# 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

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