Introduction
In this Section we formally introduce the Laplace transform. The transform is only applied to causal functions which were introduced in Section 20.1. We find the Laplace transform of many commonly occurring ‘signals’and produce a table of standard Laplace transforms.
We also consider the inverse Laplace transform. To begin with, the inverse Laplace transform is obtained ‘by inspection’ using a table of transforms. This approach is developed by employing techniques such as partial fractions and completing the square introduced in HELM booklet 3.6.
Prerequisites
 understand what a causal function is
 be able to find and use partial fractions
 be able to perform integration by parts
 be able to use the technique of completing the square
Learning Outcomes
 find the Laplace transform of many commonly occurrring causal functions

obtain the inverse Laplace transform using techniques involving
 a table of transforms
 partial fractions
 completing the square
 the first shift theorem
Contents
1 The Laplace transform1.1 The linearity property of the Laplace transformation
2 The inverse Laplace transform
2.1 Inverting through the use of partial fractions
3 The first shift theorem
3.1 Inverting using completion of the square