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# Complex analysis

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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
• 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.

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.

### 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.

## Entry requirements

There is no formal pre-requisite study, but you must have the required mathematical skills.

## 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).

## What's included

• a week-by-week study planner
• course-specific module materials
• audio and video content
• assessment details, instructions and guidance
• online tutorial access

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

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 an up-to-date version of Windows or macOS.

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.

## Teaching and assessment

• 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. If you want to participate, you’ll likely need a headset with a microphone.

### 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 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.

## Future availability

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

We expect it to start for the last time in October 2027.

### Course work includes:

4 Tutor-marked assignments (TMAs)
Examination
No residential school

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