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

Code	Subject	
MMA1009	Complex Functions of Several Variables	

Section	Sem.	Hours: C+S+L	Category	Type	
Mathematics	4	2+1+0	speciality	optional	

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 implications of students 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%).