Module Details

MA302-6-SP-CO: Complex Variables And Applications

Year: 2016/17
Department: Mathematical Sciences
Essex credit: 15
ECTS credit: 7.5
Available to Study Abroad / Exchange Students: Yes
Full Year Module Available to Study Abroad / Exchange Students for a Single Term: No
Outside Option: Yes
Pre-requisites: MA206

Supervisor: Prof Peter Higgins, email
Teaching Staff: Prof Peter Higgins, email
Contact details: Miss Claire Watts, Department Manager, Tel. 01206 873040, email

Module is taught during the following terms
Autumn Spring Summer

Module Description

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


- 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.

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.

Learning and Teaching Methods

This course runs at 3 hours per week. There are 5 lectures and one class in every fortnight. In the Summer term 3 revision lectures are given.


20 per cent Coursework Mark, 80 per cent Exam Mark


The coursework consists of 2 tests worth 10% each.

Other details

Information about coursework deadlines can be found in the "Coursework Information" section of the Current Students, Useful Information Maths web pages: Coursework and Test Information

Exam Duration and Period

2:00 during Summer Examination period.

Other information

Available to Socrates /IP students spending all relevant terms at Essex.


  • Essential Reading:
  • M. J. Albowtiz and A. S. Fokas, Complex Variables: Introduction and Applications (2nd edition), Cambridge University Press (2003)
  • Recommended Reading:
  • J. E. Marsden and M. J. Hoffman, Basic Complex Analysis, W. H. Freeman (1999)
  • T. Needham, Visual Complex Analysis, Oxford University Press (1998)
  • L. I. Volkovyskii, G. G. Lunts and I. G. Aramanovich, A Collection of Problems on Complex Analysis, Dover Publications (1991)
  • Spiegel et. al., Schaums Outlines: Complex Variables (2nd Edition), McGraw Hill
  • E. Kreyszig, Advanced Engineering Maths, Wiley, Chapters 12-15

Further information