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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, Liouville'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: Proofs and Problem Solving (MATH08059) OR Accelerated Proofs and Problem Solving (MATH08071) AND Several Variable Calculus and Differential Equations (MATH08063) OR Introductory Fields and Waves (PHYS08053)
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 2019/20, 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%

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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. Identify, manipulate and calculate with holomorphic functions.
  2. Construct and calculate with holomorphic branches of multivalued functions.
  3. Identify and construct conformal maps, and determine images under conformal maps of subsets of the extended complex plane.
  4. Compute series expansions of complex functions, and identify and classify singularities of complex functions.
  5. Apply the integral theorems, in particular to the evaluation of real sums and integrals.
Reading List
Useful reading, not essential:
(1) Sarason, Complex Function Theory, 2nd Edition 30.95 ISBN 9780821844281

(2) Bak and Newman, Complex Analysis, 3rd ed. 2010 49.99 ISBN 9781441972873

(3) Wilde, Lecture Notes in Complex Analysis, illustrated edition 46.00 ISBN 9781860946431

(4) Priestley, Introduction to Complex Analysis, 2nd edition, 31.49, ISBN 9780198525622
Additional Information
Graduate Attributes and Skills Not entered
Course organiserDr Richard Gratwick
Tel: (0131 6)51 3411
Course secretaryMiss Sarah McDonald
Tel: (0131 6)50 5043
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