Undergraduate Course: Honours Complex Variables (MATH10067)
|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
|Summary||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 to complete a project researching a topic connected with complex numbers or complex analysis and produce a written report.
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.
Information for Visiting Students
|Pre-requisites||Visiting 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?
Course Delivery Information
|Academic year 2023/24, Available to all students (SV1)
|Learning and Teaching activities (Further Info)
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
|Assessment (Further Info)
|Additional Information (Assessment)
||Coursework 25%, Examination 75%
||Hours & Minutes
|Main Exam Diet S2 (April/May)||Honours Complex Variables||3:00|
On completion of this course, the student will be able to:
- Identify, construct, and calculate with holomorphic functions, including branches of multivalued functions, and conformal maps on the extended complex plane.
- Compute series expansions of complex functions, and identify and classify singularities of complex functions.
- Apply the integral theorems, in particular to the evaluation of real sums and integrals..
- Make deductions about both specific examples and unfamiliar abstract situations using general results from the course.
- Compose technically accurate, appropriately formatted, and well-presented mathematical documents.
|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
|Graduate Attributes and Skills
|Course organiser||Dr Richard Gratwick
Tel: (0131 6)51 3411
|Course secretary||Miss Greta Mazelyte