Complex analysis 1000-135ANZ
Weierstrass theorem, Mittag-Leffler theorem, applications (1 -- 2 lectures). Runge theorem with applications (1 -- 2 lectures).
Many-valued functions, analytical extensions, monodromy (1 -- 2 lectures). Riemann surfaces. Analytical functions on Riemann surfaces.
Problems in Riemann surface theory: basic information and examples
(2 -- 3 lectures).
Fundamental notions of the theory of analytic functions in several complex variables, sets of Cauchy-Riemann equations, multi-power series expansions, analytical extensions, Cousin problems (7 -- 8 lectures).
Main fields of studies for MISMaP
astronomy
physics
Type of course
Mode
Requirements
Mathematical analysis I.2
Mathematical analysis II.1
Mathematical analysis II.2
Analytic Functions of One Complex Variable
Prerequisites (description)
Course coordinators
Learning outcomes
They can effectively write down an entire function with a prescribed countably infinite set of zeroes of goven orders.
They can effectively write down a meromorphic function with a prescribed countably infinite set of poles of given orders.
They can describe the generators of the monodromy group of an algebraic
multi-valued function w = w(z), which function becomes single-valued (or: univalent) W = W(Z) when Z is taken from the Riemann surface of the function under consideration.
They can compute, for a given power series in several complex variables,
its set of associated radii of convergence.
They can effectively produce a power series in several variables with
a beforehand prescribed set of associated radii of convergence.
They can verify if a given open set in C^n is holomorphically convex.
They know examples of domains in C^n , n > 1, in which the first
(i.e., additive) Cousin problem is not solvable.
Assessment criteria
Written examination, with taking into account student's activity and work during the semester.
Bibliography
S. Saks, A. Zygmund, Analytic Functions, Warsaw 1959 (in Polish).
F. Leja, Analytic Functions, Warsaw 1979 (in Polish).
B.W. Szabat, Introduction to Complex Analysis, Warsaw 1974 (in Polish, translation from Russian).
W. Rudin, Real and Complex Analysis, McGraw-Hill 1974.
P. Jakóbczak, M. Jarnicki, Introduction to the Theory of Holomorphic Functions of Several Complex Variables, Cracow 2002 (in Polish).
M. Skwarczyński, T. Mazur, Introductory Theorems of the Theory of Several Complex Variables, Warsaw 2001 (in Polish).
Additional information
Information on level of this course, year of study and semester when the course unit is delivered, types and amount of class hours - can be found in course structure diagrams of apropriate study programmes. This course is related to the following study programmes:
- Bachelor's degree, first cycle programme, Mathematics
- Master's degree, second cycle programme, Mathematics
Additional information (registration calendar, class conductors, localization and schedules of classes), might be available in the USOSweb system: