5. Verify that f(x)= einx is periodic with period 27 and that 1 if n = 0, 0 if n #0. 1 einx dx Use this fact to prove that if n, m > 1 we have S0 ifn+m, cos nx cos mx dx = 1 n= m, and similarly 1 So ifn+ m, sin næ sin mæ dx = 1 n= m. Finally, show that sin næ cos mx dx = 0 for any n, m. [Hint: Calculate ein#e-imx + ein" eimx and ein"e-imx - einx eima ]
5. Verify that f(x)= einx is periodic with period 27 and that 1 if n = 0, 0 if n #0. 1 einx dx Use this fact to prove that if n, m > 1 we have S0 ifn+m, cos nx cos mx dx = 1 n= m, and similarly 1 So ifn+ m, sin næ sin mæ dx = 1 n= m. Finally, show that sin næ cos mx dx = 0 for any n, m. [Hint: Calculate ein#e-imx + ein" eimx and ein"e-imx - einx eima ]
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![5. Verify that f(x) = ein¤ is periodic with period 27 and that
1 if n = 0,
0 if n +0.
1
einz dx =
27
Use this fact to prove that if n, m > 1 we have
0 ifn + m,
1 n=m,
1
cos nx cos mx dx =
and similarly
0if n+ m,
1 п%3D т.
sin næ sin mx dx =
%3D
Finally, show that
| sin næ cos mx dx = 0
for any n, m.
[Hint: Calculate einae-imr + einx eimx and einxe-imx
einx eimx ]
-](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F833481a2-df8c-4805-95a2-f24b64ba619f%2F08ed8198-89be-4197-acb8-c8c4edcfcddd%2Fjqvc5nq_processed.png&w=3840&q=75)
Transcribed Image Text:5. Verify that f(x) = ein¤ is periodic with period 27 and that
1 if n = 0,
0 if n +0.
1
einz dx =
27
Use this fact to prove that if n, m > 1 we have
0 ifn + m,
1 n=m,
1
cos nx cos mx dx =
and similarly
0if n+ m,
1 п%3D т.
sin næ sin mx dx =
%3D
Finally, show that
| sin næ cos mx dx = 0
for any n, m.
[Hint: Calculate einae-imr + einx eimx and einxe-imx
einx eimx ]
-
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