Complex Variable & Differential Equations (Ord) (U03106)

? Credit Points : 20 ? SCQF Level : 9 ? Acronym : MAT-3-CVDO

Cognate with core course for Honours degrees involving mathematics and/or statistics. For this Ordinary version there is more emphasis on the technical, rather than conceptual elements, which will be reflected by a different examination.

Syllabus summary: Power series and differential equations, systems of ODEs, separation of variables, orthogonal expansions and applications; analytic functions, contour integrals, Laurent series and residues and Fourier transform.

Entry Requirements

? Pre-requisites : MAT-2-FoC, MAT-2-SVC, MAT-2-LiA, MAT-2-MAM

? Prohibited combinations : MAT-3-CoV, similar courses from Mathematics 3 (Hons) prior to 2004-05; PHY-3-PhMath, MAT-3-CVD

Subject Areas

Home subject area

Specialist Mathematics & Statistics (Ordinary), (School of Mathematics, Schedule P)

Delivery Information

? Normal year taken : 3rd year

? Delivery Period : Full Year (Blocks 1-4)

? Contact Teaching Time : 3 hour(s) per week for 22 weeks

All of the following classes

Type	Day	Start	End	Area
Lecture	Monday	12:10	13:00	KB
Lecture	Thursday	12:10	13:00	KB

? Additional Class Information : Supervision: one hour per week (shared with other 'core' courses), at a time to be arranged with Supervisor.

Summary of Intended Learning Outcomes

The following are the learning objectives for the Honours version, MAT-3-CVD:

1. Solution of a linear system (in non-degenerate cases) using eigenpairs
2. Evaluation and application of matrix exponential (in non-degenerate cases)
3. Classification of planar linear systems (non-degenerate cases)
4. Determination of stability and classification of an equilibrium of a planar nonlinear system, by linearisation
5. Graphic use of integral of a conservative planar system
6. Acquaintance with Poincare-Bendixson Theorem
7. Acquaintance with basic partial differential equations and types of boundary conditions
8. Solution of first-order linear pde with constant coefficients
9. Solution of the wave equation by change of variable, leading to d'Alembert's solution
10. Acquaintance with notions of existence and uniqueness by example
11. Separation of variables for wave equation (finite string) and Laplace's equation (disc)
12. Handling Fourier series as orthogonal expansions, with an inner product and projection operator
13. Self-adjoint linear differential operators and their elementary spectral properties
14. The notion of completeness
15. Power series solution about a regular points of an analytic ordinary differential equation
16. Power series solution of Bessel's equation of order 0
17. Solutions of the wave equation for a circular drum
18. Knowledge of basic properties of analytic functions of a complex variable, including power-series expansions, Laurent expansions, and Liouville's theorem
19. The idea of conformal mapping, use of fractional linear transformations
20. Knowledge of the fundamental integral theorems of complex analysis
21. Ability to use residue calculus to perform definite integrals
22. Knowledge of some of the relations between analytic functions and PDE, e.g. relation to harmonic functions, the maximum principle
23. Familiarity with the Fourier integral as a tool for the study of ordinary and partial differential equations.