Introduction & Chapter Objectives

Introduction

The Fourier transform is a fundamental tool in mathematical analysis used to express functions in terms of their frequency components. It plays a central role in many fields, including signal processing, physics, and engineering, due to its ability to convert complex differential equations into simpler algebraic expressions in the frequency domain. In this chapter, we introduce the definition of the Fourier transform and its inverse, explore essential properties that allow us to manipulate and simplify expressions, and learn how to compute transforms of common functions. We then apply the Fourier transform to solve differential equations and analyze systems where periodicity and frequency behavior are important

Chapter  Objectives

At the end of this chapter, students should be able to

1. Understand the definition of the Fourier transform and its inverse.

2. Use fundamental properties such as linearity, scaling, shifting, and convolution.

3. Compute Fourier transforms of basic functions.

4. Apply the Fourier transform to solve ordinary and partial differential equations.

5. Interpret the Fourier transform in the context of frequency analysis and signal processing.