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Complex analysis

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 count towards OU qualifications

OU qualifications are modular in structure; the credits from this undergraduate-level module could count towards a certificate of higher education, diploma of higher education, foundation degree or honours degree.

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Module

Module code
M337
Credits
30
Study level
OU SCQF FHEQ
3 10 6
Study method
Distance Learning
Module cost
See Module registration
Entry requirements
See Am I ready?

Student Reviews

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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 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.

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.

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 tutor-marked assignments (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.

Future availability

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

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.

Course work includes:

4 Tutor-marked assignments (TMAs)
Examination
No residential school

Course satisfaction survey

See the satisfaction survey results for this course.


Entry

This is an OU 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 OU level 2 mathematics modules. To study this module you should have a grade 2 pass (minimum) in at least one of the following: Pure mathematics (M208), Mathematical methods, models and modelling (MST210) or its predecessor MST209, Mathematical methods (MST224), 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.

Register

Start End England fee Register
04 Oct 2014 Jun 2015 £1316.00

Registration closes 18/09/14 (places subject to availability)

Register

The deadline for financial support applications has now passed

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

Ways to pay for this module

Open University Student Budget Account

The Open University Student Budget Accounts Ltd (OUSBA) offers a convenient 'pay as you go' option to pay your OU fees, which is a secure, quick and easy way to pay. Please note that The Open University works exclusively with OUSBA and is not able to offer you credit facilities from any other provider. All credit is subject to status and proof that you can afford the repayments.

You pay the OU through OUSBA in one of the following ways:

  • Register now, pay later – OUSBA pays your module fee direct to the OU. You then repay OUSBA interest-free and in full just before your module starts. 0% APR representative. This option could give you the extra time you may need to secure the funding to repay OUSBA.
  • Pay by instalments – OUSBA calculates your annual fees and spreads them out over up to a year, enabling you to pay your fees monthly and walk away with a qualification without any further debt. APR 5.1% representative.

Read more about Open University Student Budget Accounts (OUSBA).  

Employer sponsorship

Studying with The Open University can boost your employability. OU qualifications are recognised and respected by employers for their excellence and the commitment they take to achieve one. They also value the skills that students learn and can apply in the workplace.

More than one in 10 OU students are sponsored by their employer, and over 30,000 employers have used the OU to develop staff so far. If the qualification you’ve chosen is geared towards your job or developing your career, you could approach your employer to see if they will sponsor you by paying some or all of the fees. 

  • Your employer just needs to complete a simple form to confirm how much they will be paying and we will invoice them.
  • You won’t need to get your employer to complete the form until after you’ve chosen your modules.  

Credit/debit card

You can pay part or all of your tuition fees upfront with a debit or credit card when you register for each module. 

We accept American Express, Maestro (UK only), Mastercard, Visa/Delta and Visa Electron. 

Gift vouchers

You can pay for part or all of your tuition fees with OU gift vouchers. Vouchers are currently available in the following denominations, £10, £20, £50 and £100. 

Mixed payments

We know that sometimes you may want to combine payment options. You may, for example wish to pay part of your tuition fee with a debit card and pay the remainder in instalments through an Open University Student Budget Accounts (OUSBA).

For more information about combining payment options, speak to an adviser.


Note: Your permanent address/domicile will affect your fee status and therefore the fees you are charged and any financial support available to you. The fees and funding information provided here is based upon current details for  year 1 August 2014 to 31 July 2015.
This information was provided on 16/09/2014.

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 module as it includes online activities, which you can access using a web browser.

  • If you have purchased a new desktop or laptop computer since 2008 you should have no problems completing the online activities.
  • If you’ve got a netbook, tablet or other mobile device check our Technical requirements section.
  • If you use an Apple Mac you will need OS X 10.7 or later.

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

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.

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. Find out more about our services for disabled students.