An undergraduate course in Mathematics and Statistics.

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.

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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.

Regulations

As a student of The Open University, you should be aware of the content of the Module Regulations and the Student Regulations which are available on our Essential documents website.

If you have a disability

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.

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

You will need a computer with internet access to study this course as it includes online activities, which you can access using a web browser.

• If you have purchased a new desktop or laptop computer since 2007 you should have no problems completing the online activities.
• If you’ve got a netbook, tablet or other mobile computing device check our Technical requirements section.
• If you use an Apple Mac you will need OS X 10.6 or later.

You can also visit the Technical requirements section for further computing information including the details of the support we provide.

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.

Future availability

The details given here are for the course that starts in October 2014. It will then be available once a year, in October.

Students also studied

Students who studied this course also studied at some time:

• Pure mathematics (M208)
• Mathematical methods and models (MST209)
• Exploring mathematics (MS221)
• Graphs, networks and design (MT365)

How to register

We regret that we are currently unable to accept registrations for this course. Where the course is to be presented again in the future, relevant registration information will be displayed on this page as soon as it becomes available.

Student Reviews

Distance learning

The Open University is the world’s leading provider of flexible, high quality distance learning. Unlike other universities we are not campus based. You will study in a flexible way that works for you whether you’re at home, at work or on the move. As an OU student you’ll be supported throughout your studies – your tutor or study adviser will guide and advise you, offer detailed feedback on your assignments, and help with any study issues. Tuition might be in face-to-face groups, via online tutorials, or by phone.

For more information read Distance learning explained.

Course satisfaction survey

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