Higher Engineering Mathematics

658 FOURIER SERIES

(ii)a 0 ,anandbnare called theFourier coefficients

of the series and if these can be determined,

the series of equation (1) is called theFourier

seriescorresponding tof(x).

(iii) An alternative way of writing the series is by
using the acosx+bsinx=csin (x+α) rela-
tionship introduced in Chapter 18, i.e.
f(x)=a 0 +c 1 sin (x+α 1 )+c 2 sin (2x+α 2 )

+···+cnsin (nx+αn),

wherea 0 is a constant,

c 1 =

√

(a 12 +b^21 ),...cn=

√

(a^2 n+b^2 n)

are the amplitudes of the various components,

and phase angle

αn=arctan

an

bn

(iv) For the series of equation (1): the term

(a 1 cosx+b 1 sinx)orc 1 sin (x+α 1 ) is called

thefirst harmonicor thefundamental, the

term (a 2 cos 2x+b 2 sin 2x)orc 2 sin (2x+α 2 )

is called thesecond harmonic, and so on.

For an exact representation of a complex wave, an

infinite number of terms are, in general, required. In

many practical cases, however, it is sufficient to take

the first few terms only (see Problem 2).

The sum of a Fourier series at a point of dis-

continuityis given by the arithmetic mean of the

two limiting values off(x)asxapproaches the point

of discontinuity from the two sides. For example,

for the waveform shown in Fig. 69.2, the sum of the

Fourier series at the points of discontinuity (i.e. at

π

2

,π,...is given by:

8 +(−3)

2

=

5

2

or 2

1

2

f(x)

8

−π −π/ 2 0

− 3

π/ 2 π 3 π/ 2 x

Figure 69.2

69.4 Worked problems on Fourier

series of periodic functions of

period 2π

Problem 1. Obtain a Fourier series for the

periodic functionf(x) defined as:

f(x)=

{

−k, when−π<x< 0

+k, when 0 <x<π

The function is periodic outside of this range

with period 2π.

The square wave function defined is shown in

Fig. 69.3. Sincef(x) is given by two different expres-

sions in the two halves of the range the integration

is performed in two parts, one from−πto 0 and the

other from 0 toπ.

f(x)

0

k

−k

−π π 2 π x

Figure 69.3

From Section 69.3(i):

a 0 =

1

2 π

∫π

−π

f(x)dx

=

1

2 π

[∫ 0

−π

−kdx+

∫π

0

kdx

]

=

1

2 π

{[−kx]^0 −π+[kx]π 0 }= 0

[a 0 is in fact themean valueof the waveform over

a complete period of 2πand this could have been

deduced on sight from Fig. 69.3.]

From Section 69.3(i):

an=

1

π

∫π

−π

f(x) cosnxdx

=

1

π

{∫ 0

−π

−kcosnxdx+

∫π

0

kcosnxdx

}

=

1

π

{[

−ksinnx

n

] 0

−π

+

[

ksinnx

n

]π

0

}

= 0