622 Higher Engineering Mathematics
in the range 0 to 2π.
y
0
21
1
2
Figure 67.5
8. Sketch the waveform defined by:
f (x) =
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
1 +
2x
π
, when −π < x < 0
1 −
2x
π
, when 0 < x < π
Determine the Fourier series in this range.
⎡
⎢
⎢
⎣
f (x) =
8
π 2
cos x +
1
3 2 cos 3x
+
1
5 2 cos 5x +
1
7 2 cos 7x + · · ·
⎤
⎥
⎥
⎦
9. For the Fourier series of Problem 8, deduce a
series for
π 2
8
π 2
8
= 1 +
1
3 2 +
1
5 2 +
1
7 2 +
1
9 2 + · · ·
in the range 0 to 2π.
y
0
21
1
2
Figure 67.5
8. Sketch the waveform defined by:
f (x) =
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
1 +
2x
π
, when −π < x < 0
1 −
2x
π
, when 0 < x < π
Determine the Fourier series in this range.
⎡
⎢
⎢
⎣
f (x) =
8
π 2
cos x +
1
3 2 cos 3x
+
1
5 2 cos 5x +
1
7 2 cos 7x + · · ·
⎤
⎥
⎥
⎦
9. For the Fourier series of Problem 8, deduce a
series for
π 2
8
π 2
8
= 1 +
1
3 2 +
1
5 2 +
1
7 2 +
1
9 2 + · · ·