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

Code	Subject	
MMC0004	Supplement of Complex Analysis	

Section	Semester	Hours: C+S+L	Category	Type	
Mathematics - in Romanian	5	2+1+0	speciality	optional	
Mathematics-Computer Science - in Romanian	5	2+1+0	speciality	optional	
Applied Mathematics	5	2+1+0	speciality	optional	

Teaching Staff in Charge	Prof. SALAGEAN Grigore Stefan, Ph.D., salagean@math.ubbcluj.ro
Prof. BULBOACA Teodor, Ph.D., bulboaca@math.ubbcluj.ro	
Aims	Appropriation of the basic knowledge of the theory of complex functions of a complex variable and the presentation of some applications of this theory.	
Content	1. Analytical  branches.  The index of a path. The homological version of Cauchy's theorem. 
2. Residues. Meromorphic functions. The principle of argument's variation. Rouche's theorem. The open  mapping theorem.
3. The geometric theory of holomorphic functions: Conformal mappings. Conformal automorphisms of the unit disc and  of annuli. Normal families. Montel's and Vitali's theorems. Conformal equivalence of simply connected domains. The Riemann mapping theorem. Continuity at  the boundary.
4. Properties of univalent functions on the unit disc. The class S.   
5. Infinite series and products. The theorems of Weierstrass and Mittag-Leffler.
6. Harmonic and subharmonic functions. Poisson representation of harmonic functions.	
Bibliography	1. HAMBURG, PETRE - MOCANU, PETRU - NEGOESCU, NICOLAE : Analiză matematică (Funcţii complexe), Editura Didactică şi Pedagogică, Bucureşti, 1982.
2. GAŞPAR, DUMITRU - SUCIU, NICOLAE : Analiză complexă, Editura Academiei Române, Bucureşti, 1999.
3. KRANTZ, STEVEN : Handbook of complex variables, Birkhauser Verlag, Boston, Basel, Berlin, 1999.
4. CONWAY, J. B. : Functions of one complex variable II, Graduate Texts in Mathematics, 159, Springer Verlag, New York, 1996.
5. BULBOACĂ, TEODOR - NÉMETH, SÁNDOR : Komplex Analizis, Editura Abel (Erdely Tankönyvtanács), Cluj-Napoca, 2004.
6. BULBOACĂ, TEODOR - SALAMON, JULIA : Komplex Analizis II. Feladatok és megoldások, Editura Abel (Erdely Tankönyvtanács), Cluj-Napoca, 2002.
7. MAYER, OCTAV : Teoria funcţiilor de o variabilă complexă (vol. I, II), Editura Academiei Române, Bucureşti, 1981-1990.
8. STOILOV, SIMION : Teoria funcţiilor de o variabilă complexă (vol. I, II), Editura Academiei Române, Bucureşti, 1954-1958.
9. CĂLUGĂREANU, GHEORGHE : Elemente de teoria funcţiilor de o variabilă complexă, Editura Didactică şi Pedagogică, Bucureşti, 1963.
10. MOCANU, PETRU : Funcţii complexe, Lit. Univ. Cluj, 1972.	
Assessment	Exam. Student tests during the semester; their average represents 1/3 from the final score.