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# Complex analysis

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.

## 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.

## 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
To enable you to make international comparisons, the information provided shows how OU levels correspond to the Framework for Higher Education Qualifications in England, Northern Ireland and Wales (FHEQ).
 OU FHEQ 3 6
Study method
Distance Learning
Module cost
See Module registration
Entry requirements
See Entry requirements

### Student Reviews

I want to congratulate the OU on this incredibly impressive module in Complex Analysis. The module materials are excellent, there...

This module has been very enjoyable though I found some of it conceptually challenging. Its quite broad in the number...

## 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.

## 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.

### 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.

## Future availability

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

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

## 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
No residential school

## 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).

## Register

Start End England fee Register
02 Oct 2021 Jun 2022 -

Registration now closed

This module is expected to start for the last time in October 2027.

### Study costs

There may be extra costs on top of the tuition fee, such as set books, a computer and internet access.

If your income is not more than £25,000 or you are in receipt of a qualifying benefit, you might be eligible for help with some of these costs after your module has started.

## Ways to pay for this module

### Open University Student Budget Account

The Open University Student Budget Accounts Ltd (OUSBA) offers a convenient 'pay as you go' option to pay your OU fees, which is a secure, quick and easy way to pay. Please note that The Open University works exclusively with OUSBA and is not able to offer you credit facilities from any other provider. All credit is subject to status and proof that you can afford the repayments.

You pay the OU through OUSBA in one of the following ways:

• Register now, pay later – OUSBA pays your module fee direct to the OU. You then repay OUSBA interest-free and in full just before your module starts. 0% APR representative. This option could give you the extra time you may need to secure the funding to repay OUSBA.
• Pay by instalments – OUSBA calculates your monthly fee and number of instalments based on the cost of the module you are studying. APR 5.1% representative.

#### Joint loan applications

If you feel you would be unable to obtain an OUSBA loan on your own due to credit history or affordability issues, OUSBA offers the option to apply for a joint loan application with a third party. For example, your husband, wife, partner, parent, sibling or friend. In such cases, OUSBA will be required to carry out additional affordability checks separately and/or collectively for both joint applicants who will be jointly and severally liable for loan repayments.

As additional affordability checks are required when processing joint loan applications, unfortunately, an instant decision cannot be given. On average the processing time for a joint loan application is five working days from receipt of the required documentation.

Studying with The Open University can boost your employability. OU courses are recognised and respected by employers for their excellence and the commitment they take to complete. They also value the skills that students learn and can apply in the workplace.

More than one in ten OU students are sponsored by their employer, and over 30,000 employers have used the OU to develop staff so far. If the module you’ve chosen is geared towards your job or developing your career, you could approach your employer to see if they will sponsor you by paying some or all of the fees.

• Your employer just needs to complete a simple form to confirm how much they will be paying and we will invoice them.
• You won’t need to get your employer to complete the form until after you’ve chosen your module.

### Credit/debit card

You can pay part or all of your tuition fees upfront with a debit or credit card when you register for each module.

We accept American Express, Mastercard, Visa and Visa Electron.

### Mixed payments

We know that sometimes you may want to combine payment options. For example, you may wish to pay part of your tuition fee with a debit card and pay the remainder in instalments through an Open University Student Budget Account (OUSBA).

Please note: your permanent address/domicile will affect your fee status and therefore the fees you are charged and any financial support available to you. The fees and funding information provided here is valid for modules starting before 31 July 2022. Fees normally increase annually in line with inflation and the University's strategic approach to fees.

This information was provided on 28/09/2021.

## 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

You'll need a desktop or laptop computer with an up-to-date version of 64-bit Windows 10 (note that Windows 7 is no longer supported) or macOS and broadband internet access.

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.