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# MATH6094 Complex Function Theory

## Module Overview

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.

### Aims and Objectives

#### Module Aims

This is a structured reading module, designed for MMath students in their fourth year, which 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.

#### Learning Outcomes

##### Learning Outcomes

Having successfully completed this module you will be able to:

• analyse the behaviour of a meromorphic function on the Riemann sphere, and relate it to general properties of such functions
• define Möbius transformations, explain their main properties, and illustrate them in specific examples
• recognise periodic functions, explain their main properties, and illustrate those properties in specific examples
• define elliptic functions, explain their main properties, and illustrate those properties in specific examples
• define the modular group, explain its main properties, and illustrate those properties in specific examples
• critically reflect on your own written work

### Syllabus

• The Riemann sphere • Möbius transformations • Periodic functions • Elliptic functions • The modular group

### Learning and Teaching

#### Teaching and learning methods

This module is primarily a directed reading module. The lecturer will provide a structured reading programme, based largely on the textbook by Jones and Singerman, but with references to other material as appropriate. In the timetabled sessions, the lecturer will summarise the next reading section, give solutions to problem sheets, and deal with students' questions. Students should spend their private study time in reading, attempting the problem sheets, and working on the coursework.

TypeHours
Teaching24
Independent Study126
Total study time150

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

AHLFORS L V. Complex Analysis.

G.A. Jones and D. Singerman (1987). Complex Functions: An algebraic and geometric viewpoint.

COHN H. Conformal Mapping on Riemann Surfaces.

### Assessment

#### Summative

MethodPercentage contribution
Coursework 100%

#### Repeat Information

Repeat type: Internal & External

Pre-requisites: MATH2003 AND MATH2046 AND MATH2045 AND MATH3079

### Costs

#### Costs associated with this module

Students are responsible for meeting the cost of essential textbooks, and of producing such essays, assignments, laboratory reports and dissertations as are required to fulfil the academic requirements for each programme of study.

In addition to this, students registered for this module typically also have to pay for:

##### Textbooks

This is a reading course based on the textbook Complex Functions: An algebraic and geometric viewpoint, by Gareth Jones and David Singerman. You will need a copy of this book. There are 5 copies in the Hartley library (QA331 JON) or you are welcome to buy a new, or used, copy.

Please also ensure you read the section on additional costs in the University’s Fees, Charges and Expenses Regulations in the University Calendar available at www.calendar.soton.ac.uk.

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