Module Information

Module Identifier
Module Title
Geometry of Complex Numbers
Academic Year
Semester 1
External Examiners
  • Dr Andrew Hazel (Reader - University of Manchester)
Other Staff

Course Delivery

Delivery Type Delivery length / details
Lecture 22 x 1 Hour Lectures


Assessment Type Assessment length / details Proportion
Semester Exam 2 Hours   WRITTEN EXAMINATION  100%
Supplementary Exam 2 Hours   WRITTEN EXAMINATION  100%

Learning Outcomes

On successful completion of this module students should be able to:

Identify the properties of complex functions and describe their effects on objects in the z-plan (particularly the exponential, cosine and sine, multifunctions and logarithmic functions.

Demonstrate a broad understanding of the properties of Mobius transformations and their applications.

Illustrate the basic theory of conformal mapping and be able to apply it to solve problems.

Brief description

This course aims to develop the student's grasp of the geometric signficiance of complex transformations and their mapping properties. Students will solve problems with a number of transformations, including Mobius transformations and stereographic projections, which are important in a number of contemporary contexts.


Geometry of complex arithmetic: Euler's theorem revisited, applications, Euclidean geometry.

Complex functions as transformations: the exponential function, the cosine and sine functions, multifunctions and logarithmic functions.

Mobius transformations: Inversion, the Riemann sphere, sterographic projection, Mobius transformations, visualistion and classification, decomposition, automorphism.

The geometry of complex differentiation: Local description of mapps, conformality, physical applications.

Module Skills

Skills Type Skills details
Application of Number Necessary throughout.
Communication Written answers to exercises must be clear and well-structures. Good listening skills are essential to succussful progress in this course.
Improving own Learning and Performance Students will be expected to develop their own approach to time-management in their attitude to the completion of work on time, and in doing the necessary prepration between lectures.
Information Technology Students will be set exercises involving the use of computer and library facilities.
Personal Development and Career planning Completion of tasks (problem sheets) to set deadlines will aid personal development. The course will give clear indications of the range of possible opportunities in academic research to students who successfully complete it.
Problem solving Worksheet exercises will be set and marked. These will involve the derivation of appropriate proofs and the application of these results to solve physical and mathematical problems.
Research skills Compueter classes will allow students to explore the prooperties of various conformal transformations, these will include: exponential, cosine and sine, multifunctions and logarithmic functions, stereographic projection and Mobius transformations.
Subject Specific Skills
Team work


This module is at CQFW Level 6