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2013/14 - ECM3703 - Complex Analysis Module Title Complex Analysis Credit Value: 15 Module Code ECM3703 Module Convener: Dr Robin Chapman (Coordinator) DURATION: TERM 1 2 3 DURATION: Weeks 0 11 weeks 0 Number of Students Taking Module (anticipated) 47 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. Date of Last Revision: October 9th 2013 2013/14 - ECM3703 - Complex Analysis 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 33.00 Date of Last Revision: October 9th 2013 Guided Independent Study 117.00 Placement / Study Abroad 0.00 2013/14 - ECM3703 - Complex Analysis 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 20 Written Exams 80 Practical Exams 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. Date of Last Revision: October 9th 2013 2013/14 - ECM3703 - Complex Analysis 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.ukReading list for this module: Author Title Edition Publisher Year ISBN Stewart I. & Tall D. Complex Analysis (the hitchhiker's guide to the plane) Cambridge University Press 1983 000-0-521-28763-4 Priestley H.A. Introduction to Complex Analysis Oxford University Press 2003 000-0-198-53428-0 Howie, John M Complex Analysis Springer 2003 000-1-852-33733-8 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 CREDIT VALUE 15 ECTS VALUE 7.5 PRE-REQUISITE MODULES ECM2701 CO-REQUISITE MODULES None NQF LEVEL (FHEQ) 3 (NQF level 6) AVAILABLE AS DISTANCE LEARNING No ORIGIN DATE Monday 19 November 2012 LAST REVISION DATE Wednesday 09 October 2013 KEY WORDS SEARCH Complex analysis; contour integration; residue calculus. Date of Last Revision: October 9th 2013