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.

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)
No residential school

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.


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), 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. If you have not completed M208, you may not be familiar with some of the topics towards the end of the quiz, so you should pay particularly close attention to the feedback provided in the quiz solutions.

If you have any doubt about the level of study, please speak to an adviser.

Preparatory work

There is no formal preparatory work, but you should revise your algebraic skills, and differential and integral calculus, before the module begins.

If you have a disability or additional requirement

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.

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 device with internet access to study this module as a web browser is used to access learning materials and activities. Any other computer-based activities you will need to carry out, such as word processing, using spreadsheets, taking part in online forums, and submitting files to the university for assessment, are specified in the module materials. If any additional software is needed for these tasks it will either be provided or is freely available. You may need administrative privileges to install software required for this module. Windows 10 S is not suitable as it restricts software installation to software available in the Windows Application Store.

Suitable devices are:

  • A Windows desktop or laptop running Windows 7 or later operating system
  • A Macintosh desktop or laptop running OS X 10.8 or later operating system.

Some software will not run on Linux, iOS or Android devices.

A netbook, tablet, smartphone or Linux computer that supports one of the browsers listed below may be suitable. However, these devices may not be suitable for some activities. If you intend to use one of these devices please ensure you have access to a suitable desktop or laptop device that uses the Windows or OS X operating system in case you are unable to carry out all activities on your mobile device.

Recent versions of the following browsers are suitable for carrying out web-based activities:

  • Safari
  • Chrome
  • Firefox
  • Edge

Or Internet Explorer 9 and above.

Using a browser upgraded to the latest version will maximise security when accessing the internet.

Using company or library computers may prevent you accessing some internet materials or installing additional software.

To be able to talk and listen in our online discussions you will need both a microphone and speakers/headphones.

Devices with small screens may make it difficult to view the material provided and carry out the activities. However, a device that has a resolution of at least 1024 pixels horizontally and also at least 768 pixels vertically should be adequate.

See our Skills for OU study website for further information about computing skills for study and educational deals for buying Microsoft Office software.

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 us if you want to know more about study with The Open University before you register.


The assessment details for this module can be found in the facts box above.

You can choose whether to submit your tutor-marked assignments (TMAs) on paper or online through the eTMA system. You may want to use the eTMA system for some of your assignments but submit on paper for others. This is entirely your choice.

Future availability

The details given here are for the module that starts in October 2017. It starts once a year – in October.

This course is expected to start for the last time in October 2027.

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.