Module Identifier MA21510  
Module Title COMPLEX ANALYSIS  
Academic Year 2007/2008  
Co-ordinator Professor John Gough  
Semester Semester 2  
Other staff Professor John Gough, Dr Rolf Gohm  
Pre-Requisite MA11010 , MA11110  
Mutually Exclusive MX31510  
Course delivery Lecture   19 Hours. (19 x 1 hour lectures)  
  Seminars / Tutorials   3 Hours. (3 x 1 hour example classes)  
Assessment
Assessment TypeAssessment Length/DetailsProportion
Semester Exam2 Hours (written examination)  100%
Supplementary Assessment2 Hours (written examination)  100%

Learning outcomes

On completion of this module, a student should be able to:
1. compute efficiently with complex numbers;
2. state and use the Cauchy-Riemann equations;
3. determine whether a complex-valued function is analytic;
4 evaluate contour integrals;
5 state Cauchy's theorem and describe its concequences;
6 evaluate integrals using Cauchy's integral formula;
7. expand analytic functions as a Taylor series and a Laurent series;
8. evaluate real integrals using the theory of residues.

Brief description

Complex analysis is the study of complex valued functions of a complex variable. It is, on the one hand, a fruitful area of pure mathematics exhibiting many elegant and surprising results, while, on the other, the theory has numerous applications in many branches of mathematics and engineering. The important role of complex variables in aplied mathematics, for instance, is partly due to the use of the theory of residues in the evaluation of certain real integrals and the application of conformal mapping in hydrodynamics and problems in potential theory.

Aims

The aim of the module is to study the theoretical foundations of complex variable theory and to develop skills in the application of this theory to particular problems. These skills are a necessary prerequisite to the study of some topics in other modules in the department.

Content

1. Revision of the Elementary Properties of Complex Numbers.
2. Cauchy-Riemann Equations. Analytic functions. Necessary and sufficient conditions for a function to be analytic.
3. Contour Integration. The fundamental theorem of integration.
4. Cauchy's theorem. Cauchy's integral formula, including the general version.
5. Taylor series.
6. Laurent series.
7. Theory of residues.

Reading Lists

Books
** Recommended Text
Nehari,Z (1961) Introduction to Complex Analysis Allyn and Bacon 0061011974
Wunsch, A D (1994) Complex Variables with Applications 2nd. Addison Wesley 0201122995
** Supplementary Text
Beardon, A F (1979) Complex Analysis Wiley 0471996726
Jameson, G J O (1970) A First Course on Complex Functions Chapman and Hall 0412097109
Tall, D O Functions of a Complex Variable Routledge 0710086555

Notes

This module is at CQFW Level 5