Content: The course 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.
We will start with a review of elementary complex analysis topics from MA244 Analysis 3. This includes complex differentiability, the Cauchy-Riemann equations, Cauchy's theorem, Taylor's and Liouville's theorem etc. Most of the course will be new topics. This page will be updated in due course with the exact topics, but topics from previous years have included: Winding numbers, the generalized version of Cauchy's theorem, Morera's theorem, the fundamental theorem of algebra, the identity theorem, classification of singularities, the Riemann sphere and Weierstrass-Casorati theorem, meromorphic functions, Rouche's theorem, integration by residues.
Books: This list will be updated in due course.
Stewart and Tall, Complex Analysis: (the hitchhiker's guide to the plane), (Cambridge University Press).
Conway, Functions of one complex variable, (Springer-Verlag).
Ahlfors, Complex Analysis: an introduction to the theory of analytic functions of one complex variable, (McGraw-Hill Book Co).