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

 Level, degree of commitment Advanced 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(s): Successful completion of at least 50 percent of the points from the weekly exercises. Examination type: Written or oral examination (individual examination) Language,Grading German,The grading is done with 0 to 15 points according to the examination regulations for the degree program B.Sc. Mathematics. 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

Students will

• Understand how complex-analytic methods enable them to solve problems in real analysis,
• have deepened their understanding of elementary functions through the complex point of view,
• know connections of methods of geometry, algebra, and calculus, as well as topology and number theory, and have further developed their mathematical understanding as a result,
• have learned methods and skills central to applications in computer science (e.g., coding theory), physics (e.g., quantum theory), and engineering (e.g., electrical engineering),
• have practiced mathematical ways of working (developing mathematical intuition and its formal justification, training the ability to abstract, reasoning),
• have improved their oral communication skills in exercises by practicing free speech in front of an audience and in 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.

## Applicability

Module imported from B.Sc. Mathematics.

It can be attended at FB12 in study program(s)

• B.Sc. Mathematics
• LAaG Mathematics
• BA Minor Mathematics

When studying BA Minor Mathematics, this module can be attended in the study area Compulsory Elective Modules.

The module is assigned to variant A of the BSc minor program in Mathematics.

• 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

This page describes a module according to the latest valid module guide in Winter semester 2023/24. 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:

• Winter 2016/17 (no corresponding element)
• Summer 2018 (no corresponding element)
• Winter 2018/19 (no corresponding element)
• Winter 2019/20 (no corresponding element)
• Winter 2020/21 (no corresponding element)
• Summer 2021 (no corresponding element)
• Winter 2021/22 (no corresponding element)
• Winter 2022/23
• Winter 2023/24

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.