A function of period 2L has the form f(x) = L²x², xe [-L, L]. i) Compute the Fourier series for this function. ii) By choosing an appropriate value of a show that an infinite series can be con- structed for T².
A function of period 2L has the form f(x) = L²x², xe [-L, L]. i) Compute the Fourier series for this function. ii) By choosing an appropriate value of a show that an infinite series can be con- structed for T².
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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![Question 8
A function of period 2L has the form
f(x)=L²x².
i) Compute the Fourier series for this function.
ii) By choosing an appropriate value of a show that an infinite series can be con-
structed for 7².
The following indefinite integrals might be useful.
L
n²7²
(L sin (27²)
,
J
[₁
J
² sin
² cos
x sin
x Cos
nTX
nTX
nTX
ᎡᏤᏆ .
L
dx =
dx =
dx =
dx =
L
n³73
L
n³73
L
n²π²
x € [-L, L].
L cos
2Lnxx sin
(2Lm
2LnTx cos
nπx
L
nTX
L
NTX
$(TTT)) +
-nTX COS
+ nπx sin
NTX
L
+ C
(2L² -n²²x²) sin
+C
+ (2L² − n²π²x²) cos (¹T)) + ¤
(ntx)) +
+C.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0567139a-9f93-4ba8-a710-9c19173fc9d7%2Fab3d53a3-0b16-456a-a70c-6e063993bda1%2Fqed9xf1f_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Question 8
A function of period 2L has the form
f(x)=L²x².
i) Compute the Fourier series for this function.
ii) By choosing an appropriate value of a show that an infinite series can be con-
structed for 7².
The following indefinite integrals might be useful.
L
n²7²
(L sin (27²)
,
J
[₁
J
² sin
² cos
x sin
x Cos
nTX
nTX
nTX
ᎡᏤᏆ .
L
dx =
dx =
dx =
dx =
L
n³73
L
n³73
L
n²π²
x € [-L, L].
L cos
2Lnxx sin
(2Lm
2LnTx cos
nπx
L
nTX
L
NTX
$(TTT)) +
-nTX COS
+ nπx sin
NTX
L
+ C
(2L² -n²²x²) sin
+C
+ (2L² − n²π²x²) cos (¹T)) + ¤
(ntx)) +
+C.
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