Higher Engineering Mathematics

L

Fourier series

71

Even and odd functions and half-range

Fourier series

71.1 Even and odd functions

Even functions

A functiony=f(x) is said to beeveniff(−x)=f(x)
for all values ofx. Graphs of even functions are
always symmetrical about they-axis (i.e. is a
mirror image). Two examples of even functions
arey=x^2 andy=cosxas shown in Fig. 19.25,
page 199.

Odd functions

A functiony=f(x) is said to beoddiff(−x)=
−f(x) for all values ofx. Graphs of odd functions are
alwayssymmetrical about the origin. Two exam-

ples of odd functions arey=x^3 andy=sinxas
shown in Fig. 19.26, page 200.
Many functions are neither even nor odd, two such
examples being shown in Fig. 19.27, page 200.
See also Problems 3 and 4, page 200.

71.2 Fourier cosine and Fourier sine

series

(a) Fourier cosine series

The Fourier series of aneven periodicfunction
f(x) having period 2πcontainscosine terms only
(i.e. contains no sine terms) and may contain a
constant term.

Hence f(x)=a 0 +

∑∞

n= 1

ancosnx

where a 0 =

1

2 π

∫π

−π

f(x)dx

=

1

π

∫π

0

f(x)dx

(due to symmetry)

and an=

1

π

∫π

−π

f(x) cosnxdx

=

2

π

∫π

0

f(x) cosnxdx

(b) Fourier sine series

The Fourier series of anoddperiodic functionf(x)

having period 2π contains sine terms only (i.e.

contains no constant term and no cosine terms).

Hence f(x)=

∑∞

n= 1

bnsinnx

where bn=

1

π

∫π

−π

f(x) sinnxdx

=

2

π

∫π

0

f(x)sinnxdx

Problem 1. Determine the Fourier series for

the periodic function defined by:

f(x)=

⎧

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎩

−2, when−π<x<−

π

2

2, when−

π

2

<x<

π

2

−2, when

π

2

<x<π.

and has a period of 2π

The square wave shown in Fig. 71.1 is an even

function since it is symmetrical about thef(x) axis.

Hence from para. (a) the Fourier series is given by:

f(x)=a 0 +

∑∞

n= 1

ancosnx

(i.e. the series contains no sine terms)