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.
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 considered 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 include many worked examples, problems, and exercises (all with complete solutions), as well as a module handbook with reference material, the main results, and an index.
The full content list is on the Open mathematics and statistics website.
Entry requirements
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.
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).
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 forums.
Online tutorials run throughout the module. While they’re not compulsory, we strongly encourage you to participate. Where possible, we’ll make recordings available.
Assessment
Course work includes:
4 Tutor-marked assignments (TMAs)
Examination
For this module, we may use an assessment verification process to meet the requirements of relevant accrediting, professional, statutory or regulatory bodies. As part of this process, you may be asked to attend a short post-assessment video discussion, lasting around 15 minutes. During the discussion, you’ll need to show a photo ID and talk through your answers to a small number of questions with a tutor or a member of the module team. The discussion is not graded and is only used to confirm that you completed the assessment yourself.
We regularly review the assessments in our modules, so we may update the examination method used for this module to an in-person exam or an online invigilated exam. When we are making such a change, we will make it clear on this page. If we need to make a change after you have registered, we will notify you as soon as possible. If you have additional requirements, we will support you to complete your assessments.
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 and submission section
online tutorial access
access to student forums.
We also provide physical:
module books
module handbook.
You will need
a scientific calculator (recommended, but not essential).
Computing requirements
Primary device – A desktop or laptop computer with at least 8 GB of RAM and a quad-core processor (2.4 GHz minimum speed). It’s possible to access some materials on a mobile phone, tablet or Chromebook; however, they will not be suitable as your primary device or for taking your online exam.
Peripheral devices – Headphones/earphones with a built-in microphone for online tutorials, plus an external webcam and a microphone.
Operating systems – Windows 11 or the latest supported macOS.
Internet access – A broadband or mobile connection with at least 3 Mbps upload and download speed.
Browser – Google Chrome and Microsoft Edge are recommended. Mozilla Firefox and Safari may be suitable. You may need to use Google Chrome specifically for some activities.
Our OU Study app operates on supported versions of Android and iOS.
Software – Any additional software will be provided or is generally freely available.
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 website.
Future availability
Complex analysis (M337) starts once a year – in October.
It will next start in October 2027.
We expect it to start for the last time in October 2031.