The University of Southampton
Courses

# MATH6094 Complex Function Theory

## Module Overview

This is a structured reading module, designed for MMath students in their fourth year. The module aims to introduce students to the basic concepts and techniques of Complex Function Theory, building on earlier work on complex functions, algebra, analysis and geometry at levels 2 and 3. These topics were developed in the 19th century, but have become very relevant to present day mathematics in disciplines as diverse as number theory and physics. The module will concentrate on rational and elliptic functions, viewed as meromorphic functions on the sphere and the torus. A rational function is the quotient of two polynomials. We shall characterize these as the meromorphic functions on the sphere (those whose only singularities are poles). A bijective rational function is a Möbius transformation, and these are important 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, which are cubic curves. Their deeper study leads us to the modular group, which consists of the Möbius transformations with integer coefficients, and then on to the study of modular forms. These are of great use in many topics in number theory, where they played an important role in the proof of Fermat’s Last Theorem

### Aims and Objectives

#### Module Aims

To introduce students to the basic concepts and techniques of Complex Function Theory, building of earlier work on complex functions, algebra, analysis and geometry at levels 2 and 3.

#### Learning Outcomes

##### Knowledge and Understanding

Having successfully completed this module, you will be able to demonstrate knowledge and understanding of:

• Understand the meromorphic functions on the sphere and the rational functions
• Understand Mobius transformations as bijective rational functions
• Understand elliptic functions as meromorphic functions on the torus
• Understand the relationship between lattices, elliptic functions,elliptic curves and the modular group.
##### Subject Specific Intellectual and Research Skills

Having successfully completed this module you will be able to:

• Use cross-ratios to further understand Mobius transformations
• Recall the basic properties of the Weierstrass s, ?, and P functions

### Syllabus

• The Riemann sphere • Meromorphic functions • Periodic functions • Lattices • Construction of elliptic functions • Basic properties of elliptic functions • Elliptic curves • The modular group

### Learning and Teaching

#### Teaching and learning methods

The lecturer will provide a structured week-by-week reading programme, based largely on the textbook by Jones and Singerman, but with references to other material as appropriate. Each week there will be two timetabled lectures. These will be conducted as informal classes, in which the lecturer will summarise the next reading section, give solutions to problem sheets, or deal with students' questions as appropriate. Students should spend their private study time doing the reading and attempting the problem sheets The assessment for this module will be based on a portfolio of five sets of solved problems, submitted periodically during the module. A high standard of explanation will be expected in the solutions; this will form part of the assessment criteria, in addition to the mathematical correctness of the solutions. Feedback to students will be prompt and formative.

TypeHours
Teaching24
Independent Study126
Total study time150

COHN H. Conformal Mapping on Riemann Surfaces.

KNOPP K. Theory of Functions (Vols. I and II).

AHLFORS L V. Complex Analysis.

JONES G A & SINGERMAN D (1987). Complex Functions (An algebraic and geometric viewpoint).

### Assessment

#### Summative

MethodPercentage contribution
Coursework 100%

#### Repeat Information

Repeat type: Internal & External

#### Pre-requisites

To study this module, you will need to have studied the following module(s):

CodeModule
MATH3079Metric Spaces and Topology
MATH2045Vector Calculus and Complex Variable
MATH2046Algebra and Geometry
MATH2003Group Theory