1. komponenta

Lecture type	Total
Lectures	30
Exercises	15

* Load is given in academic hour (1 academic hour = 45 minutes)

COURSE AIMS AND OBJECTIVES: The aim is to work out advanced chapters of the theory of functions of complex variable, continuing the course Complex Analysis from the 3rd year of Bsc study of Mathematics.

COURSE DESCRIPTION AND SYLLABUS:
1. Integrals of Cauchy's type; index of a curve, connectedness, homotopy and simple connectedness; Jordan's theorem; global Cauchy's theorem; theorem on residua; Rouche's theorem; Vitali's theorem; Montel's theorem; Riemann's theorem.
2. Singularities of ordinary differential equations; the equations of Fuchs' type; the hypergeometric equation; orthogonal polynomials and functions; the Bessel's functions.
3. Entire and meromorphic functions; Mittag-Leffler's theorem; Weierstrass' factorization theorem; rank, order and genus of an entire function; Hadamard's canonical factorization theorem.
4. Elliptic functions; the Weierstrass' p-function and its Laurent's series; the Eisenstein's series and the invariants g2 and g3; the discriminant; the Klein's modular function; Fourier's expansions; Mobius transformations and the modular group; fundamental domains; modular functions.
5. Riemann's surfaces; harmonoic functions; the fundamental group and coverings; the uniformization theorem.
Remark: This is a list of possible themes for this elective course. The realisation can change each year, depending on the affinity of the lecturer and on the interest of the students.

1. Matematička analiza IV/I. Funkcije kompleksne varijable, H. Kraljević, S. Kurepa, Tehnička knjiga, Zagreb, 1986.
2. Functions of one compex variable, J. B. Conway, Springer Verlag, 1973.
3. The theory of functions, E. C. Titchmarsh, Oxford, 1939.
4. Real and complex analysis, W. Rudin, McGraw-Hill, New York, 1970.
5. Conformal invariants, L. V. Ahlfors, McGraw-Hill, New York, 1973.
6. Analytic function theory Vol. 1, 2, E.Hille, Blaisdello, New York, 1963.
7. Analytic functions, A. Saks, A. Zygmund, Polskie Towarzystwo Matematyczne, Warszawa, 1952.
8. Fonctions analytiques, G. Valiron, Press Universitaire, Paris, 1954.
9. Funtionentheorie, K. Diedrich, R. Remert, Springer Verlag, 1972.
10. Modular function and Dirichlet series in number theory, T. M. Apostol, Springer Verlag, 1976.

Analiza - Regular study - Mathematics Education

Analiza - Regular study - Mathematics Education