THE UNIVERSITY of EDINBURGH

DEGREE REGULATIONS & PROGRAMMES OF STUDY 2018/2019

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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, Moebius 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: Moebius 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, Rouche's Theorem and applications to the evaluation of real integrals and of sums.

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) OR Introductory Fields and Waves (PHYS08053)
Co-requisites
Prohibited Combinations Other requirements None
Information for Visiting Students
Pre-requisitesVisiting students are advised to check that they have studied the material covered in the syllabus of each pre-requisite course before enrolling.
High Demand Course? Yes
Course Delivery Information
Academic year 2018/19, 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 )
Assessment (Further Info) Written Exam 80 %, Coursework 20 %, Practical Exam 0 %
Additional Information (Assessment) Coursework 20%, Examination 80%

Students must pass exam and course overall.
Feedback Not entered
Exam Information
Exam Diet Paper Name Hours & Minutes
Main Exam Diet S2 (April/May)Honours Complex Variables3:00
Learning Outcomes
On completion of this course, the student will be able to:
  1. Calculate proficiently with complex numbers, including solving algebraic equations.
  2. Understand the notion of holomorphicity/analyticity in its many manifestations and ability to use their basic properties with confidence.
  3. Be familiar with elementary holomorphic functions, including multivalued functions, and the ability to work with them effectively.
  4. 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.
  5. Expand meromorphic functions (especially rational functions) in series and to deduce information about their singularities.
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
KeywordsHCoV
Contacts
Course organiserDr Richard Gratwick
Tel: (0131 6)51 3411
Email: R.Gratwick@ed.ac.uk
Course secretaryMiss Sarah McDonald
Tel: (0131 6)50 5043
Email: sarah.a.mcdonald@ed.ac.uk
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