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

Level, degree of commitment in original study programme Intermediate module, required 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: 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 study course B.Sc. Mathematics.
Original study programme B.Sc. Mathematik / Mathematik Aufbaumodule (Kernfächer)
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.


Recommended Reading

  • 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.



Please note:

This page describes a module according to the latest valid module guide in Wintersemester 2022/23. 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:

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.