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1.2 Arbitrary Period and Half-Range Expansions 69

(a) (b)

(c) (d)

Figure 4 A function is given in the interval 0<x<a(heavy curve). The figure
shows: (a) the odd extension; (b) the even extension; (c) the odd periodic exten-
sion; and (d) the even periodic extension.

Definition
Letf(x)be given for 0<x<a.Theodd extensionoffis defined by

fo(x)=

{

f(x), 0 <x<a,

−f(−x), −a<x<0.

Theeven extensionoffis defined by

fe(x)=

{

f(x), 0 <x<a,

f(−x), −a<x<0.

Notice that if−a<x<0, then 0<−x<a, so the functional values on the
right are known from the given functions.
Graphically, the even extension is made by reflecting the graph in the vertical
axis. The odd extension is made by reflecting first in the vertical axis and then
in the horizontal axis (see Fig. 4).
Now the Fourier series of either extension may be calculated from the for-
mulas in Theorem 2. Sincefeis even andfois odd, we have

fe(x)∼a 0 +

∑∞

n= 1

ancos

(

nπx

a

)

, −a<x<a,

fo(x)∼

∑∞

n= 1

bnsin

(

nπx

a

)

, −a<x<a.