The aim of this module is to introduce fundamental concepts in complex analysis. The following topics are discussed: types of convergence of series of functions, Taylor series and zeroes, differentiation, complex exponential, trigonometric and logarithmic functions, Laurent series and isolated singularities, integration along a path, Cauchy’s Theorem and Integral Formula, Liouville’s Theorem, the Residue Theorem, applications and other topics.

Complex analysis is a fascinating and highly useful part of Mathematics, and has applications in, among others, Functional Analysis, Number Theory, Approximation Theory, Applied Mathematics and Engineering.

**Module information**

- 21539 324 (16) Mathematics 324; language specification: A
- Third year, first semester of the Programme in the Mathematical Sciences
- Prerequisite pass modules (PP ³ 50): Mathematics 214, 244
- Classnotes will be provided
- There are several books on the subject in the university library

**Lecturer
**

Dr A Muller, Room 1021, Department of Mathematics

**Learning opportunities**

The learning material is covered during the lecture periods. During the tutorial periods problems are solved under supervision.

**Assessment**

- Method: Obtain class mark (KP ³ 40) and pass exam (final mark PP ³ 50).
- The class mark is determined by the class test.
- Formula for final mark: PP = 0,4 KP + 0,6 EP, where EP = exam mark.
- The dates and times of the class test and exam are published on the University’s web page. For more information on exam regulations, see the Yearbook of the University, Parts 1 and 5.

**Rationale**

This module is presented within the Programme in the Mathematical Sciences. It contributes to providing a basic training in Mathematics and is one of the core modules required in order to obtain the B.Sc. degree with a major in Mathematics.

**Outcomes**

A student who passed this module should be equipped with the knowledge and comprehension of the basic concepts in metric spaces, as well as of complex analysis. Specifically such a student should have the following knowledge and skills:

- Can determine Taylor series and circles of convergence of complex functions. Can determine Laurent series and domains of convergence of complex functions.
- Can identify and classify the zeroes and isolated singularities of a complex function.
- Know and understand the properties of the complex exponential and trigonometric functions. Can determine arguments and complex logarithms.
- Can investigate the differentiability and determine the derivative of a complex function using the theory of the Cauchy Riemann equations.
- Can integrate complex functions along paths using different techniques.
- Can compute residues and use the Residue Theorem to determine complex integrals, as well as improper integrals and/or infinite series.