Complex analysis
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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 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 .
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 .
Entry requirements
There is no formal pre-requisite study, but you must have the required mathematical skills.
You can .
if you’re not sure you’re ready.
Preparatory work
You should aim to be confident and fluent with the concepts covered in the , 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).
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 may 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 10 or 11 or macOS Ventura (or higher).
- Internet access – Broadband or mobile connection.
- Browser – Google Chrome and Microsoft Edge are recommended; Mozilla Firefox and Safari may be suitable.