Introduction

In this Section we introduce the convolution of two functions $f\left(t\right),\phantom{\rule{1em}{0ex}}g\left(t\right)$ which we denote by $\left(f\ast g\right)\left(t\right)$ . The convolution is an important construct because of the convolution theorem which allows us to find the inverse Laplace transform of a product of two transformed functions:

$\phantom{\rule{2em}{0ex}}{\mathsc{L}}^{-1}\left\{F\left(s\right)G\left(s\right)\right\}=\left(f\ast g\right)\left(t\right)$

Prerequisites

• be able to find Laplace transforms and inverse Laplace transforms of simple functions
• be able to integrate by parts
• understand how to use step functions in integration

Learning Outcomes

• calculate the convolution of simple functions
• apply the convolution theorem to obtain inverse Laplace transforms