# Complex Analysis and Vector Analysis (dt. Funktionentheorie und Vektoranalysis)

 Level, degree of commitment Advanced module, depends on importing study program 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. Subject, Origin Mathematics, B.Sc. Mathematics Duration,frequency One semester, each summer semester Person in charge of the module's outline Prof. Dr. Ilka Agricola, Prof. Dr. Thomas Bauer, Prof. Dr. Oliver Goertsches, Prof. Dr. Pablo Ramacher

## Contents

• Complex differentiability, Cauchy-Riemann differential equations,
• Fundamentals of curve theory (curve length, curvature, number of turns) and curve integrals.
• Cauchy integral theorems and corollaries
• Isolated singularities, elementary holomorphic functions, meromorphic functions, Laurent series, residue theorem with applications,
• submanifolds of R^n (this topic can be treated by the lecturer alternatively in Analysis II)
• classical vector analysis (gradient, divergence, rotation), differential forms,
• integration on submanifolds, classical integral theorems (Stokes, Gauss, Ostrogradski ...), applications.

## Qualification Goals

Students will be able to,

• use complex-analytic methods to solve problems in real analysis,
• deal with complex-differentiable functions used in complex and algebraic geometry,
• apply integral theorems as a tool for describing various phenomena in mathematical physics (field theory, fluid mechanics, etc.),
• to reflect knowledge from the basic module Analysis and to consider it in connection with algebra, geometry and topology,
• to proceed according to mathematical working methods (developing mathematical intuition and its formal justification, training the ability to abstract),
• can speak freely about scientific content, both in front of an audience and in a discussion.

## Prerequisites

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

• Fischer, W., Lieb, I.: Funktionentheorie: Komplexe Analysis in einer Veränderlichen, Vieweg.
• Remmert, R., Schumacher, G.: Funktionentheorie I,II, Berlin: Springer.
• Klaus Jänich: Funktionentheorie, Springer-Verlag.
• Ilka Agricola, Thomas Friedrich: Vektoranalysis, Vieweg-Verlag 2010.