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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.


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).

Additional Resources

Year 1 regs and modules
G100 G103 GL11 G1NC

Year 2 regs and modules
G100 G103 GL11 G1NC

Year 3 regs and modules
G100 G103

Year 4 regs and modules

Archived Material
Past Exams
Core module averages