Department of Mathematics & Computer Science

MC383 Complex Analysis

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 contour 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 simple 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. Complex functions of a real variable. Complex power series and some functions defined by them. Radius of converegence, term-by-term differentiability of functions defined by a power series with positive radius of convergence. Cauchy-Riemann equations.

Contour integrals. Simple closed paths. Jordan contours. Star-shaped domains. Cauchy's integral theorem. Cauchy's integral formula. Taylor's theorem. Entire functions. Liouville's theorem and its application to the fundamental theorem of algebra. Laurent series. Singularities, poles, residues. The order of a function at a singularity. Orders of poles. The residue theorem and some techniques for calculating residues. Rouché's theorem. Use of the residue theorem to evaluate real integrals and to evaluate sums of certain real series. Partial fractions.

Reading list

Essential:

Background:

Recommended:

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

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

Details of Assessment

Author: S. J. Ambler, tel: +44 (0)116 252 3884
Last updated: 2001-09-20
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