Titlebook: Lectures on Several Complex Variables; Paul M. Gauthier Book 2014 Springer International Publishing Switzerland 2014 Cauchy integral formu

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Basics,Two striking theorems of Hartogs, which have no counterpart in a single variable, are introduced: the Hartogs theorem on separate holomorphy and the Hartogs phenomenon on the possibility of extending holomorphic functions.

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Zero Sets of Holomorphic Functions,Zeros of holomorphic functions of several variables have a much richer structure than those of a single variable. Fundamental concepts from algebra enter the scene and lead up to the Weierstrass preparation theorem, which is the best instrument for understanding the local nature of the set of zeros of a holomorphic function.

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Holomorphic Mappings,The theory of smooth mappings of several real variables will help us to develop a similar theory for smooth (holomorphic) mappings of several complex variables. As in the real case, the main features are the inverse function theorem and the implicit function theorem.