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

Code	Subject	
MMC0006	Special Topics in Complex Analysis	

Section	Sem.	Hours: C+S+L	Category	Type	
Mathematics - in Romanian	6	2+1+0	speciality	optional	
Applied Mathematics	6	2+1+0	speciality	optional	

Teaching Staff in Charge	Assoc.Prof. ANISIU Valeriu, Ph.D., anisiu@math.ubbcluj.ro	
Aims	Basic facts omitted during the main course in the theory of Complex functions and its applications are presented. The emphasis is on problem solving.

Content	Geometric properties of complex numbers. Geometric transforms.
Using complex numbers in synthetic geometry. The theorems of Ptolemeu, Pompeiu, Angheluţă, D. V. Ionescu 
Complex analytic Geometry.
Insscribed polygons. Euler@s circle, generalizations.
Moebius functions.
Classifications of Moebius transforms.
Models of non-Euclidean geometries (Lobacevski-Poincare). The Axioms.
The circle, the elliptic, parabolic and hyperbolic transforms in Poincare@s model.
The generalized Schwarz lemma.
Sets of holomorphic functions, Montel@s theorem
Univalent functions, Hurvitz@ theorem
Conformal representation, the Riemann@s theorem.
Elements of Geometric function theory.
Fractal dimensions, examples.
The dynamic of complex mappings. Atractors.

Bibliography	1. G. S. Sălăgean, Geometria planului complex, Promedia-Plus, Cluj-Napoca, 1997
2. P. T. Mocanu, T. Bulboacă, G. S. Sălăgean, Teoria geometrică a funcţiilor univalente, Casa Cărţii de Ştiinţă, Cluj-Napoca, Ed. I, 1999, Ed. II, 2006.
3. P. Hamburg, P. T. Mocanu, N. Negoescu, Analiză matematică (Funcţii complexe), Ed. Did. şi Ped., Bucureşti, 1982
4. G. Helmberg, Getting Acquainted with Fractals. DeGruyter, 2007
5. R. Deaux, Introduction to the Geometry of Complex Numbers. Dover, 2008
6. T. Andreescu, D. Andrica, Complex Numbers from A to &#8230; Z, Birkhauser, Boston, 2006
7. Krantz S.G. - Geometric Function Theory. Birkhauser, 2006
8. P. T. Mocanu, Funcţii complexe, Partea I, Lito. Universitaţii Cluj, 1972
9. R. Shakarchi, Problems and Solutions for Complex Analysis. Springer, 1999
10. M. Evgrafov, K. Bobejnov, Y. Sidorov, Recueil de problemes sur la theorie des functions analytiques, Edition Mir, Moscou, 1974

	
Assessment	Midterm test and Final Exam.