### Introduction

In this final Section on Fourier transforms we shall study briefly a number of topics such as Parsevalâ€™s theorem and the relationship between Fourier transform and Laplace transforms. In particular we shall obtain, intuitively rather than rigorously, various Fourier transforms of functions such as the unit step function which actually violate the basic conditions which guarantee the existence of Fourier transforms!

#### Prerequisites

- be aware of the definitions and simple properties of the Fourier transform and inverse Fourier transform.

#### Learning Outcomes

- use the unit impulse function (the Dirac delta function) to obtain various Fourier transforms

#### Contents

1 Parsevalâ€™s theorem2 Existence of Fourier transforms

3 Fourier transform and Laplace transforms

4 Some special Fourier transform pairs

5 Fourier transform of the unit step function