Even and odd functions and half-range Fourier series 629
Hence the half-range Fourier sine series for f (x) in the
range 0 to π is given by:
f (x) =
8
3π
sin 2x +
16
15π
sin 4x
+
24
35π
sin 6x + · · ·
or f (x)=
8
π
1
3
sin 2x +
2
(3)(5)
sin 4x
+
3
(5)(7)
sin 6x +· · ·
Now try the following exercise
Exercise 231 Further problems on
half-range Fourier series
1. Determine the half-range sine series for the
function defined by:
f (x) =
⎧
⎨
⎩
x, 0 < x <
π
2
0,
π
2
< x < π
⎡
⎢
⎢
⎢
⎢
⎢
⎣
f (x) =
2
π
sin x +
π
4
sin 2x
−
1
9
sin 3x
−
π
8
sin 4x + · · ·
⎤
⎥
⎥
⎥
⎥
⎥
⎦
2. Obtain (a) the half-range cosine series and
(b) the half-range sine series for the function
f (t ) =
⎧
⎪ ⎨
⎪ ⎩
0, 0 < t <
π
2
1,
π
2
< t < π
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
(a) f (t ) =
1
2
−
2
π
cos t
−
1
3
cos 3t
+
1
5
cos 5t − · · ·
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
(b) f (t ) =
2
π
sin t − sin 2t
+
1
3
sin 3t +
1
5
sin 5t
−
1
3
sin 6t + · · ·
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
3. Find (a) the half-range Fourier sine series and
(b) the half-range Fourier cosine series for the
function f (x) = sin 2 x in the range 0 ≤ x ≤ π.
Sketch the function within and outside of the
given range.
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
(a) f (x) =
8
π
sin x
(1)(3)
−
sin 3x
(1)(3)(5)
−
sin 5x
(3)(5)(7)
−
sin 7x
(5)(7)(9)
− · · ·
(b) f (x) =
1
2
(1 − cos 2x)
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
4. Determine the half-range Fourier cosine series
in the range x = 0 to x = π for the function
defined by:
f (x) =
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
x,
0 < x <
π
2
(π − x),
π
2
< x < π
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
f (x) =
π
4
−
2
π
cos 2x
+
cos 6x
3 2
+
cos 10x
5 2 + · · ·
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦