Complex analysis
On this page
This course develops the theory of functions of a complex variable, emphasising their geometric properties and indicating some applications. Introduction covers complex numbers; complex functions; sequences and continuity; and differentiation of complex functions. Representation formulas covers integration of complex functions; Cauchy’s theorem and Cauchy’s integral formula; Taylor series; and Laurent series. Calculus of residues covers residue calculus; winding number and the location of zeros of complex functions; analytic continuation; Euler’s gamma function and Riemann’s zeta function. Finally, Applications covers conformal mappings; fluid flows; complex analytic dynamics; Julia sets; and the Mandelbrot set. You need a sound knowledge of differentiation and integration of real functions for this course.
Fees 2012
See fees and funding options for study from September 2012.
Course facts
An undergraduate course in Mathematics and Statistics.
| About this course: | |
|---|---|
| Course code | M337 |
| Credits | 30 |
| OU Level | 3 |
| SCQF level | 10 |
| FHEQ level | 6 |
| Course work includes: |
|---|
| 4 Tutor-marked assignments (TMAs) |
| Examination |
| No residential school |
Register for the course
This course is available for study in the countries shown. Fees and financial support may vary by country.
What you will study
There is no real number whose square is –1, but mathematicians long ago invented a system of numbers, called complex numbers, in which the square root of –1 does exist. These complex numbers can be thought of as points in a plane, in which the arithmetic of complex numbers can be pictured. When the ideas of calculus are applied to functions of a complex variable a powerful and elegant theory emerges, known as complex analysis.
The course shows how complex analysis can be used to:
- determine the sums of many infinite series
- evaluate many improper integrals
- find the zeros of polynomial functions
- give information about the distribution of large prime numbers
- model fluid flow past an aerofoil
- generate certain fractal sets whose classification leads to the Mandelbrot set.
The fourteen study texts make up four blocks of work, roughly equal in length:
Introduction Complex numbers – complex functions – continuity – differentiation
Representation formulas Integration – Cauchy’s theorem – Taylor series – Laurent series
Calculus of residues Residues – zeros and extrema – analytic continuation
Applications Conformal mappings – fluid flows – the Mandelbrot set.
The texts have many worked examples, problems and exercises (all with full solutions), and there is a course handbook that includes reference material, the main results and an index. These texts are supported by CDs that teach complex analysis techniques, while another CD presents a discussion of the central role of complex analysis in mathematics. A DVD uses computer graphics to demonstrate many geometric properties of complex functions.
You will learn
Successful study of this course should enhance your skills in understanding complex mathematical texts, working with abstract concepts, constructing solutions to problems logically and communicating mathematical ideas clearly.
Entry
This is a Level 3 course. Level 3 courses build on study skills and subject knowledge acquired from studies at Levels 1 and 2. They are intended only for students who have recent experience of higher education in a related subject, preferably with the OU.
You need proficiency in algebra, trigonometry and calculus, and the mathematical maturity gained from Level 2 mathematics courses. To study this course you should have at least a grade 2 pass in Pure mathematics (M208) or Mathematical methods and models (MST209), or the equivalent.
Your regional centre will be able to tell you where you can see reference copies of the course units. There is also a diagnostic quiz that will help you to determine whether you are adequately prepared for this course.
If you have any doubt about the level of study, please seek advice from our Student Registration & Enquiry Service.
Preparatory work
There is no formal preparatory work, but you should revise your algebraic skills, and differential and integral calculus, before the course begins.
If you have a disability or additional requirement
The course should present no special difficulties, though it does include a lot of diagrams. The study materials are available on audio in DAISY Digital Talking Book format and there are transcripts of the course audio-visual material. Our Services for disabled students website has the latest information about availability.
If you are a new student, or new to courses using a computer or the internet, you will need to inform us of your particular needs as soon as possible, as some of our support services may take several weeks to arrange. Details of how to do this and our range of support services are described in our publication Meeting Your Needs.
You can also find information about accessible study materials, financial support and the Disabled Students' Allowance, equipment and other services, on our Services for disabled students website. It also includes our contact details for advice and support both before you register and while you are studying.
Study materials
What's included
Course books, CDs, DVD.
You will need
CD player and DVD player (or computer able to play DVDs). A scientific calculator would be useful but is not essential.
You require access to the internet at least once a week during the course to download course resources and assignments, and to keep up to date with course news.
Computing requirements
This course includes online computer activities – you can access these using a web browser that can play Flash and Shockwave.
You will need internet access and a computer. If you have purchased a new computer since 2005 it should meet your course computing requirements. Check our Technical Requirements section if your computer is older than this or is otherwise unusual.
Teaching and assessment
Support from your tutor
You will have a tutor who will mark and comment on your written work, and whom you can ask for advice and guidance. We may also be able to offer group tutorials or day schools that you are encouraged, but not obliged, to attend. Where your tutorials are held will depend on the distribution of students taking the course. Contact our Student Registration & Enquiry Service if you want to know more about study with The Open University before you register.
Assessment
The assessment details for this course can be found in the facts box above.
Please note that TMAs for all undergraduate mathematics and statistics courses must be submitted on paper as – due to technical reasons – we are unable to accept TMAs via our eTMA system.
Professional recognition
This course may help you to gain recognition from a professional body. You can view or download our Recognition leaflets 3.3 Professional Engineering Institutions and 3.6 Institute of Mathematics and its Applications for further information.
Students also studied
Students who studied this course also studied at some time:
Future availability
The details given here are for the course that starts in October 2012. We then expect it to be available, in alternate (even-numbered) years, in October.
Fees 2012
See fees and funding options for study from September 2012.
Course facts
An undergraduate course in Mathematics and Statistics.
| About this course: | |
|---|---|
| Course code | M337 |
| Credits | 30 |
| OU Level | 3 |
| SCQF level | 10 |
| FHEQ level | 6 |
| Course work includes: |
|---|
| 4 Tutor-marked assignments (TMAs) |
| Examination |
| No residential school |
Study explained
- Financial support
- - find out if you qualify for support with your fees with our eligibility checker.
- Study explained
- - all you need to know about distance learning with the OU.
Student Reviews
M337 is a great course - it builds nicely on the real analysis covered in M208, but although it does ...
Read more
This is a good course that should be highly recommended (if not compulsory) for any math degree. It covers all ...
Read more
Your questions
We may have already answered it in our frequently asked questions.
Or contact an adviser in our Student Registration & Enquiry Service Email or call +44(0) 845 300 60 90+44(0) 845 366 60 35
Browse the prospectus
Get a prospectus
Download or
order
Share
This PageThe University wishes to emphasise that, while every effort is made to regularly update this site, the material on it is subject to alteration or amendment in the light of changes in regulations or in policy or of financial or other necessity.