Complex Variables and Applications

The details
Mathematical Sciences
Colchester Campus
Undergraduate: Level 6
Monday 13 January 2020
Friday 20 March 2020
01 October 2019


Requisites for this module



Key module for

BSC G100 Mathematics,
BSC G102 Mathematics (Including Year Abroad),
BSC G103 Mathematics (Including Placement Year),
BSC G104 Mathematics (Including Foundation Year),
BSC G1G4 Mathematics with Computing (Including Year Abroad),
BSC G1G8 Mathematics with Computing (Including Foundation Year),
BSC G1GK Mathematics with Computing,
BSC G1IK Mathematics with Computing (Including Placement Year),
BSC G1F3 Mathematics with Physics,
BSC G1F4 Mathematics with Physics (Including Placement Year),
BSC GCF3 Mathematics with Physics (Including Year Abroad)

Module description

An introduction to complex analysis, up to and including evaluation of contour integrals using the Residue theorem.

Module aims

- Complex numbers: Cartesian and polar forms
- Lines, circles and regions in the complex plane
- Functions of a complex variable: analytic functions
- Cauchy's theorem (statement only)
- Cauchy's integral formula
- Derivatives of an analytic function
- Taylor's theorem
- Singularities : Laurent's theorem
- Residues: calculation of residues at poles
- Cauchy's residue theorem
- Jordan's lemma
- Calculation of definite integrals using residue theory.

Module learning outcomes

On successful completion of the course, students should be able to:
- express complex numbers in both cartesian and polar forms;
- identify curves and regions in the complex plane defined by simple formulae;
- determine whether and where a function is analytic;
- obtain appropriate series expansions of functions;
- evaluate residues at pole singularities;
- apply the Residue Theorem to the calculation of real integrals.

Module information

No additional information available.

Learning and teaching methods

This course runs at 3 lectures per week. In the Summer term 3 revision lectures are given.


This module does not appear to have any essential texts. To see non-essential items, please refer to the module's reading list.

Assessment items, weightings and deadlines

Coursework / exam Description Deadline Weighting
Written Exam Test 1
Written Exam Test 2
Exam 120 minutes during Summer (Main Period) (Main)

Overall assessment

Coursework Exam
20% 80%


Coursework Exam
0% 100%
Module supervisor and teaching staff
Professor Chris Saker (
Professor Christopher Saker (



External examiner

Dr Tania Clare Dunning
The University of Kent
Reader in Applied Mathematics
Available via Moodle
Of 37 hours, 33 (89.2%) hours available to students:
4 hours not recorded due to service coverage or fault;
0 hours not recorded due to opt-out by lecturer(s).


Further information
Mathematical Sciences

Disclaimer: The University makes every effort to ensure that this information on its Module Directory is accurate and up-to-date. Exceptionally it can be necessary to make changes, for example to programmes, modules, facilities or fees. Examples of such reasons might include a change of law or regulatory requirements, industrial action, lack of demand, departure of key personnel, change in government policy, or withdrawal/reduction of funding. Changes to modules may for example consist of variations to the content and method of delivery or assessment of modules and other services, to discontinue modules and other services and to merge or combine modules. The University will endeavour to keep such changes to a minimum, and will also keep students informed appropriately by updating our programme specifications and module directory.

The full Procedures, Rules and Regulations of the University governing how it operates are set out in the Charter, Statutes and Ordinances and in the University Regulations, Policy and Procedures.