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 fourteen study texts make up four blocks of work, roughly equal in length:
Introduction Complex numbers – complex functions – continuity – differentiation
Representation formulas Integration – Cauchy’s theorem – Taylor series – Laurent series
Calculus of residues Residues – zeros and extrema – analytic continuation
Applications 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. These texts are supported by CDs that teach complex analysis techniques, while another CD presents a discussion of the central role of complex analysis in mathematics. A DVD uses computer graphics to demonstrate many geometric properties of complex functions.
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.