We consider the following 27-periodic function f given by S0, if -n < x < 0, lx, if 0 1. 3- Deduce the Real Fourier Series associated to the function f.
We consider the following 27-periodic function f given by S0, if -n < x < 0, lx, if 0 1. 3- Deduce the Real Fourier Series associated to the function f.
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
Transcribed Image Text:We consider the following 2T-periodic function f given by
(0, if -T < x <0,
lx, if
f(x) =
0 < x < T.
1- Determine the Complex Fourier Series associated to the function f.
Hint : Use the fact that
(ах — 1)
xear
-eda + C, where C is constant.
а?
2- Deduce ao, an and bn for n > 1.
3- Deduce the Real Fourier Series associated to the function f.
4- Use Perseval's equality in order to show that :
1
+
1
(-1)n+1
57?
2n2
Tn²
48
n=1
n=1
n=1
Hint : Use the fact that
73
%3D
![F[sin(at)] = in (8(w + a) – 8(w – a)
F[8(1)] = 1
sin(at) = inF" (6(w+ a) – 6(w - a)].
8(t) = F='(1]
%3D
iw
F(cos(at)H(1)] =, (8(w + a) + 8(w – a)) +
a2
%3D
w2
= 278(W – b)
iw
cos(at)H(t) = F-1
(8(w + a) + 8(w – a)) +
a2
w2
= 27F'(ô(w – b)]
a
F[sin(at)H(t)] =
(8(w + a) – 8(w - a).
%3D
a – w2
F[cos(at)] = (8(w + a) + 8(W – a))
%3D
in
(8(W + a) – 8(W – a) +
a
sin(at)H(t) = F1
%3D
a2
w2
cos(at) = xF' [8(w + a) + 8(w – a)] .
Page 3
1
b+ iw
F(0-b" cos(an)H()] =
%3D
b>0
a+ iw
(b + iw)2 + a2
n!
a> 0, ne
(a + iwy
b+ iw
e-aH(1) = F-1
a+ iw
e
- b' cos(at)H(t) = F-
%3D
|(b+iw)2 + a²
n!
t"e-"H(1) = F=1|
a
(a+ iw)n-1
Fo-b" sin(at)H(t)
b>o F e" H(-1) :
%3D
%3D
a >0
(b.
+ iw)2 + a2
a - iw
a
sin(at)H(t) = F-1
e" H(-t) = F-1
iw
%3D
(b+ iw)2 + a](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8873d101-14ca-49b1-8b25-0cc260dfbfaa%2Fcc46d507-302b-46bd-91ef-876363cd4c34%2Fcx2gq5_processed.jpeg&w=3840&q=75)
Transcribed Image Text:F[sin(at)] = in (8(w + a) – 8(w – a)
F[8(1)] = 1
sin(at) = inF" (6(w+ a) – 6(w - a)].
8(t) = F='(1]
%3D
iw
F(cos(at)H(1)] =, (8(w + a) + 8(w – a)) +
a2
%3D
w2
= 278(W – b)
iw
cos(at)H(t) = F-1
(8(w + a) + 8(w – a)) +
a2
w2
= 27F'(ô(w – b)]
a
F[sin(at)H(t)] =
(8(w + a) – 8(w - a).
%3D
a – w2
F[cos(at)] = (8(w + a) + 8(W – a))
%3D
in
(8(W + a) – 8(W – a) +
a
sin(at)H(t) = F1
%3D
a2
w2
cos(at) = xF' [8(w + a) + 8(w – a)] .
Page 3
1
b+ iw
F(0-b" cos(an)H()] =
%3D
b>0
a+ iw
(b + iw)2 + a2
n!
a> 0, ne
(a + iwy
b+ iw
e-aH(1) = F-1
a+ iw
e
- b' cos(at)H(t) = F-
%3D
|(b+iw)2 + a²
n!
t"e-"H(1) = F=1|
a
(a+ iw)n-1
Fo-b" sin(at)H(t)
b>o F e" H(-1) :
%3D
%3D
a >0
(b.
+ iw)2 + a2
a - iw
a
sin(at)H(t) = F-1
e" H(-t) = F-1
iw
%3D
(b+ iw)2 + a
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