Chapter 68

Even and odd functions and

half-range Fourier series

68.1 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=cosx
as shown in Fig. 18.25, page 186.

Odd functions

A function y=f(x)is said to beodd if f(−x)=
−f(x)for all values ofx. Graphs of odd functions are
alwayssymmetrical about the origin. Two examples
of odd functions arey=x^3 andy=sinxas shown in
Fig. 18.26, page 187.
Many functions are neither even nor odd, two such
examples being shown in Fig. 18.27, page 187.
See also Problems 3 and 4, page 187.

68.2 Fourier cosine and Fourier

sine series

(a) Fourier cosine series

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

Hencef(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 sineseries

The Fourier series of anoddperiodic function f(x)

having period 2πcontains 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