Higher Engineering Mathematics

Fourier series

L

69

Fourier series for periodic functions of

period 2π

69.1 Introduction

Fourier seriesprovides a method of analysing peri-
odic functions into their constituent components.
Alternating currents and voltages, displacement,
velocity and acceleration of slider-crank mecha-
nisms and acoustic waves are typical practical exam-
ples in engineering and science where periodic
functions are involved and often requiring analysis.

69.2 Periodic functions

A function f(x) is said to be periodic if
f(x+T)=f(x) for all values ofx, whereTis some
positive number. T is the interval between two
successive repetitions and is called theperiodof
the functionsf(x). For example,y=sinxis peri-
odic inxwith period 2πsince sinx=sin (x+ 2 π)
=sin (x+ 4 π), and so on. In general, ify=sinωt
then the period of the waveform is 2π/ω. The func-
tion shown in Fig. 69.1 is also periodic of period 2π
and is defined by:

f(x)=

{

−1, when−π<x< 0

1, when 0 <x<π

f(x)

0

1

− 1

2 ππ−π− 2 π x

Figure 69.1

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. 69.1

has finite 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.

69.3 Fourier series

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

of practical significance which are defined in

the interval−π≤x≤πcan be expressed in

terms of a convergent trigonometric series of

the form:

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

+a 3 cos 3x+···+b 1 sinx

+b 2 sin 2x+b 3 sin 3x+···

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

stants, i.e.

f(x)=a 0 +

∑∞

n= 1

(ancosnx+bnsinnx) (1)

where for the range−πtoπ:

and

a 0 =

1

2 π

∫π

−π

f(x)dx

an=

1

π

∫π

−π

f(x) cosnxdx

(n=1, 2, 3,...)

bn=

1

π

∫π

−π

f(x) sinnxdx

(n=1, 2, 3,...)