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German original

Complex Analysis
(dt. Funktionentheorie (Analytische Funktionen einer komplexen Veränderlichen))

Level, degree of commitment in original study programme Intermediate module, compulsory elective module
Forms of teaching and learning,
workload
Lecture (4 SWS), recitation class (2 SWS),
270 hours (90 h attendance, 180 h private study)
Credit points,
formal requirements
9 CP
Course requirement: Successful completion of at least 50 percent of the points from the weekly exercises.
Examination type: Written or oral examination
Language,
Grading
German,
The grading is done with 0 to 15 points according to the examination regulations for study course B.Sc. Mathematics.
Original study programme Reine Exportmodule
Duration,
frequency
One semester,
irregular
Person in charge of the module's outline Prof. Dr. Thomas Bauer

Contents

  • Complex differentiability, Cauchy-Riemann differential equations
  • Power series, Taylor expansion
  • Curve integrals, Cauchy's integral theorems
  • Isolated singularities, elementary holomorphic functions, meromorphic functions, Laurent series
  • Residue theorem and applications
  • Conform maps, Möbius group
  • Normal families, Montel's theorem
  • Riemann mapping theorem

Qualification Goals

The students shall

  • understand how complex-analytical methods help to solve real-analytical problems,
  • deepen their understanding of the elementary functions through the complex viewpoint,
  • learn about the connections between methods of geometry, algebra, and analysis, as well as topology and number theory, and thus develop their mathematical understanding,
  • Learn methods and skills that are central to applications in computer science (e.g. coding theory), physics (e.g. quantum theory) and engineering (e.g. electrical engineering)
  • practice mathematical working methods (development of mathematical intuition and its formal justification, training of abstraction, proof methods),
  • improve their oral communication skills in the recitation class by practicing free speech in front of an audience and during discussion.

Prerequisites

None. The competences taught in the following modules are recommended: either Linear Algebra I and Linear Algebra II or Basic Linear Algebra, either Analysis I and Analysis II or Basic Real Analysis.


Recommended Reading

  • Fischer,W., Lieb, I.: Funktionentheorie: Komplexe Analysis in einer Veränderlichen, Vieweg; Conway, J.B.: Functions of one complex variable, Graduate Texts in Mathematics, Springer; Lang, S.: Complex analysis, Graduate Texts in Mathematics, Springer; Remmert, R., Schumacher, G.: Funktionentheorie I,II, Berlin: Springer



Please note:

This page describes a module according to the latest valid module guide in Wintersemester 2019/20. Most rules valid for a module are not covered by the examination regulations and can therefore be updated on a semesterly basis. The following versions are available in the online module guide:

The module guide contains all modules, independent of the current event offer. Please compare the current course catalogue in Marvin.

The information in this online module guide was created automatically. Legally binding is only the information in the examination regulations (Prüfungsordnung). If you notice any discrepancies or errors, we would be grateful for any advice.