Scheidemann V. Introduction to Complex Analysis in Several Variables 2ed 2023 torrent

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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.
Preface to the Second Edition
Preface to the First Edition
Register of Symbols
Elementary Theory of Several Complex Variables
Geometry of Cn
Holomorphic Functions in Several Complex Variables
Definition of a Holomorphic Function
Basic Properties of Holomorphic Functions
Partially Holomorphic Functions and the Cauchy–Riemann Differential Equations
The Cauchy Integral Formula
O( U) as a Topological Space
Locally Convex Spaces
The Compact-Open Topology on C( U,E)
The Theorems of Arzelà–Ascoli and Montel
Power Series and Taylor Series
Summable Families in Banach Spaces
Power Series
Reinhardt Domains and Laurent Expansion
Continuation on Circular and Polycircular Domains
Holomorphic Continuation
Representation-Theoretic Interpretation of the Laurent Series
Hartogs' Kugelsatz, Special Case
Biholomorphic Maps
The Inverse Function Theorem and Implicit Functions
The Riemann Mapping Problem
Cartan's Uniqueness Theorem
Analytic Sets
Elementary Properties of Analytic Sets
The Riemann Removable Singularity Theorems
Hartogs' Kugelsatz
Holomorphic Differential Forms
Multilinear Forms
Complex Differential Forms
The Exterior Derivative
The inhomogenous Cauchy–Riemann Differential Equations
Dolbeault's Lemma
The Kugelsatz of Hartogs
Continuation on Tubular Domains
Convex Hulls
Holomorphically Convex Hulls
Bochner's Theorem
Cartan–Thullen Theory
Holomorphically Convex Sets
Domains of Holomorphy
The Theorem of Cartan–Thullen
Holomorphically Convex Reinhardt Domains
Local Properties of Holomorphic Functions
Local Representation of a Holomorphic Function
Germ of a Holomorphic Function
The Algebras of Formal and of Convergent Power Series
The Weierstrass Theorems
The Weierstrass Division Formula
The Weierstrass Preparation Theorem
Algebraic Properties of C{ z1,…, zn}
Hilbert's Nullstellensatz
Germs of a Set
The Radical of an Ideal
Hilbert's Nullstellensatz for Principal Ideals
Hints to the Exercises
References