Module Identifier MA21510 Module Title COMPLEX ANALYSIS Academic Year 2001/2002 Co-ordinator Professor Arthur Davies Semester Semester 2 Pre-Requisite MA11010 , MA11110 Mutually Exclusive MX31510 Course delivery Lecture 19 x 1 hour lectures Seminars / Tutorials 3 x 1 hour example classes Assessment Exam 2 Hours (written examination) 100% Resit assessment 2 Hours (written examination) 100%

#### General 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.

#### Learning outcomes

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

#### Syllabus

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.