Content: 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.

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.

Books: Please see the Talis-aspire web page of this module for the latest recommended books.