Complex analysis is a rich subject that is 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. In studying the module, you will consolidate many of the mathematical ideas and methods that you have learned in earlier modules, and it will 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 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
Book B: Integration of complex functions
- Complex numbers
- Complex functions
Book C: Geometric methods in complex analysis
- Cauchy's Theorem
- Taylor series
- Laurent series
Book D: Applications of complex analysis
- Zeros and extrema
- 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.
Read the full content list here.
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 module may help you to gain membership of the Institute of Mathematics and its Applications (IMA). For further information, see the IMA website.
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.
There is no formal preparatory work, but you should revise your algebraic skills, and differential and integral calculus, before the module begins.
You will need
A scientific calculator would be useful but is not essential.
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
- Mac OS X 10.7 or higher
The screen of the device must have a resolution of at least 1024 pixels horizontally and 768 pixels vertically.
To join in the spoken conversation in our online rooms we recommend a headset (headphones or earphones with an integrated microphone).
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.