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Complex analysis

General data

Course ID:	1000-135ANZ
Erasmus code / ISCED:	11.133 Kod klasyfikacyjny przedmiotu składa się z trzech do pięciu cyfr, przy czym trzy pierwsze oznaczają klasyfikację dziedziny wg. Listy kodów dziedzin obowiązującej w programie Socrates/Erasmus, czwarta (dotąd na ogół 0) – ewentualne uszczegółowienie informacji o dyscyplinie, piąta – stopień zaawansowania przedmiotu ustalony na podstawie roku studiów, dla którego przedmiot jest przeznaczony. / (0541) Mathematics The ISCED (International Standard Classification of Education) code has been designed by UNESCO.
Course title:	Complex analysis
Name in Polish:	Analiza zespolona
Organizational unit:	Faculty of Mathematics, Informatics, and Mechanics
Course groups:	(in Polish) Przedmioty 4EU+ (z oferty jednostek dydaktycznych) (in Polish) Przedmioty obieralne na studiach drugiego stopnia na kierunku bioinformatyka Elective courses for 2nd stage studies in Mathematics
ECTS credit allocation (and other scores):	6.00 Basic information on ECTS credits allocation principles: the annual hourly workload of the student’s work required to achieve the expected learning outcomes for a given stage is 1500-1800h, corresponding to 60 ECTS; the student’s weekly hourly workload is 45 h; 1 ECTS point corresponds to 25-30 hours of student work needed to achieve the assumed learning outcomes; weekly student workload necessary to achieve the assumed learning outcomes allows to obtain 1.5 ECTS; work required to pass the course, which has been assigned 3 ECTS, constitutes 10% of the semester student load. view allocation of credits
Language:	English
Main fields of studies for MISMaP:	astronomy mathematics physics
Type of course:	elective courses
Requirements:	Analytic Functions of One Complex Variable 1000-134FAN Mathematical analysis I.1 1000-111bAM1a Mathematical analysis I.2 1000-112bAM2a Mathematical analysis II.1 1000-113bAM3a Mathematical analysis II.2 1000-114bAM4a
Prerequisites (description):	It is obligatory to have the full 2-year long course on Mathematical Analysis passed, as well as the one-semester course on Analytic Functions.
Mode:	Classroom
Short description:	Main topics: Weierstrass theorem, Mittag-Leffler theorem, Runge theorem. Many-valued functions, analytical extensions, monodromy. Analytical functions on Riemann surfaces; Problems in Riemann surface theory: basic information and examples. Fundamental notions of the theory of analytic functions in many complex variables, Cauchy-Riemann equations, power series expansions, analytical extensions, Cousin problems.
Full description:	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).
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).
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 methods and assessment criteria:	Written examination, with taking into account student's activity and work during the semester.

Classes in period "Summer semester 2023/24" (in progress)

Time span:	2024-02-19 - 2024-06-16	Selected timetable range: weekly course term Navigate to timetable MO CW WYK TU W TH FR
Type of class:	Classes, 30 hours more information Lecture, 30 hours more information
Coordinators:	Piotr Mormul
Group instructors:	Marcin Bobieński, Piotr Mormul
Students list:	(inaccessible to you)
Examination:	Course - Examination Lecture - Examination

Classes in period "Summer semester 2024/25" (future)

Time span:	2025-02-17 - 2025-06-08	Selected timetable range: weekly course term Navigate to timetable
Type of class:	Classes, 30 hours more information Lecture, 30 hours more information
Coordinators:	Piotr Mormul
Group instructors:	Marcin Bobieński, Piotr Mormul
Students list:	(inaccessible to you)
Examination:	Examination

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