Page 356 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 13] FOURIER SERIES 347

f (x)

4p 2

x
_ 6p _ 4p _ 2p O 2p 4p 6p
Fig. 13-7

Period ¼ 2L ¼ 2 and L ¼ . Choosing c ¼ 0, we have
1 ð cþ2L n x 1 ð 2 2
f ðxÞ cos x cos nx dx
a n ¼ L dx ¼
L c 0
2
1 2 sin nx cos nx sin nx 4
n n n 0 n
¼ ðx Þ ð2xÞ 2 þ 2 3 ¼ 2 ; n 6¼ 0
1 ð 2 2 8 2
If n ¼ 0; a 0 ¼ x dx ¼ :
0 3
1 ð cþ2L n x 1 ð 2 2
f ðxÞ sin x sin nx dx
b n ¼ L dx ¼
L c 0
2
1 2 cos nx sin nx cos nx 4

n n n 0 n
¼ ðx Þ ð2xÞ 2 þð2Þ 3 ¼
4 2 X 4 4
1
2
3 n n
Then f ðxÞ¼ x ¼ þ 2 cos nx sin nx :
n¼1
2
This is valid for 0 < x < 2 .At x ¼ 0 and x ¼ 2 the series converges to 2 .
(b)Ifthe period is not specified, the Fourier series cannot be determined uniquely in general.
1 1 1 2
13.7. Using the results of Problem 13.6, prove that þ þ þ ¼ .
1 2 2 2 3 2 6
4 2 X 4
1
At x ¼ 0the Fourier series of Problem 13.6 reduces to þ .
3 n 2
n¼1
2
2
1
By the Dirichlet conditions, the series converges at x ¼ 0to ð0 þ 4 Þ¼ 2 .
2
4 2 X 4 2 X 1 2
1
1
Then þ ¼ 2 , and so ¼ .
3 n 2 n 2 6
n¼1 n¼1
ODD AND EVEN FUNCTIONS, HALF RANGE FOURIER SERIES
13.8. Classify each of the following functions according as they are even, odd, or neither even nor odd.
2 0 < x < 3

Period ¼ 6
ðaÞ f ðxÞ¼
2 3 < x < 0
From Fig. 13-8 below it is seen that f ð xÞ¼ f ðxÞ,sothat the function is odd.
cos x 0 < x <

Period ¼ 2
ðbÞ f ðxÞ¼
0 < x < 2