2. Consider the function f(x) = 1 x on the interval [0, 1].a) In two separate graphs, sketchi) the odd extension fodd on the interval [-1,1],ii) the Fourier series associated with fodd on the interval [-3,3].b) Carefully state Fourier's theorem, including the definition of the Fourier series and the Fouriercoefficients.c) Compute the Fourier coefficients of fodd and write down the Fourier series. Give full justificationfor your answer.d) By evaluating the Fourier series for an appropriate value of x, show that12 2 223π5TT7π=2|7++

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Answered: 2. Consider the function f(x) = 1 -x on…

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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Suppose that f(t) is periodic with period -T, 7) and has the following real Fourier coefficients: a2 = 3, аз — 3, b2 = -3, b3 = 0, ao 4, a1 = 1, b, = 2, (A) Write the beginning of the real Fourier series of f(t) (through frequency 3): f(t) = 2+cost+2sint+3cos2t-3sin2t+3cos3t+0+. (B) Give the real Fourier coefficients for the following functions: (i) The derivative f'(t) ao = , a1 = -1 , az = -6 , az = -9 b, = 2 , b2 = -6 bz = (ii) The function f(t) – 2 ao = , a1 = , az = 3 , аз — 3 b1 = 2 , b2 = -3 bz = (iii) The antiderivative of (f(t) 2) (with C 0) ao = , a1 , a2 = 3/2 , аз 1 b1 =-2 b2 = 3/2 b3 = (iv) The function f(t) + 3 sin(3t) + 3 cos(2t) ao = 2 , a1 = 1 , a2 = 6 , аз — 3 b, = 2 b2 = -3 b3 = 3 (iv) The function f(2t) an =2 , a1 , a2 3 , a3 3 b, = 2 , b2 = -3 b3 = 0
A periodic function, f(x) with period 4x is defined as - 2n sx<-1 - nSX<0 2n, %3D f(x) Osx<* 2n, Sketch the graph of f(x) on the interval [-5x, 57). Determine if f(x) is an even, odd or neither even nor odd function. a) b) Find the Fourier series of f(x).
5) If f(x)= x?; f (x +4)=f (x) b. The coefficient n in this Fourier series is : 2 (-1)". (na) (-1)** . (na) (-1)- cos d) 2 a) b) 0 c)
Q Find Fourier series an [o, 2w] くxく下 o f cx)= _2π-X πくx< 2π 3 fcx):
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Determine the bn of nth partial sum of the Fourier Series of the periodic function f(x) = -9x + 6, [-, π]: bn A = A == TU (−1)n
2 Denote the Fourier series of f(z) = { -x, 0

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2. Consider the function f(x) = 1 -x on the interval [0, 1]. a) In two separate graphs, sketch i) the odd extension fodd on the interval [−1, 1], ii) the Fourier series associated with fodd on the interval [-3,3].