Introductory description

The module focuses on the properties of differentiable functions on the complex plane. Unlike real analysis, complex differentiable functions have a large number of amazing properties, and are very ``rigid'' objects. Some of these properties have been explored already in second year core. Our goal will be to push the theory further, hopefully revealing a very beautiful classical subject.

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

In the early part of the module we will see some of the complex analysis topics from MA244 Analysis III, typically in greater depth and/or generality. This includes complex differentiability, the Cauchy-Riemann equations, complex power series, Cauchy's theorem, Taylor's and Liouville's theorem etc. Most of the course will be new topics. We will cover Möbius transformations, the Riemann sphere, winding numbers, generalised versions of Cauchy's theorem, Morera's theorem, zeros of holomorphic functions, the identity theorem, the Schwarz lemma, the classification of isolated singularities, the Weierstrass-Casorati theorem, meromorphic functions, Laurent series, the residue theorem (and applications to integration), Rouché's theorem, the Weierstrass convergence theorem, Hurwitz’s theorem, Montel’s theorem, and the remarkable Riemann mapping theorem (with proof) that ties the whole module together.

Learning outcomes

By the end of the module, students should be able to:

• N/A

Indicative reading list

Jameson, A first course on complex functions, (Chapman & Hall)
Priestley, Introduction to complex analysis, (Oxford University Press).
Ahlfors, Complex Analysis: an introduction to the theory of analytic functions of one complex variable, (McGraw-Hill Book Co).
Stewart and Tall, Complex Analysis: (the hitchhiker's guide to the plane), (Cambridge University Press).
Conway, Functions of one complex variable, (Springer-Verlag).

View reading list on Talis Aspire

Subject specific skills

N/A

Transferable skills

Students will acquire key reasoning and problem solving skills which will empower them to address new problems with confidence.