2) For the Fourier series of the function f, ao + Σo acos (kx) + bxsin (kx) we have |bk| ≤ 211/11. (a)True (b) False a bO
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- 4. a) Show that exp(z) in the range 0< < is represented by the Fourier series 2 sin(2) 4 sin(4r) 6 sin(6z) exp(z) - exp (т)) 17 37 3 sin(3z) sin(x) +((1+ exp(7)) 2 105) 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)d) Write the corresponding Fourier series of f (x). A periodic function of f (x) is given by [0, f(x)={-x-7, f(x+27). - T < x < 0, 0Q Find Fourier series an [o, 2w] くxく下 o f cx)= _2π-X πくx< 2π 3 fcx):You are given that the function if f - 7please helpIf f is the Fourier series of g(x)= √3, [16-², -4 < x < 0 then 0≤ < 4 f(2)=¯ + 2 [(0) cos (1 x) + ( ) sin (7-²)] 2 What does f(-4) equal? f(-4) What does f(-2) equal? f(-2) = What does f(0) equal? What does f(1) equal? What does f(4) equal? (0) f(1) = ƒ(4) = *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]. b) Carefully state Fourier's theorem, including the definition of the Fourier series and the Fourier coefficients. c) Compute the Fourier coefficients of fodd and write down the Fourier series. Give full justification for your answer. d) By evaluating the Fourier series for an appropriate value of x, show that 1 2 2 2 2 3π 5TT 7π = 2|7 + +If f(x) is an even function on [-pi, pi], then the Fourier series of f is of of the form a0 + E-1 a, cos(jx) 2 A True В FalseSEE MORE QUESTIONSRecommended textbooks for youAdvanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, Incorporated
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