integral formulas
• Subharmonicity and its applications
• Convexity
• Hörmander's solution of the \(\bar\partial\) equation
• Solution of the Levi problem and other applications of \(\bar\partial\) techniques
• Cousin problems, cohomology, and sheaves
• The zero set of a holomorphic function
• Some harmonic analysis
• Constructive methods
• Integral formulas for solutions to the \(\bar\partial\) problem and norm estimates
• Holomorphic mappings and invariant metrics
• Manifolds
• Area measures
• Exterior algebra
• Vectors, covectors, and differential forms
• List of notation
• Bibliography
• Index