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

SUBJECT

Code
Subject
MMA1004 Special Topics of Complex Analysis
Section
Semester
Hours: C+S+L
Category
Type
Mathematics
2
2+1+0
speciality
compulsory
Teaching Staff in Charge
Prof. KOHR Gabriela, Ph.D.,  gkohrmath.ubbcluj.ro
Aims
The aim of this course is the acquirement and thoroughgoing study of certain notions and fundamental results in the theory of functions of one complex variable. The students will be involved in the research activity.
Content
1. Analytical branches. The theorems related to analytical branches for logarithm and power functions. Examples of analytical brances. Applications.
2. Index. General properties. Cauchy@s integral formulas for contours.
3. Zeros and poles for meromorphic functions. The Cauchy theorem related to zeros and poles of meromorphic functions. The argument principle. Rouché’s theorem. Applications: the open mapping theorem and Hurwitz’s theorem.
4. The topological and metric structures of the space H(Ω), where Ω is an open set in the complex plane.
5. Families of holomorphic functions. Montel’s and Vitali’s theorems.
6. Univalent functions. General properties. Applications.
7. The class S. The covering and growth theorems. The compactness of the class S. The Bieberbach theorem for the second coefficient a_2.
8. Conformal mappings. The Riemann mapping theorem. The conformal radius for simply connected domains in C. Correspondence of boundaries.
9. Conformal automorphisms of the unit disc, upper half-plane, complex plane C and extended complex plane C∞. Conformal automorphisms of annulus.
10. Decomposition of meromorphic functions through Mittag-Leffler series. The Mittag-Leffler theorem. Applications.
11. Infinite products of complex numbers. Decomposition of entire functions in canonical products. The Weierstrass theorem. Domains of holomorphy.
12. Harmonic and subharmonic functions. General properties.
References
1. Kohr, G., Mocanu, P.T., Capitole Speciale de Analizã Complexã, Presa Universitarã Clujeanã, Cluj Napoca, 2005.
2. Hamburg, P., Mocanu, P.T., Negoescu, N., Analiză Matematică (Funcţii complexe), Editura Didactică şi Pedagogică, Bucureşti, 1982.
3. Gaşpar, D., Suciu, N., Analiză Complexă, Editura Academiei Române, Bucureşti, 1999.
4. Krantz, S., Handbook of Complex Variables, Birkhäuser Verlag, Boston, Basel, Berlin, 1999.
5. Conway, J.B., Functions of One Complex Variable, vol. I, Graduate Texts in Mathematics, 159, Springer Verlag, New York, 1996.
6. Stein, E.M., Shakarchi, R., Complex Analysis, Princeton University Press, 2003.
7. Rudin, W., Real and Complex Analysis, 3nd ed., Mc. Graw-Hill, 1987.
8. Mayer, O., Teoria Funcţiilor de o Variabilă Complexă, Editura Academiei Române, Bucureşti, 1981.
9. Narasimhan, R., Nievergelt, Complex Analysis in One Variable, Second Edition, Birkhäuser, 1985.
10. Popa, E., Introducere în Teoria Funcţiilor de o Variabilã Complexã, Editura Univ. A.I. Cuza, Iaşi, 2001.
Assessment
Exam (70%)+ student activity (30%).
Links: Syllabus for all subjects
Romanian version for this subject
Rtf format for this subject