Page 355 - Schaum's Outline of Theory and Problems of Advanced Calculus

346 FOURIER SERIES [CHAP. 13

f (x)
Period

3
x
_ _ _
15 10 5 5 10 15
Fig. 13-6

(a)Period ¼ 2L ¼ 10 and L ¼ 5. Choose the interval c to c þ 2L as 5to 5,sothat c ¼ 5. Then
1 ð cþ2L n x 1 ð 5 n x
f ðxÞ cos f ðxÞ cos dx
a n ¼ dx ¼
L c L 5 5 5
ð 0 ð 5 ð 5
1 n x n x 3 n x
ð0Þ cos ð3Þ cos cos dx
5 5 5 0 5 5 0 5
¼ dx þ dx ¼
5
3 5 n x
sin ¼ 0 if n 6¼ 0
5 n 5 0
¼
3 ð 5 0 x 3 ð 5
cos dx ¼ 3:
If n ¼ 0; a n ¼ a 0 ¼ dx ¼
5 0 5 5 0
1 ð cþ2L n x 1 ð 5 n x
f ðxÞ sin f ðxÞ sin dx
b n ¼ dx ¼
L c L 5 5 5
1 ð 0 n x ð 5 n x 3 ð 5 n x
ð0Þ sin ð3Þ sin sin dx
5 5 5 0 5 5 0 5
¼ dx þ dx ¼

3 5 n x 5 3ð1 cos n Þ
cos
¼ ¼
5 n 5 0 n
(b) The corresponding Fourier series is
1
1
a 0 X n x n x 3 X 3ð1 cos n Þ n x
a n cos þ b n sin sin
2 þ L L ¼ 2 þ n 5
n¼1 n¼1
3 6 x 1 3 x 1 5 x
sin sin sin
¼ þ þ þ þ
2 5 3 5 5 5
(c) Since f ðxÞ satisfies the Dirichlet conditions, we can say that the series converges to f ðxÞ at all points of
continuity and to f ðx þ 0Þþ f ðx 0Þ at points of discontinuity. At x ¼ 5, 0, and 5, which are points
2
of discontinuity, the series converges to ð3 þ 0Þ=2 ¼ 3=2as seen from the graph. If we redefine f ðxÞ as
follows,
8
3=2 x ¼ 5
>
>
>
> 0 5 < x < 0
<
3=2 x ¼ 0 Period ¼ 10
f ðxÞ¼
>
> 3 0 < x < 5
>
>
3=2 x ¼ 5
:
then the series will converge to f ðxÞ for 5 @ x @ 5.
2
13.6. Expand f ðxÞ¼ x ; 0 < x < 2 in a Fourier series if (a) the period is 2 ,(b) the period is not
specified.
(a) The graph of f ðxÞ with period 2 is shown in Fig. 13-7 below.