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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
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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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
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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