# Chapter 5Overview of Fourier Series

### Chapter Learning Objectives

• Appreciate that Fourier series are the mathematical form for periodic physical phenomena.
• Learn to use Fourier series to represent periodic physical phenomena in engineering analysis.
• Learn the required conditions for deriving Fourier series.
• Appreciate the principle of using Fourier series derived from the function for one period to apply the same Fourier series for other periods.
• Derive the mathematical expressions of Fourier series representing common physical phenomena.
• Understand the convergence of Fourier series of continuous periodic functions.
• Understand the convergence of Fourier series of piecewise continuous functions.
• Understand the convergence of Fourier series at discontinuities

## 5.1 Introduction

We learned in Chapter 2 that functions are used to represent physical phenomena in engineering analysis. In Section 2.4.2, we learned that step functions can be used to represent physical phenomena that exist at the inception of a specific time or location in spectra, and that impulsive functions describe phenomena of extremely short duration in time or extremely small extent in space. In this chapter, we will present another useful function that represents physical phenomena of periodic nature. Being periodical in their existence, these functions start and end at specific instants in time or location in space. They are expressed in the form of infinite series that is called the Fourier series. The functions ...

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