- UCAS course code
- GG14
- UCAS institution code
- M20
Bachelor of Science (BSc)
BSc Computer Science and Mathematics
Duration: 3 years
Year of entry: 2025
UCAS course code: GG14 / Institution code: M20
- Key features:
Scholarships available
- Typical A-level offer: A*A*A including specific subjects
- Typical contextual A-level offer: AAA including specific subjects
- Refugee/care-experienced offer: AAB including specific subjects
- Typical International Baccalaureate offer: 38 points overall with 7,7,6 at HL, including specific requirements
Course unit details:
2P1: Complex Analysis
| Unit code | MATH29141 |
|---|---|
| Credit rating | 10 |
| Unit level | Level 2 |
| Teaching period(s) | Semester 1 |
| Available as a free choice unit? | No |
Overview
This course introduces the analysis of complex functions of a complex variable. Complex differentiability is a very strong condition, and differentiable (or holomorphic or regular) functions have many strong properties. Integration is along paths in the complex plane. The central result of this spectacularly beautiful part of mathematics is Cauchy’s Theorem guaranteeing that certain integrals along closed paths are zero. After studying properties of isolated singularities of functions, Cauchy’s theorem leads to useful techniques for evaluating real integrals based on the ‘calculus of residues’ (problems unsolvable by other means).
Syllabus:
A. Complex functions: Domains and paths in the complex plane. Differentiation and the Cauchy-Riemann equations; holomorphic functions. Path integrals and the Fundamental Theorem of contour integration.
B. Power series: Power series and their radius of convergence. Derivatives and integrals of power series. Elementary functions such as exp, sin, cos, sinh, cosh and log.
C. Cauchy’s Theorem and Formula: Winding numbers, Cauchy’s theorem, Cauchy’s integral formula and the estimation lemma.
D. Taylor and Laurent series: The Cauchy-Taylor theorem, Liouville’s theorem, Laurent’s theorem and calculation of Laurent series.
E. Residues: Isolated singularities, poles and their residues; Cauchy’s residue theorem. Applications of the residue theorem to the evaluation of trigonometric integrals, integrals over the real line, and summation of series.
Pre/co-requisites
| Unit title | Unit code | Requirement type | Description |
|---|---|---|---|
| Mathematical Foundations & Analysis | MATH11121 | Pre-Requisite | Compulsory |
Aims
The unit aims to introduce the basic ideas of complex analysis, with particular emphasis on contour integration, Cauchy’s Theorem and the calculus of residues.
The course is only available to students on the BSc/MMathPhys Mathematics & Physics programmes and the BSc Computer Science and Mathematics programme.
Learning outcomes
On succesful completion of the course, students will be able to:
- Use the Cauchy-Riemann Theorem it to decide whether a given function is holomorphic.
- Define power series, determine their properties, and use them to define elementary holomorphic functions.
- Use the Cauchy-Taylor Theorem and Laurent's Theorem to expand a holomorphic function as a power series on a disc or on an annulus.
- Define and calculate complex integrals using a variety of methods (in particular the Fundamental Theorem and the Cauchy Residue Theorem) and apply to the evaluation of some real integrals.
Teaching and learning methods
This course will be delivered together with the first half of MATH34011 (which is 20 credits). It will involve 3 hours of lectures plus an examples class and a tutorial each week for 6 weeks.
The course is only available to students on the BSc/MMathPhys Mathematics & Physics programmes and the BSc Computer Science and Mathematics programme.
Assessment methods
| Method | Weight |
|---|---|
| Other | 20% |
| Written exam | 80% |
Online coursework test (using stack)
Test mid-way through semester; Stack feedback available after test is closed
20% Weighting
Written exam
General feedback provided after exam is marked
80% Weighting
Feedback methods
Online coursework test (using stack)
Test mid-way through semester; Stack feedback available after test is closed
20% Weighting
Written exam
General feedback provided after exam is marked
80% Weighting
Recommended reading
Ian Stewart and David Tall, Complex Analysis, Cambridge University Press, 1983.
Study hours
| Scheduled activity hours | |
|---|---|
| Lectures | 18 |
| Practical classes & workshops | 6 |
| Tutorials | 6 |
| Independent study hours | |
|---|---|
| Independent study | 70 |
Teaching staff
| Staff member | Role |
|---|---|
| James Montaldi | Unit coordinator |
| Michael Coleman | Unit coordinator |