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Complex analysis

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.

Modules count towards OU qualifications

OU qualifications are modular in structure; the credits from this undergraduate 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

Credits

  • Credits measure the student workload required for the successful completion of a module or qualification.
  • One credit represents about 10 hours of study over the duration of the course.
  • You are awarded credits after you have successfully completed a module.
  • For example, if you study a 60-credit module and successfully pass it, you will be awarded 60 credits.
30

Study level

Across the UK, there are two parallel frameworks for higher education qualifications, the Framework for Higher Education Qualifications in England, Northern Ireland and Wales (FHEQ) and the Scottish Credit and Qualifications Framework (SCQF). These define a hierarchy of levels and describe the achievement expected at each level. The information provided shows how OU module levels correspond to these frameworks.
Level of Study
OU SCQF FHEQ
3 10 6

Study method

Module cost

Entry requirements

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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 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.

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

Throughout your module studies, you’ll get help and support from your assigned module tutor. They’ll help you by:

  • Marking your assignments (TMAs) and providing detailed feedback for you to improve.
  • Guiding you to additional learning resources.
  • Providing individual guidance, whether that’s for general study skills or specific module content.
  • Facilitating online discussions between your fellow students, in the dedicated module and tutor group forums.

Module tutors also run online tutorials throughout the module. Where possible, recordings of online tutorials will be made available to students. While these tutorials won’t be compulsory for you to complete the module, you’re strongly encouraged to take part.

Assessment

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

Future availability

Complex analysis (M337) starts once a year – in October.

This page describes the module that will start in October 2024.

We expect it to start for the last time in October 2031.

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.

Course work includes:

4 Tutor-marked assignments (TMAs)
Examination


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.

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).

Register

Start End England fee Register
05 Oct 2024 Jun 2025 £1818.00

Registration closes 05/09/24 (places subject to availability)

Register
This module is expected to start for the last time in October 2031.

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In the meantime, explore our overview of Fees and funding.

Can you study an Access module for free?

Depending on eligibility and availability of places, you could apply to study your Access module for free.

To qualify, you must:

  1. be resident in England
  2. have a household income of not more than £25,000 (or be in receipt of a qualifying benefit)
  3. have not completed one year or more on any full-time undergraduate programme at FHEQ level 4 or above or successfully completed 30 credits or more of OU study within the last 10 years

How to apply to study an Access module for free

Once you've started the registration process, either online or over the phone, we'll contact you about your payment options. This will include instructions on how you can apply to study for free if you are eligible and funded places are still available.

If you're unsure if you meet the criteria to study for free, you can check with one of our friendly advisers on +44 (0)300 303 0069, or you can request a call back.

Not eligible to study for free?

Don't worry! We offer a choice of flexible ways to help spread the cost of your Access module. The most popular options include:

  • monthly payments through OUSBA
  • part-time tuition fee loan (you'll need to be registered on a qualification for this option)

To explore all the options available to you, visit Fees and Funding.

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

You’ll need broadband internet access and a desktop or laptop computer with an up-to-date version of Windows (10 or 11) or macOS Ventura or higher.

Any additional software will be provided or is generally freely available.

To join in spoken conversations in tutorials, we recommend a wired headset (headphones/earphones with a built-in microphone).

Our module websites comply with web standards, and any modern browser is suitable for most activities.

Our OU Study mobile app will operate on all current, supported versions of Android and iOS. It’s not available on Kindle.

It’s also possible to access some module materials on a mobile phone, tablet device or Chromebook. However, as you may be asked to install additional software or use certain applications, you’ll also require a desktop or laptop, as described above.

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.