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DEGREE REGULATIONS & PROGRAMMES OF STUDY 2015/2016

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DRPS : Course Catalogue : School of Mathematics : Mathematics

Undergraduate Course: Honours Complex Variables (MATH10067)

Course Outline
SchoolSchool of Mathematics CollegeCollege of Science and Engineering
Credit level (Normal year taken)SCQF Level 10 (Year 3 Undergraduate) AvailabilityAvailable to all students
SCQF Credits20 ECTS Credits10
SummaryCore 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, infinite products and Weierstrass's factorisation 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: 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: Complex integration; Fundamental theorem of calculus; Cauchy's Theorem; Evaluation of integrals.

Cauchy's Integral Formulae and applications: Liouville's Theorem and the fundamental theorem of algebra; Maximum principle; Taylor's Theorem; More evaluation of integrals.

Laurent expansions and the residue theorem: The Laurent expansion and isolated singularities; The residue theorem and calculation of residues; Evaluation of integrals and sums; Counting zeros and ples of meromorphic functions; The Argument principle; Rouché's theorem.

Analytic functions: Analytic continuation; Infinite products; Weierstrass's factor theorem; The gamma function.

Skills: The content will be chosen appropriate to the learning outcomes.
Entry Requirements (not applicable to Visiting Students)
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
Information for Visiting Students
Pre-requisitesNone
High Demand Course? Yes
Course Delivery Information
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 Variables3:00
Learning Outcomes
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.
Reading List
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
Additional Information
Graduate Attributes and Skills Not entered
Study Abroad Not Applicable.
KeywordsHCoV
Contacts
Course organiserProf José Figueroa-O'Farrill
Tel: (0131 6)50 5066
Email:
Course secretaryMr Brett Herriot
Tel: (0131 6)50 4885
Email:
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© Copyright 2015 The University of Edinburgh - 27 July 2015 11:35 am