The University of Southampton
Courses

# MATH6162 Integral Transform Methods

## Module Overview

Many classes of problems are difficult to solve in their original domain. An integral transform maps the problem from its original domain into a new domain in which solution is easier. The solution is then mapped back to the original domain with the inverse of the integral transform. This module will provide a systematic mathematical treatment of the theory of integral transforms and its varied applications in applied mathematics and engineering.

### Aims and Objectives

#### Learning Outcomes

##### Knowledge and Understanding

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

• Be confident in the use of complex variable theory and contour integration.
• Be able to demonstrate knowledge of a range of applications of these methods.
##### Subject Specific Intellectual and Research Skills

Having successfully completed this module you will be able to:

• Understand how integral transforms can be used to solve a variety of differential equations.

### Syllabus

• Analyticity of complex functions; Taylor and Laurent series • Contour integration of functions and multifunctions • Integration contours, including semi-circles, segments, box contours and keyhole contours • Use of complex methods for evaluation of real integrals • Laplace transforms and inverse Laplace transforms • Application of Laplace transforms to PDEs and to closed loop circuits (Nyquist stability theory) • Fourier transforms and their applications to PDEs • Fourier sine and cosine transforms and their applications to PDEs • Other examples of integral transforms (e.g. Hankel) and their applications • Fredholm theory (if time permits) • Higher-dimensional examples of integral transforms and their uses • Sturm-Liouville theory

### Learning and Teaching

#### Teaching and learning methods

The lecturer will provide a structured week-by-week study programme, based largely on the notes provided. Each week there will be three hours of lectures. There will be tutorials every week, each lasting one hour. The tutorial classes will be used to study problems illustrating the lecture material. Students should spend their private study time studying the lecture notes and working through these problem sets.

TypeHours
Independent Study120
Teaching60
Total study time180

H A Priestley. An introduction to complex analysis.

C Wylie and L C Barrett. Advanced Engineering Mathematics.

L Debnath. Integral transforms and their applications.

R V Churchill and J W Brown. Complex Variables and Applications.

M D Greenberg. Advanced Engineering Mathematics.

### Assessment

#### Summative

MethodPercentage contribution
Class Test 10%
Coursework 10%
Examination  (2 hours) 80%

#### Referral

MethodPercentage contribution
Exam 100%

#### Repeat Information

Repeat type: Internal & External