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
Co-requisites:

Staff
Supervisor: Prof Peter Higginsi, email peteh@essex.ac.uk
Teaching Staff: Prof Peter Higginsi, email peteh@essex.ac.uk
Contact details: Miss Claire Watts, Departmental Administrator, Tel. 01206 873040, email cmwatts@essex.ac.uk

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.

Syllabus

- 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.

Assessment

20 per cent Coursework Mark, 80 per cent Exam Mark

Coursework

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.

Bibliography

  • 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