### Introduction

The Laplace transformation is a technique employed primarily to solve constant coefficient ordinary differential equations. It is also used in modelling engineering systems. In this section we look at those functions to which the Laplace transformation is normally applied; so-called
**
causal
**
or
**
one-sided functions
**
. These are functions
$f\left(t\right)$
of a single variable
$t$
such that
$f\left(t\right)=0$
if
$t<0$
. In particular we consider the simplest causal function: the unit step function (often called the Heaviside function)
$u\left(t\right)$
:

$\phantom{\rule{2em}{0ex}}u\left(t\right)=\left\{\begin{array}{cc}1\phantom{\rule{1em}{0ex}}\hfill & \text{if}\phantom{\rule{1em}{0ex}}t\ge 0\hfill \\ 0\phantom{\rule{1em}{0ex}}\hfill & \text{if}\phantom{\rule{1em}{0ex}}t<0\hfill \end{array}\right.$

We then use this function to show how signals (functions of time $t$ ) may be ‘switched on’ and ‘switched off’.

#### Prerequisites

- understand what a function is
- be able to integrate simple functions

#### Learning Outcomes

- explain what a causal function is
- be able to apply the step function to ‘switch on’ and ‘switch off’ signals