Let f(x) be a function of variable x defined on an interval 0 to T. It can be represented by the Fourier series, given by '2na (2nt f(x) = +2 an cos T + b, sin T where 2nn u) du, bn 2 2nn :I $(u)du, an = F(u) cos| (u) sin -u ) du. T ao = T Answer the following questions. 1. A function ƒ has period T =2 and is defined by 1 0
Let f(x) be a function of variable x defined on an interval 0 to T. It can be represented by the Fourier series, given by '2na (2nt f(x) = +2 an cos T + b, sin T where 2nn u) du, bn 2 2nn :I $(u)du, an = F(u) cos| (u) sin -u ) du. T ao = T Answer the following questions. 1. A function ƒ has period T =2 and is defined by 1 0
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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Transcribed Image Text:Let f(x) be a function of variable x defined on an interval 0 to T. It can be represented
by the Fourier series, given by
2nn
(2nt
f(x) =
+> an cos
2
T *) + b, sin
T
where
2na
2nt
| f(u) cos
u) du, bu =
I f(u) sin
f(u)du, an
-u) du.
do =
T
Answer the following questions.
1. A function ƒ has period T = 2 and is defined by
1
0<t<1
F(t) = {
-1 1<t< 2
Find the Fourier series representation of this function.
2. Explain the meaning of Gibbs phenomenon, using the function f defined above
as an example. Show how the errors behave as t gets closer to one of the points
t = -1,t= 0, t = 1, ..
3. Verify the result shown using 'integration by parts'.
1
1
| t sin(nt) dt = --
t cos(nt) +
sin(nt)
n2
n
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