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10.4: EVEN AND ODD FUNCTIONS
KIAM HEONG KWA
1.
The Fourier Cosine and Sine Series
It is often desired to expand in a Fourier series of period 2
L
a func
tion
f
(
x
) originally deﬁned only on either one of the intervals (0
,L
),
(0
,L
], [0
,L
), and [0
,L
]. One way this can be done is the following.
We ﬁrst extend
f
(
x
) to a new function onto either one of the intervals
(

L,L
), (

L,L
], [

L,L
), and [

L,L
]. Then the new function is ex
tended periodically onto the real line
R
. Of particular simplicity and
applicability to us later are the following two extensions:
•
The even extension of
f
(
x
)
.
Let
(1.1)
g
(
x
) =
(
f
(
x
)
if 0
≤
x
≤
L,
f
(

x
) if

L < x <
0
.
Then we require that
g
(
x
+ 2
L
) =
g
(
x
) for all
x
∈
R
. The
function
g
(
x
) is called the
even extension
of
f
(
x
).
•
The odd extension of
f
(
x
)
.
Let
(1.2)
h
(
x
) =
f
(
x
)
if 0
< x < L,
0
if
x
= 0
,L,

f
(

x
) if

L < x <
0
.
Then we require that
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