# MA3B8 Complex Analysis

**Lecturer:** Xue-Mei Li

**Term(s):** Term 2

**Status for Mathematics students:** List A

**Commitment:** 30 one-hour lectures

**Assessment:** 3-hour examination

**Prerequisites:** MA225 Differentiation MA244 Analysis III, and MA231 Vector Analysis. MA3F1 Introduction to Topology would be helpful but not essential.

**Leads To: **MA475 Riemann Surfaces.

**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 Vector Analysis. 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 vector analysis. This includes complex differentiability, the Cauchy-Riemann equations, Cauchy's theorem, Taylor's and Liouville's theorem, Laurent expansions. Most of the course will be new topics: 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**:

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).