Use the integration theorem to show that Cos nx 효(72-3a2) =D Σ(-1)”-1! -T
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Solve c and d, using the table might help
![1. The Fourier series for the function f(x) = ;x, -T < x <T is
sin nx
E(-1)"
n-1
n
n=1
a) Sketch the graph of the function f, which is the pointwise sum of the
series on R.
b) Use Parseval's formula to show that En=1 72
1
c) Use the integration theorem to show that
COS nx
2(r² – 3x²) = E(-1)"-1
-T < x < T
n2
n=1
and
sin nx
ba(n² – a²) = E(-1)*-1
n3
-T < x < T
n=1
d) Discuss whether the sum of each Fourier series in c) is piecewise
continuous, continuous, piecewise C' or C on R. Sketch the graphs
of the sum functions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc1e46184-5bb3-4339-852b-bb3ef2e0784b%2Ff5f342cb-9175-423e-b831-3bd6d643446a%2Fafsafx_processed.png&w=3840&q=75)
Transcribed Image Text:1. The Fourier series for the function f(x) = ;x, -T < x <T is
sin nx
E(-1)"
n-1
n
n=1
a) Sketch the graph of the function f, which is the pointwise sum of the
series on R.
b) Use Parseval's formula to show that En=1 72
1
c) Use the integration theorem to show that
COS nx
2(r² – 3x²) = E(-1)"-1
-T < x < T
n2
n=1
and
sin nx
ba(n² – a²) = E(-1)*-1
n3
-T < x < T
n=1
d) Discuss whether the sum of each Fourier series in c) is piecewise
continuous, continuous, piecewise C' or C on R. Sketch the graphs
of the sum functions.
![Table 5.1: Examples of Fourier series
f(x)
zao +2(am cos nx + b, sin nx)
n=1
i)
x, 0<x < T
(-1)n+1 sin nx
n
n=1
sin nx
ii)
(T – x), 0< x < T
n=1
sin(2n – 1)x
2n – 1
ii)
7, 0< x < T
n=1
Σ
(-1)ª-1 Cos(2n – 1)x
2n – 1
iv)
T, n < x < T
uf > * > 0
n=1
COS nx
v)
(7? – 32), 0 < x < +
E(-1)"-1.
n2
n=1
COs nx
vi) (T – 2)? – RT², 0 < x < T
n2
23D1
cos (2n – 1)x
(2n – 1)2
vii) 7(T – 2), 0< x < ™
n=1
(-1)n-1 Sin(2n – 1)x
(2n – 1)2
0 < x < }T
TI,
liT(T – x), T < x < T
viii)
n=1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc1e46184-5bb3-4339-852b-bb3ef2e0784b%2Ff5f342cb-9175-423e-b831-3bd6d643446a%2Flusyh4_processed.png&w=3840&q=75)
Transcribed Image Text:Table 5.1: Examples of Fourier series
f(x)
zao +2(am cos nx + b, sin nx)
n=1
i)
x, 0<x < T
(-1)n+1 sin nx
n
n=1
sin nx
ii)
(T – x), 0< x < T
n=1
sin(2n – 1)x
2n – 1
ii)
7, 0< x < T
n=1
Σ
(-1)ª-1 Cos(2n – 1)x
2n – 1
iv)
T, n < x < T
uf > * > 0
n=1
COS nx
v)
(7? – 32), 0 < x < +
E(-1)"-1.
n2
n=1
COs nx
vi) (T – 2)? – RT², 0 < x < T
n2
23D1
cos (2n – 1)x
(2n – 1)2
vii) 7(T – 2), 0< x < ™
n=1
(-1)n-1 Sin(2n – 1)x
(2n – 1)2
0 < x < }T
TI,
liT(T – x), T < x < T
viii)
n=1
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