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)
Examination
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 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 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

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.

Study materials

What's included

Module books.

You will need

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.

Assessment

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.

If you have a disability

The OU strives to make all aspects of study accessible to everyone. The Accessibility Statement below outlines what studying this module involves. You should use this information to inform your study preparations and any discussions with us about how we can meet your needs.

Mode of study

Printed materials are provided for the core module text. All of this module's study materials are also online; this includes PDFs of any printed materials, plus some items which are only provided online. Online-only materials include audio/video clips (with transcripts/subtitles). Online materials also include links to external resources, online forums and online tutorial rooms.

Tuition strategy

This module provides a series of face-to-face learning events. Each face-to-face learning event offers an online (or other) alternative. Although not compulsory, attendance at tutorials will help you consolidate your learning.

Working with others

You will not be required to work with other students on this module. However, there is opportunity to discuss with others on the lively module forums, which are closely monitored by an experienced academic. We encourage forum discussions as they can greatly benefit your learning process.

Mathematical and scientific expressions and notations

Mathematical and scientific symbols and expressions are used throughout the module and you will be required to use such notation within assessment.

Diagrams and other visual content

The study materials contains a considerable number of figures. Interpreting these figures, and reproducing similar figures, is an important part of the study of this module, and is assessed. Figure descriptions are provided for most figures.

Specialist reading material

In this module you will be working with specialist reading material which includes mathematical notation. This is delivered both online and in printed form, and also via bespoke software.

Assessment

This module has Tutor-Marked Assignments (TMAs), which must be submitted online via the OU electronic TMA system or by post, on paper, and an Exam that must be taken at an exam centre.

Feedback

You will receive feedback from your tutor on your submitted Tutor-Marked Assignments (TMAs). This will help you to reflect on your TMA performance. You should refer to it to help you prepare for your next assignment.

Schedule

All University modules are structured according to a set timetable and you will need time-management skills to keep your studies on track. You will be supported in developing these skills.

Specialist software

This module uses specialist mathematical symbols that are not covered by standard accessibility tools, and it may not be fully accessible as a result. These mathematical symbols can be created using software packages such as Microsoft Word or LaTeX, or they can be drawn by hand.

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.