"Babes-Bolyai" University of Cluj-Napoca
Faculty of Mathematics and Computer Science

Complex functions of several variables
Hours: C+S+L
Analiză Reală şi Complexă - în limba engleză
Teaching Staff in Charge
Assoc.Prof. KOHR Gabriela, Ph.D., gkohr@math.ubbcluj.ro
The aim of this course is an introduction in the theory of holomorphic functions of several complex variables
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 unit ball and 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. Pseudoconvex domains. Hartogs pseudoconvexity and Levi pseudoconvexity. General properties. Necessary and sufficient conditions of pseudoconvexity. The equivalence of these notions in the case of domains with differentiable boundary.
7. The equivalence between pseudoconvexity and holomorphic convexity.
8. Subclasses of biholomorphic mappings on the unit ball of C^n: starlike and convex mappings. Growth, covering and distortion results. Examples.
1. I. Graham, G. Kohr, Geometric Function Theory in One and Higher Dimensions, Marcel Dekker Inc, New York, 2003.
2. B. Chabat, Introduction a l'Analyse Complexe, vol.II, Ed. MIR, Moscou, 1990.
3. R.C. Gunning, Introduction to Holomorphic Functions of Several Variables, vol.I, Wadsworth & Brooks/Cole, Monterey, 1990.
4. L. Hormander, An Introduction to Complex Analysis in Several Variables, Second Edition, North-Holland, Amsterdam, 1973.
5. L. Kaup, B. Kaup, Holomorphic Functions of Several Variables, Walter de Gruyter & Co.,
Berlin-New York, 1983.
6. S. Kobayashi, Hyperbolic Manifolds and Holomorphic Mappings, Marcel Dekker Inc, New York, 1970.
7. R. Narasimhan, Several Complex Variables, The University of Chicago Press, 1971.
8. M. Range, Holomorphic Functions and Integral Representations in Several Complex Variables, Springer-Verlag, New York, 1986.
9. W. Rudin, Function Theory in the Unit Ball of C^n, Springer-Verlag, New York, 1980.
10. J. Wermer, Banach Algebras and Several Complex Variables, Graduate Texts in Math., Springer-Verlag, New York, Heidelberg, Berlin, 1976.