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
Semester
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.,  anisiumath.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.

References
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 … 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.

Links: Syllabus for all subjects
Romanian version for this subject
Rtf format for this subject