Lecture notes for Several Complex Variables Spring 1991 by Charles L. Epstein Chapter 1 One complex variable, for adults 1.1 Introduction 1.2 Review of Functional Analysis 1.3 The Cauchy integral formula, holomorphic functions 1.4 Elementary facts about analytic functions of one variable 1.5 The Runge approximation theorem, the holomorphic convex hull 1.6 Solving the ∂–equation 1.7 The Mittag–Leffler and Weierstraß Theorems, domains of holomorphy Chapter 2 Elementary properties of holomorphic functions in several variables 2.1 Holomorphy for functions of several variables 2.2 The Cauchy formula for polydiscs and its elementary consequences 2.3 Hartogs’ Theorem on separately holomorphic functions 2.4 Solving the ∂–equation in a polydisc and holomorphic extension 2.5 Local solution of the ∂–equation for p, q–forms 2.6 Power series and Reinhardt domains 2.7 Domains of holomorphy and holomorphic convexity 2.8 Pseudoconvexity, the ball versus the polydisc 2.9 CR-structures and the Lewy extension theorem 2.10 The Weierstraß preparation theorem Chapter 3 The complete metric approach to the ∂ –problem 3.0 Introduction 3.1 The geometry of the unit ball 3.2 Analysis of Bergman Laplacian 3.3 Polyhomogeneous conormal distributions 3.4 A simple model for blow–ups 3.5 Parabolic blow–ups for the model problem 3.6 The Θ–tangent bundle and parabolic blow–ups Version: 0.2; Revised:3-22-90; Run: February 5, 1998
Chapter 1 One complex variable, for adults1.1 Introduction1.2
Review of Functional Analysis1.3 The Cauchy integral formula,
holomorphic functions1.4 Elementary facts about analytic functions
of one variable1.5 The Runge approximation theorem, the holomorphic
convex hull1.6 Solving the @{equation1.7 The Mittag{Leer and
Weierstra Theorems, domains of holomorphy
Chapter 2 Elementary properties of holomorphic functions in
several variables2.1 Holomorphy for functions of several
variables2.2 The Cauchy formula for polydiscs and its elementary
consequences2.3 Hartogs Theorem on separately holomorphic
functions2.4 Solving the @{equation in a polydisc and holomorphic
extension2.5 Local solution of the @{equation for p; q{forms2.6
Power series and Reinhardt domains2.7 Domains of holomorphy and
holomorphic convexity2.8 Pseudoconvexity, the ball versus the
polydisc2.9 CR-structures and the Lewy extension theorem
2.10 The Weierstra preparation theorem
Chapter 3 The complete metric approach to the @{problem3.0
Introduction3.1 The geometry of the unit ball3.2 Analysis of
Bergman Laplacian3.3 Polyhomogeneous conormal distributions3.4 A
simple model for blow{ups3.5 Parabolic blow{ups for the model
problem3.6 The {tangent bundle and parabolic blow{ups
Version: 0.2; Revised:3-22-90; Run: February 5, 1998
