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. There are transcripts of the 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

A computing device with a browser and broadband internet access is required for this module. Any modern browser will be suitable for most computer activities. Functionality may be limited on mobile devices.

Any additional software will be provided, or is generally freely available. However, some activities may have more specific requirements. For this reason, you will need to be able to install and run additional software on a device that meets the requirements below.

A desktop or laptop computer with either:

  • Windows 7 or higher
  • macOS 10.7 or higher

The screen of the device must have a resolution of at least 1024 pixels horizontally and 768 pixels vertically.

To participate in our online-discussion area you will need both a microphone and speakers/headphones.

Our Skills for OU study website has further information including computing skills for study, computer security, acquiring a computer and Microsoft software offers for students.

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

Complex analysis starts once a year – in October. This page describes the module that will start in October 2018. We expect it to start for the last time in October 2027.

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.