Babes-Bolyai University of Cluj-Napoca
Faculty of Mathematics and Computer Science
Study Cycle: Master
SUBJECT

Code	Subject	
MT263	Complex Functions of Several Variables	

Section	Semester	Hours: C+S+L	Category	Type	
Real and Compex Analysis - in English	1	2+1+1		compulsory	

Teaching Staff in Charge	Prof. KOHR Gabriela, Ph.D., gkohr@math.ubbcluj.ro	
Aims	The aim of this course is to introduce and thoroughgoing study certain fundamental results in the theory of functions of several complex variables. The introduction in certain special topics of geometric function theory of several complex variables. The students will be involved in the research activity.  	
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. Locally biholomorphic mappings. 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 functions and plurisubharmonic functions. Properties.
4. Holomorphic extension. Domains of holomorphy. Properties and examples of domains of holomorphy. Holomorphic convexity. Properties. The equivalence between the notions of domain of holomorphy and holomorphic convexity. 
5. Pseudoconvex domains. Hartogs' and Levi's pseudoconvexity. General properties. 
6. Subclasses of biholomorphic mappings on the unit ball in C&#8319;: starlike and convex mappings. Growth and covering results. Coefficient bounds. Examples. Applications which have parametric representation. 
7. Introduction in the theory of differential subordination chains in C&#8319;. The Loewner differential equation on the unit ball in C&#8319;. Applications. 	
Bibliography	1. Kohr, G., Basic Topics in Holomorphic Functions of Several Complex Variables, Cluj University Press, Cluj-Napoca, 2003.
2. Graham, I., Kohr, G., Geometric Function Theory in One and Higher Dimensions, Marcel 
Dekker Inc, New York, 2003.
3. Scheidemann, V., Introduction to Complex Analysis in Several Variables, Birkhäuser Verlag, Basel-Boston-Berlin, 2005.
4. Gunning, R.C., Introduction to Holomorphic Functions of Several Variables, vol.I. Function Theory, Wadsworth & Brooks/Cole, Monterey, CA, 1990.
5. Hörmander, L., An Introduction to Complex Analysis in Several Variables, Second Edition. North-Holland Publ. Co., Amsterdam-London, 1973.
6. Chabat, B., Introduction &agrave; l'Analyse Complexe, vol. II, Edition MIR, Moscou, 1990. 
7. Krantz, S.G., Function Theory of Several Complex Variables, Reprint of the 1992 Edition, AMS Chelsea Publishing, Providence, Rhode Island, 2001. 
8. Narasimhan, R., Several Complex Variables, The University of Chicago Press, Chicago, 1971.
9. Range, M., Holomorphic Functions and Integral Representations in Several Complex Variables Springer-Verlag, New York, 1986.
10. Rudin, W., Function Theory in the Unit Ball of C&#8319;, Springer-Verlag, New York, 1980.
	
Assessment	Exam (70%)+ student activity (30%).