# DEGREE REGULATIONS & PROGRAMMES OF STUDY 2015/2016

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# Undergraduate Course: Honours Complex Variables (MATH10067)

 School School of Mathematics College College of Science and Engineering Credit level (Normal year taken) SCQF Level 10 (Year 3 Undergraduate) Availability Available to all students SCQF Credits 20 ECTS Credits 10 Summary Core course for Honours Degrees involving Mathematics. This is a first course in complex analysis. Topics are: Analytic functions, Möbius transformations and the Riemann sphere, complex integration, series expansions, the residue calculus and its applications. In the 'skills' section of this course we will work on Mathematical reading and writing, although the skills involved are widely applicable to reading and writing technical and non-technical reports. Students will then use these skills complete a group project re- searching a topic connected with complex numbers or complex analysis and produce a written report. Course description Holomorphic functions: complex numbers, algebraic, geometric and topological properties of the complex plane, functions of a complex variable, differentiability and holomorphicity, Cauchy-Riemann equations, harmonic functions. Examples: polynomials, rational functions, exponential and related functions. Multivalued functions: the logarithm and complex powers, branches and an example of a Riemann surface. Holomorphic functions as mappings: Möbius transformations, the extended complex plane and the Riemann sphere, the cross-ratio. How to visualise functions of a complex variable. Complex integration: contour integrals, independence of path, the Cauchy Integral Theorem, the Cauchy Integral Formulae, Morera's Theorem, Lioville's Theorem and its applications, the Maximum modulus principle. Series expansions: holomorphic functions as analytic functions, Taylor and Laurent series, zeros, singularities, analytic continuation. Residue theory: the Cauchy Residue Theorem, the argument principle, Rouché's Theorem and applications to the evaluation of real integrals and of sums. Skills: The content will be chosen appropriate to the learning outcomes.
 Pre-requisites Students MUST have passed: Several Variable Calculus and Differential Equations (MATH08063) Co-requisites Prohibited Combinations Other requirements Students must not have taken : MATH10033 Complex Variable & Differential Equations or MATH10001 Complex Variable
 Pre-requisites None High Demand Course? Yes
 Academic year 2015/16, Available to all students (SV1) Quota:  None Course Start Semester 2 Timetable Timetable Learning and Teaching activities (Further Info) Total Hours: 200 ( Lecture Hours 35, Seminar/Tutorial Hours 10, Supervised Practical/Workshop/Studio Hours 10, Summative Assessment Hours 3, Programme Level Learning and Teaching Hours 4, Directed Learning and Independent Learning Hours 138 ) Additional Information (Learning and Teaching) Students must pass exam and course overall. Assessment (Further Info) Written Exam 80 %, Coursework 20 %, Practical Exam 0 % Additional Information (Assessment) Coursework 20%, Examination 80% Feedback Not entered Exam Information Exam Diet Paper Name Hours & Minutes Main Exam Diet S2 (April/May) Honours Complex Variables 3:00
 On completion of this course, the student will be able to: Ability to calculate proficiently with complex numbers, including solving algebraic equations.A good understanding of the notion of holomorphicity/analyticity in its many manifestations and ability to use their basic properties with confidence.Familiarity with elementary holomorphic functions, including multivalued functions, and the ability to work with them effectively.Ability to apply the basic integral theorems to the evaluation of sums and integrals and to the determination of number of poles and zeros of meromorphic functions.Ability to expand meromorphic functions (especially rational functions) in series and to deduce information about their singularities.
 Useful reading, not essential: (1) Saranson, Complex Function theory, 2nd Edition £30.95 ISBN 9780821844281: http://bookshop.blackwell.co.uk/jsp/id/Complex_Function_Theory/9780821844281 (2) Bak and Newman, Complex Analysis, 3rd ed. 2010 £49.99 ISBN 9781441972873: http://bookshop.blackwell.co.uk/jsp/id/Complex_Analysis/9781441972873 (3) Wilde, Lecture Notes in Complex Analysis, illustrated edition £46.00 ISBN 9781860946431: http://bookshop.blackwell.co.uk/jsp/id/Lecture_Notes_on_Complex_Analysis/9781860946431
 Graduate Attributes and Skills Not entered Study Abroad Not Applicable. Keywords HCoV
 Course organiser Prof José Figueroa-O'Farrill Tel: (0131 6)50 5066 Email: j.m.figueroa@ed.ac.uk Course secretary Mr Thomas Robinson Tel: (0131 6)50 4885 Email: Thomas.Robinson@ed.ac.uk
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