Page 359 - Schaum's Outline of Theory and Problems of Advanced Calculus

350 FOURIER SERIES [CHAP. 13

1 ð 1 cosðn þ 1Þx cosðn 1Þx

n þ 1 n 1
¼ fsinðx þ nxÞþ sinðx nxÞg ¼ þ
0 0
1 1 cosðn þ 1Þ cosðn 1Þ 1 1 1 þ cos n 1 þ cos n

n þ 1 n 1 n þ 1 n 1
¼ þ ¼
if n 6¼ 1:
2ð1 þ cos n Þ
2
¼
ðn 1Þ
2
2 ð 2 sin x
For n ¼ 1; a 1 ¼ sin x cos xdx ¼ ¼ 0:
0 2 0
2 ð 2 4
For n ¼ 0; a 0 ¼ sin xdx ¼ ð cos xÞ ¼ :
0 0
2 2 X
1
Then f ðxÞ¼ ð1 þ cos n Þ cos nx
2
n 1
n¼2

2 4 cos 2x cos 4x cos 6x
2 1 4 1 6 1
¼ 2 þ 2 þ 2 þ
13.12. Expand f ðxÞ¼ x; 0 < x < 2, in a half range (a) sine series, (b) cosine series.
(a) Extend the definition of the given function to that of the odd function of period 4 shown in Fig. 13-12
below. This is sometimes called the odd extension of f ðxÞ. Then 2L ¼ 4; L ¼ 2.
f (x)

O
x
_ 6 _ 4 _ 2 2 4 6

Fig. 13-12

Thus a n ¼ 0 and

2 ð L n x 2 ð 2 n x
f ðxÞ sin x sin dx
b n ¼ dx ¼
L 0 L 2 0 2

2
2 n x 4 n x 4
cos sin cos n
n 2 n 2 0 n
¼ðxÞ ð1Þ 2 2 ¼
4 n x
1
X
Then f ðxÞ¼ cos n sin
n 2
n¼1

4 x 1 2 x 1 3 x
sin sin sin
2 2 2 3 2
¼ þ
(b) Extend the definition of f ðxÞ to that of the even function of period 4 shown in Fig. 13-13 below. This is
the even extension of f ðxÞ. Then 2L ¼ 4; L ¼ 2.