Chapter 66

Fourier series for periodic

functions of period 2π

66.1 Introduction

Fourier seriesprovides a method of analysing periodic
functions into their constituent components. Alternat-
ing currents and voltages, displacement, velocity and
acceleration of slider-crank mechanisms and acoustic
waves are typical practical examples in engineering and
science where periodic functions are involved and often
requiring analysis.

66.2 Periodicfunctions

A function f(x) is said to be periodic if
f(x+T)=f(x) for all values of x,whereT is
some positive number.Tis the interval between two
successive repetitions and is called the period of
the functions f(x). For example, y=sinx is peri-
odic in x with period 2π since sinx=sin(x+ 2 π)
=sin(x+ 4 π), and so on. In general, ify=sinωtthen
the period of the waveform is 2π/ω. The function
shown in Fig. 66.1 is also periodic of period 2πand is
defined by:

f(x)=

{

− 1 , when −π<x< 0

1 , when 0<x<π

If a graph of a function has no sudden jumps or breaks
it is called acontinuous function, examples being the
graphs of sine and cosine functions. However, other
graphs make finite jumps at a point or points in the
interval. The square wave shown in Fig. 66.1 has finite

x

f(x)

0 2 

1

21

22  2 

Figure 66.1

discontinuities atx=π,2π,3π, and so on. A great

advantage of Fourier series over other series is that it

can be applied to functions which are discontinuous as

well as those which are continuous.

66.3 Fourier series

(i) The basis of a Fourier series is that all func-

tions of practical significance which are defined in

the interval −π≤x≤π can be expressed in

terms of a convergent trigonometric series of the

form:

f(x)=a 0 +a 1 cosx+a 2 cos2x

+a 3 cos3x+···+b 1 sinx

+b 2 sin2x+b 3 sin3x+···

when a 0 ,a 1 ,a 2 ,...b 1 ,b 2 ,... are real con-

stants, i.e.