[The University of Leicester]

Department of Mathematics



Next: MA3131 Group Theory Up: ModuleGuide03-04 Previous: MA3101 Abstract Algebra

MA3121 Complex Analysis

MA3121 Complex Analysis

Credits: 20 Convenor: Dr. A. Baranov Semester: 2 (weeks 15 to 26)
Prerequisites: essential: MA1151, MA2101 desirable: MA2151
Assessment: Coursework: 10% Examination: 90%
Lectures: 36 Problem Classes: 10
Tutorials: none Private Study: 104
Labs: none Seminars: none
Project: none Other: none
Surgeries: none Total: 150

Subject Knowledge

Aims

This module aims to introduce to the theory of functions of a complex variable and to familiarize with its applications.

Learning Outcomes

Students should know and understand the key concepts of this module: holomorphic functions, path integrals, Taylor and Laurent series, singularities and residues of complex functions. Students should be able to explain the main proofs given in the lectures and be able to determine whether a complex function is differentiable, define and evaluate path integrals, find Taylor and Laurent expansions of complex functions, calculate residues and use the residue theorem to evaluate real integrals and sums of real series.

Methods

Lectures and problem classes.

Assessment

Marked problem sheets, written examination.

Subject Skills

Aims

To develop problem solving skills, written communication skills.

Learning Outcomes

Students will be able to use the techniques taught within the module to solve problems and be able to present written arguments in a coherent and logical form.

Methods

Class sessions, coursework, exam.

Assessment

Marked problem sheets, exam.

Explanation of Pre-requisites

The student will be assumed to be familiar with the general notion of continuity of a real function as well as other basic concepts from real analysis, such as differentiability of real functions, power series and integration.

Course Description

In many ways, the subject of Complex Analysis is aesthetically more pleasing than Real Analysis, several of the results being ``cleaner'' than their real counterparts. In this course, we begin with the study of analogues for complex functions of familiar properties of real functions, though differences in the two theories emerge as we proceed. Cauchy's theory of complex integration is developed, culminating in a number of remarkable results and strikingly beautiful applications. Towards the end of the course, the results from complex integration theory are used to evaluate certain real integrals and to sum certain real infinite series.

Syllabus

Review of complex numbers. Holomorphic functions. Cauchy-Riemann equations. Power series, radius of convergence, Taylor series. Path integrals, path length, smooth curves, closed chains. Cauchy's theorem and Cauchy's integral formula. Liouville's theorem. Fundamental theorem of algebra. Homologous and homotopic curves. Winding numbers. Types of singularities (removable, pole, essential). Laurent series and residues. Partial fractions. The residue theorem and its applications for evaluating real integrals and sums of certain real series.

Reading list

Recommended:

J. Bak and D. Newman, Complex Analyis, Springer.

I. Stewart and D. Tall, Complex Analyis, Cambridge University Press.

H.A. Priestley, Introduction to Complex Analysis, Oxford University Press.

Resources

Problem sheets, handouts, lecture rooms.

Module Evaluation

Module questionnaires, module review, year review.

Next: MA3131 Group Theory Up: ModuleGuide03-04 Previous: MA3101 Abstract Algebra

[University Home] [MCS Home] [University Index A-Z] [University Search] [University Help]

Author: C. D. Coman, tel: +44 (0)116 252 3902
Last updated: 2004-02-21
MCS Web Maintainer
This document has been approved by the Head of Department.
© University of Leicester.