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+2+0	speciality	compulsory	

Teaching Staff in Charge	Prof. KOHR Gabriela, Ph.D., gkohr@math.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. The Cauchy 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&#8217;s theorem. Applications: the open mapping theorem and Hurwitz&#8217;s theorem. 
4. The topological and metric structures of the space H(&#937;).
5. Families of holomorphic functions. Montel&#8217;s and Vitali&#8217;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&#8734;. Conformal automorphisms of annulus. 
10. Decomposition of meromorphic functions into 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. 
	
Bibliography	1. Kohr, G., Mocanu, P.T., Capitole Speciale de Analiz&atilde; Complex&atilde;, Presa Universitar&atilde; Clujean&atilde;, 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. Berenstein, C.A., Gay, R., Complex Variables: An Introduction, Springer-Verlag New York, 1991. 
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 Variabil&atilde; Complex&atilde;, Editura Univ. A.I. Cuza, Iaşi, 2001.
	
Assessment	Exam (70%)+ student activity (30%).