# Functions of a Complex Variable - MA6517

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## Module delivery information

Location Term Level1 Credits (ECTS)2 Current Convenor3 2020 to 2021
Canterbury
Autumn 6 15 (7.5) DR AF Loureiro

## Overview

Revision of complex numbers, the complex plane, de Moivre's and Euler's theorems, roots of unity, triangle inequality

Sequences and limits: Convergence of a sequence in the complex plane. Absolute convergence of complex series. Criteria for convergence. Power series, radius of convergence

Complex functions: Domains, continuity, complex differentiation. Differentiation of power series. Complex exponential and logarithm, trigonometric, hyperbolic functions. Cauchy-Riemann equations

Complex Integration: Jordan curves, winding numbers. Cauchy's Theorem. Analytic functions. Liouville's Theorem, Maximum Modulus Theorem

Singularities of functions: poles, classification of singularities. Residues. Laurent expansions. Applications of Cauchy's theorem. The residue theorem. Evaluation of real integrals.

Possible additional topics may include Rouche’s Theorem, other proofs of the Fundamental Theorem of Algebra, conformal mappings, Mobius mappings, elementary Riemann surfaces, and harmonic functions.

42 hours

## Method of assessment

80% examination, 20% coursework

H.A. Priestley, Introduction to Complex Analysis, Oxford University Press, 2003
M.R. Spiegel, Complex Variables, McGraw-Hill, 1964
J.H. Mathews & R.W Howell, Complex Analysis for Mathematics and Engineering, Jones and Bartlett 5th ed., 2006
I Stewart & D Tall, Complex Analysis, Cambridge, 2004

See the library reading list for this module (Canterbury)

## Learning outcomes

The intended subject specific learning outcomes:
On successfully completing the module students will be able to:
1 demonstrate systematic understanding of key aspects of complex analysis;
2 demonstrate the capability to deploy established approaches accurately to analyse and solve problems using a reasonable level of skill in calculation and manipulation of the material in the following areas: power series, analytic functions, contour integrals, singularities, residues, Taylor and Laurent series, the residue theorem;
3 apply key aspects of complex analysis in well-defined contexts, showing judgement in the selection and application of tools and techniques.

The intended generic learning outcomes:
On successfully completing the module students will be able to:
1 manage their own learning and make use of appropriate resources;
2 understand logical arguments, identifying the assumptions made and the conclusions drawn;
3 communicate straightforward arguments and conclusions reasonably accurately and clearly;
4 manage their time and use their organisational skills to plan and implement efficient and effective modes of working;
5 solve problems relating to qualitative and quantitative information;
6 make competent use of information technology skills such as online resources (Moodle), internet communication;
7 communicate technical material competently;
8 demonstrate an increased level of skill in numeracy and computation;
9 demonstrate the acquisition of the study skills needed for continuing professional development.

## Notes

1. Credit level 6. Higher level module usually taken in Stage 3 of an undergraduate degree.
2. ECTS credits are recognised throughout the EU and allow you to transfer credit easily from one university to another.
3. The named convenor is the convenor for the current academic session.