Chapter 68
Even and odd functions and
half-range Fourier series
68.1 Even and odd functions
Even functions
A function y = f (x) is said to be even if f (−x) = f (x)
for all values of x. Graphs of even functions are always
symmetrical about the y-axis (i.e. is a mirror image).
Two examples of even functions are y= x 2 and y = cos x
as shown in Fig. 18.25, page 186.
Odd functions
A function y = f (x) is said to be odd if f (−x) =
− f (x) for all values of x. Graphs of odd functions are
always symmetrical about the origin. Two examples
of odd functions are y = x 3 and y = sin x as 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 2π contains cosine terms only
(i.e. contains no sine terms) and may contain a constant
term.
Hence f (x) = a 0 +
∞
n=1
a n cos nx
where a 0 =
1
2π
π
−π
f (x) dx
=
1
π
π
0
f (x)dx
(due to symmetry)
and
a n =
1
π
π
−π
f (x) cos nx dx
=
2
π
π
0
f (x)cos nx dx
(b) Fourier sine series
The Fourier series of an odd periodic function f (x)
having period 2π contains sine terms only (i.e. contains
no constant term and no cosine terms).
Hence f (x) =
∞
n=1
b n sin nx
where
b n =
1
π
π
−π
f (x) sin nx dx
=
2
π
π
0
f (x) sin nx dx