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# Meromorphic Functions on the Riemann Sphere

## Module MA5942

This Module is offered by Department of Mathematics.

This module handbook serves to describe contents, learning outcome, methods and examination type as well as linking to current dates for courses and module examination in the respective sections.

### Basic Information

MA5942 is a semester module in German or English language at Master’s level which is offered irregular.

This Module is included in the following catalogues within the study programs in physics.

• Catalogue of non-physics elective courses
150 h 45 h 5 CP

### Content, Learning Outcome and Preconditions

#### Content

Meromorphic functions on the Riemann sphere; ocvering maps; Moebius transformations; simply and doubly periodic functions; elliptic functions and their topologies; meromorphic extensions; Riemann surfaces

#### Learning Outcome

At the end of the lectures, students have an understanding of the concept of meromorphic functions on the Riemann sphere. They are able to construct covering maps and are knowledgeable about elliptic functions and the underlying topologies. Students are able to construct meromorphic extensions and understand the concept of Riemann surface.

#### Preconditions

MA1001/MA0001 Analysis 1, MA1002/MA0002 Analysis 2, MA1101/MA0004 Linear Algebra 1, MA1102/MA0005 Linear Algebra 2 and Discrete Structures, MA2006 Complex Analysis

### Courses, Learning and Teaching Methods and Literature

#### Courses and Schedule

VO 2 Meromorphic Functions und Riemann Surfaces [MA5942] Massopust, P. Tue, 14:15–15:45, MI 03.10.011
UE 1 Meromorphic Functions und Riemann Surfaces (Exercise Session) [MA5942] Massopust, P. Tue, 12:15–13:45, MI 02.08.020

#### Learning and Teaching Methods

The module is offered as lectures and an exercise session. The contents in the lectures are conveyed to the students by oral presentation and discussion. The lectures are also intended to motivate students for an independent acquisition of the topics and the study of the literature. Problem sets and their suggested solutions are distributed reflecting the material presented in the lectures. These problem sets are intended to deepen the understanding of the subject and to give students the opportunity to use the acquired knowledge from the lectures to solve related problems.

Tablet

#### Literature

E. Peschl, Funktionentheorie, Band I, B.I. Hochschultaschenbücher, Mannheim.
G. Jones und D. Singerman, Complex Functions: An algebraic and geometric viewpoint, Cambridge University.
O. Forster, Riemann Surfaces, Springer Verlag.
K. Lamotke, Riemannsche Flächen, Springer Verlag.
R. Mirinda, Algebraic Curves and Riemann Surfaces, AMS GSM 5.

### Module Exam

#### Description of exams and course work

The module examination is based on a 30-minute oral exam. Students demonstrate that they have acquired fundamental knowledge of definitions and main mathematical tools from the theory of covering maps, complex periodic functions, elliptic functions, and Riemann surfaces as presented in the course. The students are expected to be able to derive the methods, explain their properties, and apply them to specific examples.

#### Exam Repetition

The exam may be repeated at the end of the semester.

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