A half range Fourier series is a Fourier series defined on an interval
[
0
,
L
]
instead of the more common
[
−
L
,
L
]
, with the implication that the analyzed function
f
(
x
)
,
x
∈
[
0
,
L
]
should be extended to
[
−
L
,
0
]
as either an even (f(-x)=f(x)) or odd function (f(-x)=-f(x)). This allows the expansion of the function in a series solely of sines (odd) or cosines (even). The choice between odd and even is typically motivated by boundary conditions associated with a differential equation satisfied by
f
(
x
)
.
Example
Calculate the half range Fourier sine series for the function
f
(
x
)
=
cos
(
x
)
where
0
<
x
<
π
.
Since we are calculating a sine series,
a
n
=
0
∀
n
Now,
b
n
=
2
π
∫
0
π
cos
(
x
)
sin
(
n
x
)
d
x
=
2
n
(
(
−
1
)
n
+
1
)
π
(
n
2
−
1
)
∀
n
≥
2
When n is odd,
b
n
=
0
When n is even,
b
n
=
4
n
π
(
n
2
−
1
)
thus
b
2
k
=
8
k
π
(
4
k
2
−
1
)
With the special case
b
1
=
0
, hence the required Fourier sine series is
cos
(
x
)
=
8
π
∑
n
=
1
∞
n
(
4
n
2
−
1
)
sin
(
2
n
x
)
