# INTRODUCTION TO COMPLEX ANALYSIS - 2020/1

Module code: MAT2052

## Module Overview

This module provides an introduction to the theory of functions of a complex variable, known as Complex Analysis, which is a fundamental topic in modern mathematics and is widely used in many branches of mathematics and physics.

### Module provider

Mathematics

PRINSLOO Andrea (Maths)

### Module cap (Maximum number of students): N/A

Independent Learning Hours: 106

Lecture Hours: 33

Tutorial Hours: 11

Semester 2

None.

## Module content

Indicative content includes:

• Complex functions on subsets of the complex plane;

• Differentiation of complex functions, the Cauchy-Riemann equations and harmonic functions;

• Common holomorphic complex functions and branches of multifunctions;

• Paths and homotopies on subsets of the complex plane;

• Contour integration;

• Cauchy's theorem and Cauchy's integral formulae, and their applications;

• Liouville’s theorem and the Fundamental Theorem of Algebra;

• Taylor and Laurent series expansions;

• Singularities of complex functions, and residues at simple and higher order poles;

• Cauchy's residue theorem and its applications.

## Assessment pattern

Assessment type Unit of assessment Weighting
School-timetabled exam/test In-Semester Test 50 mins 20
Examination Examination (2 hour) 80

N/A

## Assessment Strategy

The assessment strategy is designed to provide students with the opportunity to demonstrate:

• an understanding of and the ability to interpret and manipulate mathematical statements;

• subject knowledge through the recall of key definitions, theorems and their proofs;

• analytical ability through the solution of unseen problems in the in-semester test and examination.

The summative assessment for this module consists of:

• one two hour examination at the end of Semester 2 – worth 80% of the module mark;

• one in-semester test – worth 20% of the module mark.

The formative assessment and feedback for this module includes:

• written feedback on two marked pieces of unassessed coursework;

• verbal feedback at weekly tutorial sessions.

## Module aims

• The aim of this module is to introduce students to the theory of complex functions of a complex variable -- including key concepts such as complex differentiation, integration along contours on the complex plane, series expansions of complex functions, and the calculus of residues. At the end of the module, students should have gained a thorough understanding of the theory of complex functions and should be able to apply this knowledge in a variety of contexts.

## Learning outcomes

 Attributes Developed 001 Quote, derive or apply the Cauchy-Riemann equations. KCT 002 Compute contour integrals of continuous complex functions. KCT 003 Quote, derive or apply some or all of the following results: Cauchy's theorem, Cauchy's integral formulae and Liouville's theorem. KCT 004 Quote, derive or apply Taylor's theorem or Laurent's theorem; and compute Taylor or Laurent series expansions of complex functions. KCT 005 Identify and classify the singularities of complex functions, and compute the residues of simple or higher order poles. KCT 006 Quote, derive or apply Cauchy's residue theorem, and use the calculus of residues to compute real integrals. KCT

Attributes Developed

C - Cognitive/analytical

K - Subject knowledge

T - Transferable skills

P - Professional/Practical skills

## Methods of Teaching / Learning

The learning and teaching strategy is designed to introduce students to the theory of complex functions of a complex variable, and its applications.

The learning and teaching methods include:

• 3 hours of lectures per week x 11 weeks – material supplementing the module notes will be presented on blackboards/whiteboards in these lectures, with Q & A opportunities for students.

• 1 hour of tutorials per week x 11 weeks – students will work on exercises in this interactive tutorial session, with Q & A opportunities and partial solutions presented on blackboards/whiteboards.

Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include in-class tests where one or more of these are an assessment on the module. In-class tests are scheduled/organised separately to taught content and will be published on to student personal timetables, where they apply to taken modules, as soon as they are finalised by central administration. This will usually be after the initial publication of the teaching timetable for the relevant semester.

Upon accessing the reading list, please search for the module using the module code: MAT2052

None.

## Programmes this module appears in

Programme Semester Classification Qualifying conditions
Mathematics with Music BSc (Hons) 2 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics MMath 2 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics with Statistics MMath 2 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics with Statistics BSc (Hons) 2 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics BSc (Hons) 2 Optional A weighted aggregate mark of 40% is required to pass the module
Financial Mathematics BSc (Hons) 2 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics and Physics BSc (Hons) 2 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics and Physics MPhys 2 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics and Physics MMath 2 Optional A weighted aggregate mark of 40% is required to pass the module

Please note that the information detailed within this record is accurate at the time of publishing and may be subject to change. This record contains information for the most up to date version of the programme / module for the 2020/1 academic year.