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Complex analysis

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This module 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 module.

Modules at Level 3 assume that you are suitably prepared for study at this level. If you want to take a single module to satisfy your career development needs or pursue particular interests, you don’t need to start at Level 1 but you do need to have adequately prepared yourself for OU study in some other way. Check with our Student Registration & Enquiry Service to make sure that you are sufficiently prepared.

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04 Oct 2014 Jun 2015
£440.00
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Registration opens on 24/04/14

This course is expected to start for the last time in October 2017.
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04 Oct 2014 Jun 2015
£865.00
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Registration opens on 24/04/14

This course is expected to start for the last time in October 2017.
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04 Oct 2014 Jun 2015
£1316.00
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Registration opens on 24/04/14

This course is expected to start for the last time in October 2017.
Start End Fee Register
04 Oct 2014 Jun 2015
£1316.00
Choose country above

Registration opens on 24/04/14

This course is expected to start for the last time in October 2017.
Start End Fee Register
04 Oct 2014 Jun 2015
£440.00
Choose country above

Registration opens on 24/04/14

This course is expected to start for the last time in October 2017.
Start End Fee Register
04 Oct 2014 Jun 2015
£1316.00
Choose country above

Registration opens on 24/04/14

See below for information about part time tuition fee loans available for study towards a qualification.

This course is expected to start for the last time in October 2017.
Start End Fee Register
04 Oct 2014 Jun 2015
£440.00
Choose country above

Registration opens on 24/04/14

This course is expected to start for the last time in October 2017.
Start End Fee Register
04 Oct 2014 Jun 2015
£440.00
Choose country above

Registration opens on 24/04/14

This course is expected to start for the last time in October 2017.
Start End Fee Register
04 Oct 2014 Jun 2015
£985.00
Choose country above

Registration opens on 24/04/14

This course is expected to start for the last time in October 2017.
Start End Fee Register
04 Oct 2014 Jun 2015
£440.00
Choose country above

Registration opens on 24/04/14

This course is expected to start for the last time in October 2017.

*Fees 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 module 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 module 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 module 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 module. Level 3 modules 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 modules. To study this module you should have at least a grade 2 pass in Pure mathematics (M208), Mathematical methods, models and modelling (MST210) (planned for October 2014), or its predecessor MST209, or the equivalent.  

There is a diagnostic quiz that will help you to determine whether you are adequately prepared for this module. Your regional centre will also be able to tell you where you can see reference copies of the module units.

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 module 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 module 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 module audio-visual material. Our Services for disabled students website has the latest information about availability.

If you have particular study requirements please tell us as soon as possible, as some of our support services may take several weeks to arrange. Visit our Services for disabled students website for more information, including:

  • help to determine your study requirements and how to request the support that you need  
  • Disabled Students' Allowances (DSAs)
  • using a computer for OU study
  • equipment and other support services that we offer
  • examination arrangements
  • how to contact us for advice and support both before you register and while you are studying.

Study materials

What's included

Module 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 module to download module resources and assignments, and to keep up to date with module 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 module.  

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 module can be found in the facts box above.

Please note that TMAs for all undergraduate mathematics and statistics modules must be submitted on paper as – due to technical reasons – we are unable to accept TMAs via our eTMA system.

Professional recognition

This module may help you to gain membership of the Institute of Mathematics and its Applications (IMA). For further information, see the IMA website.

Future availability

The details given here are for the module that starts in October 2014. We expect it to be available once a year.

Students also studied

Students who studied this course also studied at some time:

How to register

To register a place on this course return to the top of the page and use the Click to register button.

Student Reviews

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“This is an excellent rigorous course. The material is well presented often with a geometric flavour. Basic concepts are presented ...”
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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.

Are you already an OU student ? Go to StudentHome for information on choosing your next module.
Course facts
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

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