# Mathematics

## ECM3703 - Complex Analysis (2015)

MODULE TITLE CREDIT VALUE Complex Analysis 15 ECM3703 Dr Gihan Marasingha (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 11 0 0
 Number of Students Taking Module (anticipated) 29
DESCRIPTION - summary of the module content

Complex analysis is one of the most beautiful and complete theories in mathematics.  Invented in the 19th century by Gauss, Cauchy and Riemann, it has developed into a powerful tool, indispensable to all mathematicians, pure and applied. The skill of computing integrals by means of residue calculus is a major tool in integration and it is an invaluable tool in physics and engineering. In this module, you will see the theory developed in a logical way, emphasizing the wide range of useful applications. You will learn to develop complex analysis in a logical and satisfying way that provides insight into the geometric and topological foundations of the subject. This skill is useful to engineers as it helps you to understand that taking a different viewpoint may render a difficult problem easy.

Prerequisite module: ECM2701 or equivalent

AIMS - intentions of the module

The main aim of this module is to inspire a genuine engagement with complex analysis and its applications in other branches of mathematics

INTENDED LEARNING OUTCOMES (ILOs) (see assessment section below for how ILOs will be assessed)

On successful completion of this module, you should be able to:

Module Specific Skills and Knowledge:
1 understand basic topological concepts like connectedness;
2 comprehend the theory of holomorphic functions, their power-series and integrals;
3 compute contour integrals and to apply this to real analysis.
Discipline Specific Skills and Knowledge:
4 compute integrals by means of residue calculus which is a major tool in integration and an invaluable tool in physics, engineering etc;
5 recognise a couple of useful techniques for the computation of integrals with complex methods.
Personal and Key Transferable/ Employment Skills and  Knowledge:
6 appreciate that taking a different viewpoint may render a difficult problem easy;
7 value the use of theoretical knowledge in concrete problems.

SYLLABUS PLAN - summary of the structure and academic content of the module

- geometry of the complex plane;
- review of functions of a complex variable: polynomials, rational functions, elementary transcendental functions;
- open and closed sets in the complex plane, continuous curves, path connectedness, domains;
- regular functions on a domain;
- continuity, differentiability, Cauchy-Riemann equations;
- complex power series;
- radius of convergence;
- properties of power series, including differentiation within circle of convergence;
- piecewise continuously differentiable curves;
- contour integrals;
- primitives;
- existence of primitives for simply connected domains;
- Cauchy's Theorem;
- Cauchy integral formulae;
- Taylor series and Laurent series;
- isolated singularities;
- poles, removable and essential singularities;
- maximum modulus principle, Liouville's theorem, fundamental theorem of algebra, meromorphic functions, residue theorem;
- Rouche's theorem, principle of the argument;
- applications to definite integrals, summation of series and the location of zeros.

LEARNING AND TEACHING
LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)
 Scheduled Learning & Teaching Activities Guided Independent Study Placement / Study Abroad 33 117 0
DETAILS OF LEARNING ACTIVITIES AND TEACHING METHODS
 Category Hours of study time Description Scheduled learning and teaching activities 33 Lectures/example classes Guided independent study 117 Working on set mathematical problems

ASSESSMENT
FORMATIVE ASSESSMENT - for feedback and development purposes; does not count towards module grade
Form of Assessment Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Not applicable

SUMMATIVE ASSESSMENT (% of credit)
 Coursework Written Exams 20 80
DETAILS OF SUMMATIVE ASSESSMENT
Form of Assessment % of Credit Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Written exam – closed book 80 2 hours All Specific comments by markers and general comments on website
Coursework – example sheets 20 60 hours All Specific comments by markers and general comments on website

DETAILS OF RE-ASSESSMENT (where required by referral or deferral)
Original Form of Assessment Form of Re-assessment ILOs Re-assessed Time Scale for Re-reassessment
All above Written exam (100%) All August Ref/Def period

RE-ASSESSMENT NOTES

If a module is normally assessed entirely by coursework, all referred/deferred assessments will normally be by assignment.

If a module is normally assessed by examination or examination plus coursework, referred and deferred assessment will normally be by examination. For referrals, only the examination will count, a mark of 40% being awarded if the examination is passed. For deferrals, candidates will be awarded the higher of the deferred examination mark or the deferred examination mark combined with the original coursework mark.

RESOURCES
INDICATIVE LEARNING RESOURCES - The following list is offered as an indication of the type & level of
information that you are expected to consult. Further guidance will be provided by the Module Convener

ELE – http://vle.exeter.ac.uk

Reading list for this module:

Type Author Title Edition Publisher Year ISBN Search
Set Howie, John M Complex Analysis Springer 2003 000-1-852-33733-8 [Library]
Set Priestley H.A. Introduction to Complex Analysis Oxford University Press 2003 000-0-198-53428-0 [Library]
Set Spiegel M.R. Schaum's outline of theory and problems of complex variables : with an introduction to conformal mapping and its appreciation. McGraw-Hill 1981 000-0-070-84382-1 [Library]
Set Stewart I. & Tall D. Complex Analysis (the hitchhiker's guide to the plane) Cambridge University Press 1983 000-0-521-28763-4 [Library]
CREDIT VALUE ECTS VALUE 15 7.5
PRE-REQUISITE MODULES ECM2701
NQF LEVEL (FHEQ) AVAILABLE AS DISTANCE LEARNING 6 No Friday 09 January 2015 Friday 09 January 2015
KEY WORDS SEARCH Complex analysis; contour integration; residue calculus.