MM549: Topology and Complex Analysis

Study Board of Science

Teaching language: Danish or English depending on the teacher, but English if international students are enrolled
EKA: N300050102
Assessment: Second examiner: External
Grading: 7-point grading scale
Offered in: Odense
Offered in: Spring
Level: Bachelor

STADS ID (UVA): N300050101
ECTS value: 10

Date of Approval: 01-11-2022


Duration: 1 semester

Version: Archive

Entry requirements

None

Academic preconditions

Students taking the course are expected to know the content of the course Mathematical analysis in MM533.

Course introduction

Topological properties are applied in many areas of mathematics and the first objective of this course is to develop further the concepts, which are introduced in Mathematical and Numerical Analysis, in more advanced settings. The second objective of the course is to give the students a fundamental knowledge of the theory of analytic functions, which will enable them to use this important theory in other areas of Mathematics and Applied Mathematics.

The course builds on the knowledge acquired in the courses Calculus and Mathematical and Numerical Analysis, and gives an academic basis for studying the topics probability theory, measure and integration and Banach spaces and Hilbert- and Banach Spaces, that are part of the degree.

In relation to the competence profile of the degree it is the explicit focus of the course to:
  • have a fundamental understanding of the theory of topological spaces, complete metric spaces, function spaces, normal topological spaces and its applications.
  • have a fundamental understanding of the theory of analytic functions and its applications
  • be able to use the calculation of residues to compute important types of integrals
  • be able to expand the most important holomorphic functions into power series and expand meromorphic functions into Laurent series

Expected learning outcome

The learning objective of the course is that the student demonstrates the ability to:
  • in a written exam, apply the concepts and ideas, from the course syllabus, in concrete mathematical examples.
  • formulate the written presentation in a mathematically correct way and argue for the validity of achieved results in stringent mathematical language.

Content

The following main topics are contained in the course:
  • Topological spaces, including construction methods and concepts of continuity,  compactness and connectedness. 
  • Complete metric spaces
  • Function spaces
  • Normal topological spaces.
  • Power series
  • Analytic functions.
  • Cauchy's integral theorem and integral formulas.
  • The fundamental theorem of algebra.
  • Taylor- and Laurent series of analytic functions.
  • Poles and zeroes. The residue theorem and its applications to compute definite integrals.

Literature

See itslearning for syllabus lists and additional literature references.

Examination regulations

Exam element a)

Timing

June

Tests

Written exam

EKA

N300050102

Assessment

Second examiner: External

Grading

7-point grading scale

Identification

Full name and SDU username

Language

Normally, the same as teaching language

Duration

3 hours

Examination aids

All common aids are allowed e.g. books, notes, computer programmes which do not use internet etc. 

Internet is not allowed during the exam. However, you may visit the course site in itslearning to open system "DE-Digital Exam". If you wish to use course materials from itslearning, you must download the materials to your computer the day before the exam. During the exam you cannot be sure that all course materials is accessible in itslearning.    

ECTS value

10

Indicative number of lessons

92 hours per semester

Teaching Method

Teaching activities result in an estimated indicative distribution of the work effort of an average student in the following way:

  • Intro phase (lectures) - 56 hours
  • Training phase: 36 hours
  • Study phase: 20

Lectures will introduce general concepts and theory and exercise sessions will be devoted to learning the material in depth.
Other study activities include studying the course material and preparing the weekly exercises, individually or through group work.

Teacher responsible

Name E-mail Department
Jens Kaad kaad@imada.sdu.dk Analyse

Timetable

Administrative Unit

Institut for Matematik og Datalogi (matematik)

Team at Educational Law & Registration

NAT

Offered in

Odense

Recommended course of study

Profile Education Semester Offer period

Transition rules

Transitional arrangements describe how a course replaces another course when changes are made to the course of study. 
If a transitional arrangement has been made for a course, it will be stated in the list. 
See transitional arrangements for all courses at the Faculty of Science.