Function Theory of Several Complex Variables

Aims at: Masters mathematics

Prerequisits: Analysis courses from the Bachelor Mathematics

Goals: Familiarity with the basic concepts in the theory of analytic functions of several complex variables, as well as with a number of fundamental results of the subject.

Contents: Analyticity in several variables, domains of convergence of power series (Reinhardt domains), Cauchy-Riemann equations, Hartogs' extension theorem, domains of holomorphy, pseudoconvexity and the Levi problem. Description of zero sets (Weierstrass preparation theorem), holomorphic mappings. Plurisubharmonic functions, L^2 and Kernel methods for solving inhomogeneous Cauchy Riemann equations. Cousin problems and cohomology.

Semester: II

Form: Course (2 h. per week) If only few students are attending, we switch possibly to a reading course

Euro credits: 6 EC Possiblity to extend with maximal 3 EC

Examination

Take Home execises

date      exercises                       worth       hand in date   set
Sept 5    Chapter 1  no 8,9,12,18,24,29    10        September 19  (1)

Sept 12   Chapter 2. no 2,7,11,15,25,26    10        September 26  (2)

Material Covered

• 5 sep. Chapter 1. (Globally)
• 12 sep Chapter 2. Sec. 3, 4,5,6,8.

The Oral Examination

Literature:
• J. Korevaar-J. Wiegerinck: Lecture notes several complex variables, PS-version or PDF-version

  or New beta-version (under development) This is basically the old version, with some clarifications, ameliorations, and in state of the art latex.

Additional reading:
• S. Krantz: Function theory of several complex variables , AMS-chelsea 1992
• L. Hörmander: Complex analysis in several variables, North Holland 1973
• K Fritzsche - H. Grauert: From Holomorphic functions to complex manifolds Springer, 2002.

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