Page 377 - Schaum's Outline of Theory and Problems of Advanced Calculus

368 FOURIER INTEGRALS [CHAP. 14

r
ffiffiffiffiffiffiffiffi
2 1 p 0 @ @ 1
ffiffiffi ð
Let f ðxÞ cos xdx ¼ Fð Þ and choose Fð Þ¼ 2= ð1 Þ . Then by Problem
14.3, 0 0 > 1
1
r ffiffiffi ð r ffiffiffi ð r ffiffiffi
2 1 2 2

f ðxÞ¼ Fð Þ cos xd ¼ ð1 Þ cos xd
0 0
2 ð 1 2ð1 cos xÞ
¼ ð1 Þ cos xd ¼ 2
0 x
ð 2
sin u
1
14.5. Use Problem 14.4 to show that 2 du ¼ .
0 u 2
As obtained in Problem 14.4,
2 ð 1 1 cos x 1 0 @ @ 1
0 x 2 cos xdx ¼ 0 > 1
Taking the limit as ! 0þ,we find
ð
1 1 cos x
0 x 2 dx ¼ 2
ð 2 ð 2
1 1 sin u
But this integral can be written as 2 sin ðx=2Þ dx which becomes 2 du on letting x ¼ 2u,sothat
2
0 x 0 u
the required result follows.
ð
1 cos x
x
14.6. Show that 2 d ¼ e ; x A 0.
0 þ 1 2
Let f ðxÞ¼ e x in the Fourier integral theorem
ð ð
2 1 1
cos xd f ðuÞ cos udu
f ðxÞ¼
0 0
2 ð 1 ð 1
Then cos xd e u cos udu ¼ e x
0 0
ð
1 1
But by Problem 12.22, Chapter 12, we have e u cos udu ¼ 2 . Then
0 þ 1
2 ð 1 cos x x ð 1 cos x x
d ¼ e or e
2
2
0 þ 1 0 þ 1 d ¼ 2
PARSEVAL’S IDENTITY

14.7. Verify Parseval’s identity for Fourier integrals for the Fourier transforms of Problem 14.1.
We must show that
ð ð
1 1
2 2
f f ðxÞg dx ¼ fFð Þg d
1 1
(
r ffiffiffi
1 jxj < a 2 sin a
:
0 jxj > a
where f ðxÞ¼ and Fð Þ¼