MA2003: Complex Analysis
| School | Cardiff School of Mathematics |
| Department Code | MATHS |
| Module Code | MA2003 |
| External Subject Code | 100405 |
| Number of Credits | 10 |
| Level | L5 |
| Language of Delivery | English |
| Module Leader | Professor Marco Marletta |
| Semester | Spring Semester |
| Academic Year | 2024/5 |
Outline Description of Module
A lecture-based module, providing an exposition of the basic theory and methods of complex analysis which are fundamental in mathematics and many of its applications.
The course shows how the concepts of differentiation and integration of real functions can be extended to complex functions. Complex functions map complex numbers to complex numbers. For a special subset of these functions it is possible to define a derivative. These differentiable complex functions have particularly nice properties. The real integral between two points x1 and x2 on the real axis is generalised to a complex integral along a path between two points z1 and z2 in the complex plane. These integrals are called contour integrals. Theorems of Cauchy show how some contour integrals of differentiable complex functions can be evaluated in a beautiful and simple way using methods known as the residue calculus. The residue calculus can be used to evaluate real integrals.
This course is essential for all mathematics students.
Pre-requisite: MA1005 and MA1006
On completion of the module a student should be able to
· determine whether a complex function is differentiable
· perform integration in the complex plane
· evaluate residues
· apply Cauchy's theorem
· use contour integrals to evaluate real integrals
· reproduce the proofs of the main theorems of the modul
How the module will be delivered
You will be guided through learning activities appropriate to your module, which may include:
· Weekly face to face classes (e.g. labs, lectures, exercise classes)
· Electronic resources to support the learning (e.g. videos, exercise sheets, lecture notes, quizzes)
Students are also expected to undertake at least 50 hours of self-guided study throughout the duration of the module, including preparation of formative assessments.
Skills that will be practised and developed
Skills: Contour integration and residue calculus.
Transferable Skills: Logical analysis and abstract reasoning.
How the module will be assessed
Exam Spring Semester (weighting 90%); duration 2 hours
Computer Assessed homework (weighting 10%).
Assessment Breakdown
| Type | % | Title | Duration(hrs) |
|---|---|---|---|
| Exam - Spring Semester | 90 | Complex Analysis | 2 |
| Written Assessment | 3 | Computer-Assessed Homework 1 | N/A |
| Written Assessment | 3 | Computer-Assessed Homework 2 | N/A |
| Written Assessment | 4 | Computer-Assessed Homework 3 | N/A |
Syllabus content
· Revision of complex numbers and elementary functions of a complex variable
· Differentiability and the Cauchy-Riemann equations
· integration in the complex plane
· Cauchy's theorem
· the application of contour integrals to evaluate real integrals
· Rouche's theorem with applications (time permitting)