### Introduction

In this Section we introduce the second shift theorem which simplifies the determination of Laplace and inverse Laplace transforms in some complicated cases.

Then we obtain the Laplace transform of derivatives of causal functions. This will allow us, in the next Section, to apply the Laplace transform in the solution of ordinary differential equations.

Finally, we introduce the delta function and obtain its Laplace transform. The delta function is often needed to model the effect on a system of a forcing function which acts for a very short time.

#### Prerequisites

- be able to find Laplace transforms and inverse Laplace transforms of simple causal functions
- be familiar with integration by parts
- understand what an initial-value problem is
- have experience of the first shift theorem

#### Learning Outcomes

- use the second shift theorem to obtain Laplace transforms and inverse Laplace transforms
- find the Laplace transform of the derivative of a causal function

#### Contents

1 The second shift theorem2 The Laplace transform of a derivative

3 The delta function (or impulse function)

3.1 The Laplace transform of the delta function