# Complex Analysis, Short Course

This Complex Analysis module at The Open University UK 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 Complex Analysis module at The Open University UK.

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.

## Detailed Programme Facts

• Programme intensity Full-time
• Credits
30 alternative credits
• Languages
• English
• Delivery mode
Online

## Programme Structure

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.

## English Language Requirements

This programme requires students to demonstrate proficiency in English.

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.

## Tuition Fee

• ### EU/EEA Applies to you

1464 GBP/full
Tuition Fee
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## Funding

Check the programme website for information about funding options.

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