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, Möbius 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 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 
Holomorphic functions: complex numbers, algebraic, geometric and topological properties of the complex plane, functions of a complex variable, differentiability and holomorphicity, CauchyRiemann 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: Möbius transformations, the extended complex plane and the Riemann sphere, the crossratio. 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, Rouché'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)
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 
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 Variables  3:00  
Learning Outcomes
On completion of this course, the student will be able to:
 Ability to calculate proficiently with complex numbers, including solving algebraic equations.
 A good understanding of the notion of holomorphicity/analyticity in its many manifestations and ability to use their basic properties with confidence.
 Familiarity with elementary holomorphic functions, including multivalued functions, and the ability to work with them effectively.
 Ability to 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.
 Ability to 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 
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  Mr Thomas Robinson
Tel: (0131 6)50 4885
Email: Thomas.Robinson@ed.ac.uk 

