### 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
1. a table of transforms
2. partial fractions
3. completing the square
4. the first shift theorem