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Theory of analytic functions

Code: 45620
ECTS: 5.0
Lecturers in charge: prof. dr. sc. Boris Širola - Lectures
Lecturers: prof. dr. sc. Boris Širola - Exercises
English level:

1,0,0

All teaching activities will be held in Croatian. However, foreign students in mixed groups will have the opportunity to attend additional office hours with the lecturer and teaching assistants in English to help master the course materials. Additionally, the lecturer will refer foreign students to the corresponding literature in English, as well as give them the possibility of taking the associated exams in English.
Load:

1. komponenta

Lecture typeTotal
Lectures 30
Exercises 15
* Load is given in academic hour (1 academic hour = 45 minutes)
Description:
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.
Literature:
  1. H. Kraljević, S. Kurepa: Matematička analiza IV/I. Funkcije kompleksne varijable
  2. J. B. Conway: Functions of one compex variable
  3. E. C. Titchmarsh: The theory of functions
  4. W. Rudin: Real and complex analysis
  5. L. V. Ahlfors: Conformal invariants
  6. E.Hille: Analytic function theory Vol. 1, 2
  7. A. Saks, A. Zygmund: Analytic functions
  8. G. Valiron: Fonctions analytiques
  9. K. Diedrich, R. Remert: Funtionentheorie
  10. T. M. Apostol: Modular function and Dirichlet series in number theory
1. semester
Izborni predmet 1, 2 - Regular study - Theoretical Mathematics

2. semester
Izborni predmet 1, 2 - Regular study - Theoretical Mathematics

3. semester
Izborni predmet 3, 4 - Regular study - Theoretical Mathematics

4. semester
Izborni predmet 3, 4 - Regular study - Theoretical Mathematics
Consultations schedule: