Undergraduate Course: Honours Complex Variables (MATH10067)
Course Outline
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 nontechnical 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 CauchyRiemann 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)

Entry Requirements (not applicable to Visiting Students)
Prerequisites 
Students MUST have passed:
Several Variable Calculus and Differential Equations (MATH08063)

Corequisites  
Prohibited Combinations  
Other requirements  Students must not have taken :
MATH10033 Complex Variable & Differential Equations or
MATH10001 Complex Variable 
Information for Visiting Students
Prerequisites  None 
Course Delivery Information

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  
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. 
Keywords  HCoV 
Contacts
Course organiser  Prof José FigueroaO'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 

