	Department of Mathematics & Computer Science

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MA3121 Complex Analysis

MA3121(=MC383) Complex Analysis

Credits: 20

Convenor: Dr A. Baranov

Semester: 2

Prerequisites:	essential: MA1151(=MC146), MA2101(=MC248), MA2151(=MC240)
Assessment:	Regular coursework: 10%	Three hour exam: 90%

Lectures:	36	Problem Classes:	10
Tutorials:	none	Private Study:	104
Labs:	none	Seminars:	none
Project:	none	Other:	none
Surgeries:	none	Total:	150

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.

Aims

To help the student to develop an appreciation of the rigorous development of this remarkable subject, and an understanding of the fundamental results of the subject.

Objectives

To enable the student to :

Determine whether a complex function is differentiable.

Define and evaluate path integrals.

Prove and use Cauchy's integral theorem and Cauchy's integral formula.

Prove the Fundamental Theorem of Algebra.

Prove Taylor's theorem and appreciate that a function which is differentiable in a neighbourhood of a point has a Taylor series expansion about that point.

Prove Laurent's theorem and appreciate that a function which is differentiable in a punctured neighbourhood of a point has a Laurent expansion about that point.

Determine whether a function has a singularity at a given point, and the nature and order of singularities.

Calculate residues and use the residue theorem to evaluate integrals around closed paths.

Use the residue theorem to evaluate certain real integrals, to evaluate the sum of certain real series and to assist in certain partial fraction decompositions.

Transferable Skills

This course should assist the student in developing skills of rigorous and precise mathematical writing and expression.

Syllabus

Review of complex numbers. Basic topological concepts. 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

Essential:

Background:

Details of Assessment

The final assessment of this module will consist of 10% coursework and 90% from a three hour examination during the Summer exam period. The 10% coursework contribution will be determined by students' solutions to coursework problems. The examination paper will contain 8 questions with full marks on the paper obtainable from 5 complete answers.