Domains of Holomorphy and Pseudoconvexity
In 1906 F. Hartogs discovered the first example exhibiting the remarkable extension properties of holomorphic functions in more than one variable. It is this phenomenon, more than anything else, which distinguishes function theory in several variables from the classical one-variable theory. Hartogs’ discovery marks the beginning of a genuine several-variable theory, in which fundamental new concepts like domains of holomorphy and the various notions of convexity used to characterize them have become indispensable. In particular, the property now generally referred to as “pseudoconvexity” originates with Hartogs, and even today it still is one of the richest sources of intriguing phenomena and deep questions in complex analysis. (See, for example, the remarks at the end of §2.8.) We will say more about this in Chapter VII.
KeywordsPseudoconvex Domain Subharmonic Function Plurisubharmonic Function Holomorphic Extension Analytic Disc
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