 Название: Introduction to Complex Analysis in Several Variables, Second Edition
Автор: Volker Scheidemann
Издательство: Springer/Birkhauser
Серия: Compact Textbooks in Mathematics
Год: 2023
Страниц: 239
Язык: английский
Формат: pdf (true), epub
Размер: 16.9 MB
This book gives a comprehensive introduction to complex analysis in several variables. While it focusses on a number of topics in complex analysis rather than trying to cover as much material as possible, references to other parts of mathematics such as functional analysis or algebras are made to help broaden the view and the understanding of the chosen topics. A major focus are extension phenomena alien to the one-dimensional theory, which are expressed in the famous Hartog's Kugelsatz, the theorem of Cartan-Thullen, and Bochner's theorem. The book aims primarily at students starting to work in the field of complex analysis in several variables and instructors preparing a course. To that end, a lot of examples and supporting exercises are provided throughout the text. This second edition includes hints and suggestions for the solution of the provided exercises, with various degrees of support.
In this chapter we give an introduction to analytic sets. Roughly speaking, analytic sets are sets whose elements are locally solutions of holomorphic systems of equations. We will introduce the notion of codimension of an analytic set and show that in most cases analytic sets are “thin enough” that holomorphic functions defined outside an analytic set can be extended across the analytic set.
Contents:
1. Elementary Theory of Several Complex Variables
2. Continuation on Circular and Polycircular Domains
3. Biholomorphic Maps
4. Analytic Sets
5. Hartogs’ Kugelsatz
6. Continuation on Tubular Domains
7. Cartan–Thullen Theory
8. Local Properties of Holomorphic Functions
9. Hints to the Exercises
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Introduction to Complex Analysis in Several Variables, Second Edition 
