German original

# 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 Translation missing. German original: Studienleistung: Erreichen von mindestens 50 Prozent der Punkte aus den wöchentlich zu bearbeitenden Übungsaufgaben. Prüfungsleistung: Klausur oder mündliche Prüfung 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

Translation is missing. Here is the German original:

Keine. Empfohlen werden die Kompetenzen, die in den Basismodulen vermittelt werden.

• 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.