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MA3B8 Complex Analysis

Lecturer: Peter Topping

Term(s): Term 1

Status for Mathematics students: List A

Commitment: 30 one-hour lectures

Assessment: 3 hour examination (100%).

Formal registration prerequisites: None

Assumed knowledge:

Please note that MA258 Mathematical Analysis III is NOT equivalent to MA244 Analysis III for the purposes of this course.

Useful knowledge: The "assumed knowledge" (and their prerequisites) will be enough.

Synergies: This course connects with virtually every other domain in both pure and applied mathematics.

Leads to: The following modules have this module listed as assumed knowledge or useful background:

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 III. 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).

Additional Resources