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As we have seen, Fourier series are constructed from orthogonal functions. Generalizations of Fourier series are of great interest and utility both from theoretical and applied viewpoints.
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CHAP. 13]
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FOURIER SERIES 13.1. Graph each of the following functions. & 3 0<x<5 Period 10 a f x 3 5 < x < 0
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3 _ 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 x is not de ned at x 0; 5; 5; 10; 10; 15; 15, and so on. These values are the discontinuities of f x .
& b f x
sin x 0
0@x@  < x < 2
Period 2
f (x) Period
_ 3p
_ 2p
Fig. 13-4 Refer to Fig. 13-4 above. Note that f x is de ned for all x and is continuous everywhere.
8 >0 < c f x 1 > : 0
0@x<2 2@x<4 4@x<6
f (x)
Period 6
Period
_ 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 x is de ned for all x and is discontinuous at x 2; 4; 8; 10; 14; . . . .
344 L 13.2. Prove
FOURIER SERIES
[CHAP. 13
kx dx L
L cos
kx dx 0 L
if k 1; 2; 3; . . . .
 kx L kx L  L cos k L cos k 0 dx cos L k L  L k k L L L  kx L kx  L L dx sin sin k sin k 0 cos L k L  L k k L
13.3. Prove
& L mx nx mx nx 0 cos dx sin dx (a) cos sin L L L L L L L L mx nx (b) cos dx 0 sin L L L L
m 6 n m n
where m and n can assume any of the values 1; 2; 3; . . . .
(a) From trigonometry: cos A cos B 1 fcos A B cos A B g; sin A sin B 1 fcos A B cos 2 2 A B g: Then, if m 6 n, by Problem 13.2, ' L & mx nx 1 L m n x m n x cos dx cos cos cos dx 0 L L 2 L L L L Similarly, if m 6 n, ' L & mx nx 1 L m n x m n x sin dx cos dx 0 sin cos L L 2 L L L L If m n, we have L  L  2nx dx L 1 cos L L L  L  mx nx 1 L 2nx sin dx dx L sin 1 cos L L 2 L L L cos mx nx 1 cos dx L L 2
Note that if m n these integrals are equal to 2L and 0 respectively. (b) We have sin A cos B 1 fsin A B sin A B g. Then by Problem 13.2, if m 6 n, 2 ' L & mx nx 1 L m n x m n x sin sin cos dx sin dx 0 L L 2 L L L L If m n, L sin
mx nx 1 cos dx L L 2
L sin
2nx dx 0 L
The results of parts (a) and (b) remain valid even when the limits of integration L; L are replaced by c; c 2L, respectively.
13.4. If the series A
1 X n 1
an cos
for n 1; 2; 3; . . . ; 1 L nx dx; f x cos a an L L L
nx nx bn sin converges uniformly to f x in L; L , show that L L 1 b bn L L
f x sin
nx dx; L
c A
a0 : 2
CHAP. 13]
FOURIER SERIES
(a) Multiplying f x A by cos
1 X n 1
an cos
nx nx bn sin L L
mx and integrating from L to L, using Problem 13.3, we have L L L mx mx dx A dx f x cos cos L L L L ' L L 1 & X mx nx mx nx cos dx bn sin dx an cos cos L L L L L L n 1 am L 1 L L
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