# BSc Computer Science and Mathematics

Year of entry: 2023

## Course unit details:2P2: Complex Analysis

Unit code MATH29142 10 Level 2 Semester 2 Department of Mathematics No

### Overview

This course introduces the calculus of complex functions of a complex variable.  Complex  differentiability is a very strong condition and differentiable functions have many strong properties.  Integration is along paths in the complex plane. The central result of this spectacularly beautiful part of mathematics is Cauchy’s Theorem guaranteeing that certain integrals along closed paths are zero.  This striking result leads to useful techniques for evaluating real integrals based on the ‘calculus of residues’.

### Pre/co-requisites

Unit title Unit code Requirement type Description
Foundations of Pure Mathematics A MATH10101 Pre-Requisite Compulsory
Calculus and Vectors A MATH10121 Pre-Requisite Compulsory
Real Analysis A MATH20101 Pre-Requisite Compulsory

### Aims

The unit aims to introduce the basic ideas of complex analysis, with particular emphasis on Cauchy’s Theorem and the calculus of residues

### Learning outcomes

On succesful completion of the course, students will be able to:

• Prove the Cauchy-Reimann Theorem and its converse and use them decide whether a given function is holomorphic
• Use power series to define a holomorphic function and calculate its radius of convergence, and perform computations with such series. Define elementary holomorphic functions such as sin. cos. sinh. cosh. exp. log.
• Define the complex integral and use a variety of methods (the Fundamental Theorem and the Cauchy Residue Theorem) to calculate the complex integral of a given function
• Use Taylor's Theorem and Laurent's Theorem to expand a holomorphic function in terms of power series on a disc and Laurent series on an annulus respectively
• Identify the location and nature of a singularity of a function and, in case of poles. Calculate the order and the residue
• Apply techniques from complex analysis to deduce results in other areas of mathematics. Including proving the Fundamental Theorem of Algebra and calculating infinite real integrals, trigonometric integrals and the summation of series

### Teaching and learning methods

2 hours of lectures and 1 hour tutorial per week, for weeks 1-11 of semester 2.

### Assessment methods

Method Weight
Other 20%
Written exam 80%

In-class coursework test
Test mid-way through semester; marked scripts returned within 15 days
20% Weighting

Written exam
General feedback provided after exam is marked
80% Weighting

Ian Stewart and David Tall, Complex Analysis, Cambridge University Press, 1983.

### Study hours

Scheduled activity hours
Lectures 22
Tutorials 11
Independent study hours
Independent study 67

### Teaching staff

Staff member Role
Michael Livesey Unit coordinator