Faculty of Mathematics and Computer Science

Functions of several complex variables |

Code |
Semes-ter |
Hours: C+S+L |
Credits |
Type |
Section |

Teaching Staff in Charge |

Prof. SALAGEAN Grigore Stefan, Ph.D., salagean@math.ubbcluj.ro |

Aims |

Appropriation of the basic knowledge of the theory of complex functions of several complex variables and the presentation of some applications of this theory. |

Content |

1. Elementary properties of holomorphic functions of several complex variables. Convergence properties of power series. The Cauchy-Riemann equations. The Cauchy integral formula for polydisc.
2. Holomorphic mappings. General properties. 3. The automorphisms of the Euclidean unit ball and the unit polydisc. 4. Convexity and pseudoconvexity. Classes of domains in the n-dimensional complex space. Domains of holomorphy. 5. Some integral formulas. The Bochner-Martinelli formula. Applications of Cauchy theory. 6. Geometric theory of biholomorphic mappings in several complex variables: Biholomorphic mappings. General properties. Starlike mappings, convex mappings, spirallike mappings and close-to starlike mappings on the unit ball of the n-dimensional complex space. 7. Sufficient criteria of univalence on the unit ball of the n-dimensional complex space. Applications. |

References |

1. C. Andreian Cazacu, Teoria Functiilor de mai multe Variabile Complexe , Ed. Acad. Rom., Bucharest, 1965.
2. B. Chabat, Introduction a l'Analyse Complexe, vol. II, Ed. MIR, Moscou, 1990. 3. S. Gong, Convex and Starlike Mappings in Several Complex Variables, Kluwer Acad. Publ., 1998. 4. I. Graham, G. Kohr, Topics in Geometric Function Theory of One and Higher Dimensions, Marcel Dekker Inc., 2002 (sub tipar). 5. R.C. Gunning, Introduction to Holomorphic Functions of Several Variables, vol.I, Wadsworth and Brooks/Cole Math. Series, 1990. 6. L. Kaup, B. Kaup, Holomorphic Functions of Several Variables, de Gruyter Studies in Math., Berlin, New York, 1983. 7. G. Kohr, P. Liczberski, Univalent Mappings of Several Complex Variables, Cluj Univ. Press , Cluj-Napoca, 1998. 8. S.G. Krantz, Function Theory of Several Complex Variables, Second Ed., Wadsworth & Brooks/Cole Advanced Books& Software, California, 1992. 9. M. Range, Holomorphic Functions and Integral Representations in Several Complex Variables, Springer Verlag, New York, 1986. 10. W. Rudin, Function Theory in the Unit Ball of C^n, Springer Verlag, Berlin, New York, 1980. |

Assessment |

Exam |