(3) Find the coefficients of the half-range even (cosine), odd (sine) and periodicextensions Fourier series for the function:f(x) =[0 if z € [0, 1]2 if x = (1,1] Answer(4) Find thL = 1, ao cos = 2, an cosdo perbn sin == 2, an per=4(−(−1)" + cosNT-0 bn pernt4 sin 2NTCOST)=2(-1+(-1)")NT

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Answered: (3) Find the coefficients of the…

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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Consider the periodic function defined within one period by the formula: 1 P(x)=x+1-(x+) if {(x) = x + ² = ( x + ² ) ² /10 (ii) sxs (b)(i) Present the first four terms of the series plus the zero term in the explicit form. Find the Fourier spectrum 4 = √²+ b² where a,, and b, are the amplitudes of the cosine and sine Fourier harmonics respectively and plot it against n in the log-log scale.
Find the Fourier series for the function f defined on [–1,1] by 3 3 7* +7 x1/3 Graph f and the first ten terms of the Fourier series to make sure that your solution is accurate. Include your file.
2. (a) Find the Fourier series for the function f(t) = t2 on the interval [-7,7].
Find the Fourier transform of (x) = {13x (13x if|x| 1
Write the Fourier series representation of the periodic function f(t) if in one period f(1) A f(t) = t, -1
Find odd and even function (half range) Fourier series of function below f(x) = { 2-x 0
Consider the function f(x) = -x, -L< x < L; f (x + 2L) = f(x) (a) Sketch the graph of the given function for three periods. (b) Find the Fourier series for the given function.
Without computing any Fourier series, address the following statements regarding the functions below:
Find the trigonometric Fourier series for the function f(x): [-T, π] → R given by the expression: J sinc if x = [-T, 0] 0 if x = (0, π] f(x) = (-1)" O FS(x) = -²/+n=2 (n²-1) π O FS(x) = +-2 1+(-1)^ cos(nx) - sinx. n=2 (²+1) -cos(nx) - sinx. 1+(-1)" cos(nx) + sinx. FS(x) = - +En-2 (n²-1) π 1-(-1)" FS(x) = ² + n-2 (n²-1) π п FS(x) = - +-2 2π -sin(nữ) — sin 2. 1+(-1)^ (n-1) T -cos(nx) + sinx. W
Find the trigonometric Fourier series for the function f(x): [-π/2, π/2] → R given by the expression: f(x) = ²2 FS(x) FS(x) = sinh n π FS(x) = FS(x)=sinh 7|1+2 2 sinh n T (-1)^ + Σn=1 n²+1 (-1)" inh 71 2n=1 n²+1 sin T π 1 + ΣΩ + -(cos 2nx (-1)" n=1 n²+1 + ΣΩ sin 2nx) -(cos 2nx + n sin 2nx) (cos 2nx n sin 2nx (-1)" (cos 2nx n+1 >] -n sin 2nx) 2nx)].
Suppose that f(t) is periodic with period [-n, 7) and has the following real Fourier coefficients: a1 = 1, b1 an = 4, a2 = 3, az = 3, 2, b, = -3, bz = 0, (A) Write the beginning of the real Fourier series of f(t) (through frequency 3): f(t) = (B) Give the real Fourier coefficients for the following functions: (i) The derivative f' (t) , a1 = a2 = -6 az = b1 = , b2 = -6 b3 = (ii) The function f(t) – 2 , a1 = , a2 = 3 , az = 3 b, = , b2 = , b3 = (iii) The antiderivative of (f(t) – 2) (with C 0) an = , a1 = , a2 3/2 , аз — b, = b2 = 3/2 , b3 = (iv) The function f(t) + 3 sin(3t) +3 cos(2t) an = , a1 = 1 , a2 = 6 , az = 3 b, = 2 , b2 = -3 bz = 3 (iv) The function f(2t) an = , a1 = , a2 = , аз — b1 = b2 = bz = 0
5. Convert the function f(x)=eux and f(x+2) = f(x) defined in the interval (-π < x < π) to the complex Fouriér Series.

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(3) Find the coefficients of the half-range even (cosine), odd (sine) and periodic extensions Fourier series for the function: f(x) = (0 if x € [0,1] 2 if x € (1,1]