Page 353 - Schaum's Outline of Theory and Problems of Advanced Calculus

344 FOURIER SERIES [CHAP. 13

L k x L k x
ð ð
13.2. Prove sin dx ¼ cos dx ¼ 0 if k ¼ 1; 2; 3; ... .
L L L L
ð L k x L L L L
sin dx ¼ cos k x ¼ cos k þ cosð k Þ¼ 0
L L k L L k k
L k x L k x L L
ð L
cos dx ¼ sin ¼ sin k sinð k Þ¼ 0
L L k L L k k

ð L m x n x ð L m x n x 0 m 6¼ n
13.3. Prove (a) cos cos dx ¼ sin sin dx ¼
L L L L L L L m ¼ n
ð L m x n x
(b) sin cos dx ¼ 0
L L L
where m and n can assume any of the values 1; 2; 3; ... .
1
1
(a)From trigonometry: cos A cos B ¼ fcosðA BÞþ cosðA þ BÞg; sin A sin B ¼ fcosðA BÞ cos
2 2
ðA þ BÞg:
Then, if m 6¼ n,byProblem 13.2,
ð L m x n x 1 ð L ðm nÞ x ðm þ nÞ x
cos cos dx ¼ cos þ cos dx ¼ 0
L L L 2 L L L
Similarly, if m 6¼ n,
ð L m x n x 1 ð L ðm nÞ x ðm þ nÞ x
sin sin dx ¼ cos cos dx ¼ 0
L L L 2 L L L
If m ¼ n,we have
L m x n x 1 L 2n x
ð ð
cos cos dx ¼ 1 þ cos dx ¼ L
L L L 2 L L
L m x n x 1 L 2n x
ð ð
sin sin dx ¼ 1 cos dx ¼ L
L L L 2 L L
Note that if m ¼ n these integrals are equal to 2L and 0 respectively.
1
(b)We have sin A cos B ¼ fsinðA BÞþ sinðA þ BÞg. Then by Problem 13.2, if m 6¼ n,
2
L m x n x 1 L ðm nÞ x ðm þ nÞ x
ð ð
sin cos dx ¼ sin þ sin dx ¼ 0
L L L 2 L L L
If m ¼ n,
L m x n x 1 L 2n x
ð ð
sin cos dx ¼ sin dx ¼ 0
L L L 2 L L
The results of parts (a) and (b) remain valid even when the limits of integration L; L are replaced
by c; c þ 2L, respectively.

X n x n x
1
a n cos þ b n sin converges uniformly to f ðxÞ in ð L; LÞ, show that
L L
13.4. If the series A þ
n¼1
for n ¼ 1; 2; 3; ... ;
1 ð L n x 1 ð L n x a 0
f ðxÞ cos dx; f ðxÞ sin dx; :
ðaÞ a n ¼ ðbÞ b n ¼ ðcÞ A ¼
L L L L L L 2