### Introduction

Fourier transforms have for a long time been a basic tool of applied mathematics, particularly for solving differential equations (especially partial differential equations) and also in conjunction with integral equations.

There are really three Fourier transforms, the Fourier Sine and Fourier Cosine transforms and a complex form which is usually referred to as
**
the
**
Fourier transform.

The last of these transforms in particular has extensive applications in Science and Engineering, for example in physical optics, chemistry (e.g. in connection with Nuclear Magnetic Resonance and Crystallography), Electronic Communications Theory and more general Linear Systems Theory.

#### Prerequisites

- be familiar with basic Fourier series, particularly in the complex form

#### Learning Outcomes

- calculate simple Fourier transforms from the definition
- state how the Fourier transform of a function (signal) depends on whether that function is even or odd or neither

#### Contents

1 The Fourier transform2 Informal derivation of the Fourier transform

3 Existence of the Fourier transform

4 Basic properties of the Fourier transform

4.1 Real and imaginary parts of a Fourier transform

4.2 Polar form of a Fourier transform