This is a structured reading module designed for MMath students in their fourth year. The module introduces the basic concepts and techniques of Complex Function Theory based on rational and elliptic functions, viewed as meromorphic functions on the sphere and the torus.
A rational function is the quotient of two polynomials and can be characterised as the meromorphic functions (holomorphic functions whose only singularities are poles) on the sphere. A bijective rational function is a Möbius transformation, and they play an important role in geometry.
Elliptic functions are doubly-periodic meromorphic functions defined on the complex plane, and these may be viewed as meromorphic functions on the torus. Elliptic functions lead naturally to the study of elliptic curves, the modular group (Möbius transformations with integer coefficients) and modular forms. These in turn appear in Number Theory, where they play a crucial role in the proof of Fermat’s Last Theorem.
Pre-requisites: (MATH2003 AND MATH2049) AND (MATH2045 OR MATH3088)