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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
Course typeStandard AvailabilityAvailable to all students
Credit level (Normal year taken)SCQF Level 10 (Year 3 Undergraduate) Credits20
Home subject areaMathematics Other subject areaNone
Course website None Taught in Gaelic?No
Course descriptionCore 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.
Entry Requirements (not applicable to Visiting Students)
Pre-requisites Students MUST have passed: Several Variable Calculus and Differential Equations (MATH08063)
Prohibited Combinations Other requirements Students must not have taken :
MATH10033 Complex Variable & Differential Equations or
MATH10001 Complex Variable
Additional Costs None
Information for Visiting Students
Displayed in Visiting Students Prospectus?No
Course Delivery Information
Delivery period: 2013/14 Semester 2, Available to all students (SV1) Learn enabled:  Yes Quota:  None
Web Timetable Web Timetable
Course Start Date 13/01/2014
Breakdown of 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 Notes Students must pass exam and course overall.
Breakdown of Assessment Methods (Further Info) Written Exam 80 %, Coursework 20 %, Practical Exam 0 %
Exam Information
Exam Diet Paper Name Hours:Minutes
Main Exam Diet S2 (April/May)Honours Complex Variables3:00
Summary of Intended 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
9. Improved ability to write mathematics clearly and to professional
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.
Assessment Information
See 'Breakdown of Assessment Methods' and 'Additional Notes' above.
Special Arrangements
Additional Information
Academic description Not entered
Syllabus 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)
Transferable skills Not entered
Reading list Useful books are:
(1) Complex Analysis, Stein & Shakarchi, Princeton University Press, 2003.
(2) Introduction to Complex Analysis, Priestley, OUP, 2003.
(3) Complex Variables, introduction and applications, Ablowitz and Fokas, CUP, 2003.
(4) Complex Variables and their applications, Osborne, Addison-Wesley, 1999.
(5) Lecture Notes on Complex Analysis, Wilde, Imperial College Press, 2006.
Study Abroad Not Applicable.
Study Pattern See 'Breakdown of Learning and Teaching activities' above.
Course organiserProf Jose Figueroa-Ofarrill
Tel: (0131 6)50 5066
Course secretaryMrs Kathryn Mcphail
Tel: (0131 6)50 4885
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