Complex analysis is a rich subject of foundational importance in mathematics and science. This module develops the theory of functions of a complex variable, emphasising their geometric properties and indicating some applications. Studying this module will consolidate many of the mathematical ideas and methods you learned in earlier modules and set you in good stead for tackling further fields of study in mathematics, engineering and physics.
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This module is expected to start for the last time in October 2031. |
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 module consists of thirteen units split between four books:
Book A: Complex numbers and functions
- Complex numbers
- Complex functions
- Continuity
- Differentiation
Book B: Integration of complex functions
- Integration
- Cauchy’s Theorem
- Taylor series
- Laurent series
Book C: Geometric methods in complex analysis
- Residues
- Zeros and extrema
- Conformal mappings
Book D: Applications of complex analysis
- 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.
You can find the full content list on the Open mathematics and statistics website.
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
There is no formal pre-requisite study, but you must have the required mathematical skills.
You can check you’re ready for M337 and see the topics it covers here.
Talk to an advisor if you’re not sure you’re ready.
Preparatory work
You should aim to be confident and fluent with the concepts covered in the Are you ready? quiz here, and follow the advice in the quiz.
The key topics to revise include:
- complex numbers and algebra
- differential and integral calculus.
One of the following is ideal preparation: Pure mathematics (M208), Mathematical methods, models and modelling (MST210), Mathematical methods (MST224).
If you have a disability
The OU strives to make all aspects of study accessible to everyone and this Accessibility Statement outlines what studying M337 involves. You should use this information to inform your study preparations and any discussions with us about how we can meet your needs.
To find out more about what kind of support and adjustments might be available, contact us or visit our disability support pages.
Study materials
What's included
You’ll have access to a module website, which includes:
- a week-by-week study planner
- course-specific module materials
- audio and video content
- assessment details, instructions and guidance
- online tutorial access
- access to student and tutor group forums.
You’ll be provided with printed books covering the content of the module, including explanations, examples and activities to aid your understanding of the concepts and associated skills and techniques. You’ll also receive a printed module handbook.
You will need
A scientific calculator would be useful but is not essential.
Computing requirements
- Primary device – A desktop or laptop computer. It’s possible to access some materials on a mobile phone, tablet or Chromebook; however, they will not be suitable as your primary device.
- Peripheral device – Headphones/earphones with a built-in microphone for online tutorials.
- Our OU Study app operates on supported versions of Android and iOS.
- Operating systems – Windows 11 or latest supported macOS. Microsoft will no longer support Windows 10 as of 14 October 2025.
- Internet access – Broadband or mobile connection.
- Browser – Google Chrome and Microsoft Edge are recommended. Mozilla Firefox and Safari may be suitable.
Teaching and assessment
Support from your tutor
You’ll get help and support from an assigned tutor throughout your module.
They’ll help by:
- marking your assignments and offering detailed feedback to help you improve
- providing individual guidance, whether that’s for general study skills or specific module content
- guiding you to additional learning resources
- facilitating online discussions between your fellow students in the dedicated module and tutor group forums.
Online tutorials run throughout the module. Where possible, we’ll make recordings available. While they’re not compulsory, we strongly encourage you to participate.
Assessment
The assessment details for this module can be found in the facts box.
We’re using a new examination verification process for this module. We may ask you to attend a 15-minute post-exam video discussion, where you’ll present a photo ID and discuss your answers to a small number of questions with a tutor or member of the module team. The discussion isn’t graded; it’s only to verify that you completed the exam yourself.
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.
Students also studied
Students who studied this module also studied at some time:
Future availability
Complex analysis (M337) starts once a year – in October.
This page describes the module that will start in October 2026.
We expect it to start for the last time in October 2031.
How to register
We regret that we are currently unable to accept registrations for this module. Where the module is to be presented again in the future, relevant registration information will be displayed on this page as soon as it becomes available.
Regulations
As a student of The Open University, you should be aware of the content of the academic regulations which are available on our
Student Policies and Regulations website.