Module Information

Module Identifier
Module Title
Integral Transforms
Academic Year
Semester 1
Mutually Exclusive
MA21510 or MT21510
External Examiners
  • Dr Andrew Hazel (Reader - University of Manchester)
Other Staff

Course Delivery

Delivery Type Delivery length / details
Lecture 22 x 1 Hour Lectures


Assessment Type Assessment length / details Proportion
Semester Exam 2 Hours   Written Examination  100%
Supplementary Exam 2 Hours   Written Examination  100%

Learning Outcomes

On successful completion of this module students should be able to:

1. Construct various integral transforms and be able to define Fourier, Laplace and Mellin transforms.
2. State the main properties of these three transforms and evaluate their inverses.
3. Make use of the transforms to solve linear ODEs and PDEs.

Brief description

In their original formulation, many problems in mathematics appear intractable. The application of an integral transform to an equation often acts to simplify its solution. This module will explain the basic ideas behind integral transforms, generalizing the idea of Fourier transforms to include Laplace and Mellin transforms. Their use will be demonstrated in the solution of both ordinary and partial differential equations. The idea of generalized functions will be introduced, and their properties and applications described.


• Brief description of complex integration: Cauchy's theorem, the residue theorem. Principal value integrals. Computations of standard integrals using contour integrals.
• Distributions and the Dirac delta function.
• The Fourier transform. Linearity, dilation, translation, convolution, derivatives, the inversion formula.
• The Laplace transform. Linearity, dilation, convolution, derivatives, the inversion formula.
• The Mellin transform. Linearity, dilation, convolution, derivatives, the inversion formula.
• Applications of integral transforms to (partial) differential equations, integral equations, difference equations, solid and fluid mechanics.

Module Skills

Skills Type Skills details
Application of Number Throughout
Communication Written answers to questions must be clear and well-structured, and should communicate students’ understanding
Improving own Learning and Performance Students are expected to develop their own approach to time-management regarding the completion of Example sheets on time, assimilation of feedback, and preparation between lectures.
Information Technology Use of blackboard
Problem solving Throughout
Research skills
Subject Specific Skills Students will be encouraged to independently find and assimilate useful resources.
Team work Students will be encouraged to work together on questions in workshops and on Example sheets.


This module is at CQFW Level 6