# DEGREE REGULATIONS & PROGRAMMES OF STUDY 2014/2015 Archive for reference only THIS PAGE IS OUT OF DATE

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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, contour integrals, Laurent series, residues, integral transforms, conformal mapping, Weierstrass's factorization theorem. 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 First properties of holomorphic functions (5h): The complex plane; Algebraic properties of C; Simple subsets of C; Functions of a complex variable; complex differentiability and the Cauchy-Riemann equations; Holomorphic functions as mappings; Complex logarithms and roots; Power series. Complex integration, Cauchy's theorem and its consequences (4h): Complex integration; Fundamental theorem of calculus; Cauchy's Theorem; Evaluation of integrals. Cauchy's Integral Formulae and applications (5h): Liouville's Theorem and the fundamental theorem of algebra; Maximum principle.; Taylor's Theorem; More evaluation of integrals; Counting zeros and poles of meromorphic functions; Rouché's theorem. Laurent expansions and the residue theorem (4h): The Laurent expansion and isolated singularities; The residue theorem and calculation of residues; Evaluation of integrals. Applications to PDEs (3h): The Fourier and Laplace transforms and their inverses; Fresnel integrals; Applications. Conformal mapping (5h): Basic facts; Examples of conformal mapping; Conformal maps of the unit disc; Advanced mappings, and some applications. Analytic functions (6h): Analytic continuation; Infinite products; Weierstrass's factor theorem; The gamma function. Skills: The content will be chosen appropriate to the learning outcomes. (10h)
 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
 Academic year 2014/15, 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
 1. Knowledge of basic properties of analytic functions of a complex variable, including power series expansions, Laurent expansions, and Liouville's theorem. 2. Knowledge of the fundamental integral theorems of complex analysis and its applications: counting zeros and poles of meromorphic functions, and Rouché's theorem. 3. Ability to use residue calculus to perform definite integrals. 4. Knowledge of some of the relations between analytic functions and PDE, e.g. relation to harmonic functions, the maximum principle. 5. Familiarity with the Fourier and Laplace transforms. 6. The idea of conformal mapping, use of fractional linear transformations and more advanced mappings. Knowledge of applications of conformal mappings. 7. Knowledge of analytic functions: analytic continuation, infinite products, Weierstrass's factor theorem, and the gamma function. 8. Improved ability to read and understand sustained mathematical arguments. 9. Improved ability to write mathematics clearly and to professional standard. 10. A good understanding of the importance and conventions of citation and reference in mathematical writing. 11. Improved general skills in reading and writing technical material. 12. Deeper understanding of some representative material related to the lectured part of the course.
 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 Mrs Kathryn Mcphail Tel: (0131 6)50 4885 Email: k.mcphail@ed.ac.uk
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