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
- MA244 Analysis III
- MA259 Multivariable Calculus
- MA260 Norms Metrics and Topologies or MA222 Metric Spaces
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:
- MA3D1 Fluid Dynamics
- MA475 Riemann Surfaces
- MA4H9 Modular Forms
- MA4L6 Analytic Number Theory
- MA426 Elliptic Curves
- MA4L7 Algebraic Curves
- MA448 Hyperbolic Geometry
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).