Subject: Complex Functions of Several Variables	Code: MT263	

Semester	 Hours: C+S+L	Type	 Section	
2	2+2+0	compulsory	Real and Compex Analysis - in English	

Teaching Staff in Charge	Prof. KOHR Gabriela, Ph.D., gkohr@math.ubbcluj.ro	
Aims	The aim of this course is to give an introduction in the theory of holomorphic functions of several complex variables	
Content	1. Holomorphic functions of several complex variables. General properties. Equivalent definitions, the integral representation formula on polydisc, power series. The identity theorem for holomorphic function, maximum principle, sequences of holomorphic functions. Hartogs' theorem.
2. Holomorphic mappings. General properties. Biholomorphic mappings. The equivalence between univalence and biholomorphy. Cartan uniqueness results. The biholomorphic automorphisms of the Euclidean unit ball and the unit polydisc.
3. Pluriharmonic and plurisubharmonic functions. Properties.
4. Domains of holomorphy. Properties and examples of domains of holomorphy. Holomorphic convexity. Properties. The equivalence of these notions. 
5. Subclasses of biholomorphic mappings on the unit ball in C^n: starlike and convex mappings. Growth, covering and distortion results. Examples	
Selective References	1. KOHR, GABRIELA: Basic Topics in Holomorphic Functions of Several Complex Variables. Cluj-Napoca: Cluj University Press, 2003.
2. GRAHAM, IAN - KOHR, GABRIELA: Geometric Function Theory in One and Higher Dimensions. New York: Marcel Dekker Inc, 2003.
3. CHABAT, BORIS V: Introduction a l'Analyse Complexe. Vol.II. Moscou: Ed. MIR, 1990.
4. GUNNING, ROBERT C:  Introduction to Holomorphic Functions of Several Variables. vol.I. Monterey: Wadsworth & Brooks/Cole, 1990.
5. HORMANDER, LARS: An Introduction to Complex Analysis in Several Variables. Second Edition. Amsterdam-London: North-Holland Publ. Co., 1973.
6. KAUP, LUDGER - KAUP, BURCHARD: Holomorphic Functions of Several Variables. Berlin-New York: Walter de Gruyter & Co., 1983. 
7. KRANTZ, STEVEN G: Function Theory of Several Complex Variables. Reprint of the 1992 Edition. Rhode Island: AMS Chelsea Publishing, Providence, 2001. 
8. NARASIMHAN, RAGHAVAN: Several Complex Variables. Chicago: The University of Chicago Press, 1971.
9. RANGE, MICHAEL: Holomorphic Functions and Integral Representations in Several Complex Variables. New York: Springer-Verlag, 1986.
10. RUDIN, WALTER: Function Theory in the Unit Ball of C^n. New York: Springer-Verlag, 1980. 	
Assessment	Exam (70%)+ student activity (30%)