Course: ASO: Integral Transforms (2025-26)

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Lecturer: Profile: Andreas Muench

Course information

General prerequisites:

This course is highly recommended for Differential Equations 2.

Course term: Hilary

Course lecture information: 8 lectures

Course overview:

The Laplace and Fourier Transforms aim to take a differential equation in a function and turn it into an algebraic equation involving its transform. Such an equation can then be solved by algebraic manipulation, and the original solution determined by recognizing its transform or applying various inversion methods.
The Dirac delta-function, which is handled particularly well by transforms, is a means of rigorously dealing with ideas such as instantaneous impulse and point masses, which cannot be properly modelled using functions in the normal sense of the word. \(\delta\) is an example of a distribution or generalized function and the course provides something of an introduction to these generalized functions and their calculus.

Learning outcomes:

Students will gain a range of techniques employing the Laplace and Fourier Transforms in the solution of ordinary and partial differential equations. They will also have an appreciation of generalized functions, their calculus and applications.

Course synopsis:

Motivation for a "function'' with the properties the Dirac delta-function. Test functions. Continuous functions are determined by \(\phi\ \rightarrow \int f \phi\). Distributions and \(\delta\) as a distribution. Differentiating distributions. (3 lectures)
Theory of Fourier and Laplace transforms, inversion, convolution. Inversion of some standard Fourier and Laplace transforms via contour integration.
Use of Fourier and Laplace transforms in solving ordinary differential equations, with some examples including \(\delta\).
Use of Fourier and Laplace transforms in solving partial differential equations; in particular, use of Fourier transform in solving Laplace's equation and the Heat equation. (5 lectures)

Section outline

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- Select activity Announcements
  
  Announcements Forum
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  Reading List URL
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Course Materials
- Select activity Sheets 1 and 2 (single document)
  
  Sheets 1 and 2 (single document) Assignment
  
  Sheets 1 and 2 as a single document, to cover two tutorials. See the sheet for what to cover when.
  
  Additional note regarding problem sheet scheduling for HT2026:
  
  Please note that this year (2026), Integral transform will start with just one lecture in HT1 and end a bit later, in HT5, with lecture 8. Due to this slight shift, not all material that is required for the later problems may have been covered by the end of week 4, which would be the normal schedule. Moreover, the lectures are late in the week. Tutors may want to take this into account for their tutorial scheduling.
- Select activity Chapter on Distributions
  
  Chapter on Distributions Assignment
  
  Background reading on distributions, from the book Practical Applied Mathematics.
- Select activity Lecture Notes for ASO Integral Transforms
  
  Lecture Notes for ASO Integral Transforms Assignment
  
  Opened: Tuesday, 6 January 2026, 12:00 AM
  
  The full lecture notes for the course, by Richard Earl/Sam Howison. You will need this to fill in details from lectures. It also has many examples different from those in lectures.
- Select activity ASO_IT_slides
  
  ASO_IT_slides File
  
  The slides for the HT2025 lectures

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Section outline

General

Course Materials