Page 352 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 13] FOURIER SERIES 343

Solved Problems

FOURIER SERIES

13.1. Graph each of the following functions.
3 0 < x < 5

Period ¼ 10
ðaÞ f ðxÞ¼
3 5 < x < 0
f (x)
Period
3
x
_ _ _ _ _
25 20 15 10 5 0 3 5 10 15 20 25

Fig. 13-3

Since the period is 10, that portion of the graph in 5 < x < 5 (indicated heavy in Fig. 13-3 above) is
extended periodically outside this range (indicated dashed). Note that f ðxÞ is not defined at
x ¼ 0; 5; 5; 10; 10; 15; 15, and so on. These values are the discontinuities of f ðxÞ.

sin x 0 @ x @
Period ¼ 2
ðbÞ f ðxÞ¼
0 < x < 2
f (x)
Period
x
_ _ _
3p 2p p 0 p 2p 3p 4p
Fig. 13-4
Refer to Fig. 13-4 above. Note that f ðxÞ is defined for all x and is continuous everywhere.

8
0 0 @ x < 2
>
<
1 2 @ x < 4 Period ¼ 6
ðcÞ f ðxÞ¼
>
0 4 @ x < 6
:
f (x)
Period
1
x
_ _ _ _ _ _
12 10 8 6 4 2 0 2 4 6 8 10 12 14
Fig. 13-5

Refer to Fig. 13-5 above. Note that f ðxÞ is defined for all x and is discontinuous at x ¼ 2; 4; 8;
10; 14; .. . .