This entry is from Winter semester 2022/23 and might be obsolete. You can find a current equivalent here.

# 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 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

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.

## Applicability

This module is an export-only module, and can as such only be used in other study programs, not in B.Sc. Mathematics.