This entry is from Winter semester 2020/21 and might be obsolete. You can find a current equivalent here.

# 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 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. Pablo Ramacher

## Contents

• Complex differentiability, Cauchy Riemann differential equations,
• Basic theory of curves (curve length, curvature, winding number) and curve integrals
• Cauchy integral theorems and consequences
• Isolated singularities, elementary holomorphic functions, meromorphic functions, Laurent series, residue theorem with applications,
• Submanifolds of R^n, classical vector analysis (gradient, divergence, rotation), differential forms,
• Integration on submanifolds, classical integral theorems (Stokes, Gauss, Ostrogradski ...), applications

## Qualification Goals

The students shall:

• learn complex-analytic methods for solving real-analytic problems,
• practice the handling of complex-differentiable functions, which are used in complex and algebraic geometry,
• be able to use integral theorems as a tool for describing various phenomena of mathematical physics (field theory, fluid mechanics, etc.),
• deepen the knowledge from the basic module Analysis and learn connections to algebra, geometry and topology,
• practice mathematical working methods (development of mathematical intuition and its formal justification, training of abstraction skills),
• improve their oral communication skills in the exercises by practicing free speech in front of an audience and during 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.