Use the definition or existing results given by the textbook to compute the following Fourier transforms. Show details of your calculations: (a) F{e-al=}, a > 0.

Q: 10. Solve the following IVP using Fourier Transform: -00 0, u(x,0) = e, -0 <r < oo.

A: Given ut=uxx, -∞&lt;x&lt;∞, t&gt;0u(x,0)=e-x2, -∞&lt;x&lt;∞

Q: 3. Evaluate Laplace transform to + 5 sin7t . 3 2

A: We have to find the Laplace transform .

Q: Find the Fourier Sine Transform of e-t and hence find the Fourier Cosine Transform of te-t.

A: Fourier sine transform of e-t is ∫0∞e-tsinωtdt. Calculate it:  ∫0∞e-tsinωtdt=∫0∞e-teiωt-e-iωt2idt…

Q: (11) The Fourier cosine transform of the function f(x) e; x>0, is

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Q: Show that the Fourier transform of f (x)= e*/2 is e

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Q: Find the Fourier transform of f(x) without using the Fourier transform table. f(x) = e-l×l. (-0 <x<…

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Q: Find the sine transform of a function of x which is equal to sin x for 0 <x<a and 0 for x2a.

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Q: Find the first three Fourier-Legendre coefficients co, C1, C2 for the function f(x) = cos Show your…

A: We will use the legendre polynomial of degree 0,1 and 2 to calculate c_0,c_1 and c_2 respectively.

Q: si |x| < a 1. f(x) = [1 0 en otro caso ||

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Q: Find the Fourier sine expansion of f(x)=x² (3L– 2x) on [0, L].

A: The function f(x) is a rule that associates or fixes a number to the particular point x. If y=f(x),…

Q: The best approximation of f(x) = cos(8 + x) by a Fourier polynomial of degree 1 is

A: The Fourier Polynomial of degree n is Fn(x)=a0 + ∑k=1nakcos(kx) +∑k=1nbksin(kx). Then, the Fourier…

Q: Find Fourier transform for the following image 0 1 0 1 0 1 0 1 1

A: The Fourier transform is Fk, l=1N∑i=0N-1  ∑j=0N-1 fi, jeι2πkiN+ljN. Expand the series: Fk, l=13∑i=02…

Q: 1. Find the Fourier transform of this rectangular pulse X (w) and show that sin πωτ X(w) = A² 2.…

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Q: IS kept contact with ice at 0°C' and if initially the rod is submerged in hot water at 100°C. Use…

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Q: find the fourier transform of the function St)= -at 2.

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Q: solve the following problem by definition of the Laplace transform and by direct formula:

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Q: ) Determine the complex exponential Fourier transform for the following periodic signal shown. x(t)…

A: To find : Complex exponential Fourier transform for given signal

Q: Calculate the Fourier transform of f(x) = (1 – 2)e-(z-a)?/2_

A: Calculate Fourier transform of e-x-a2/2:  ∫-∞∞e-x-a2/2·e-jωxdx=e-jaω∫-∞∞e-x-a2/2·e-jωx-adx-a…

Q: Find the Fourier transform of f(x) = sin x. In what sense does the Fourier transform exist?

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Q: (k) f(x) =X[a.b](x) eiaz,

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Q: Compute the Fourier transform of the following function: f(x) = e-a(r-x0)²+iqx (5) where a > 0 and q…

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Q: Find the Fourier Cosine transform of e-t² Hint: Use the integral formula √π So e-u² du = 2

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Q: b-F(w)=e"w - t

A: According to the definition of modulus function f(x)=|x|, the value of function f(x) is x, if x≥0…

Q: Compute the Fourier Transform of the following function (3 , –n < x < o (1 , 0<x< TI f(x) =

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Q: f(x)= 0, x 1

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Q: b) Find the Fourier sine transform of f(x) = 12π, 0 < x < 6π

A: Fourier coefficients represent the amplitude of each harmonically-related sinusoid present in the…

Q: Use Theorem 7.4.2 to evaluate the given Laplace transform. Do not evaluate the convolution integral…

A: Let's find.

Q: Write the general equations for the following: a. Laplace Transform b. Fourier Transform

A: The Laplace and Fourier transformation are two important integral transformations with kernel one.…

Q: f(x)= == ¹xcl! Obtain the Fourier transform of

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Q: Find the cosine transform of a function of x which is unity for 0 <x < a and zero sin ap, for x 2 a.…

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Q: Find the fourier transform of following signals a. x() = cos (2xf.1) b. x(1) = sin (27f 1) c. For…

A: Calculate Fourier transform of cos22πfct: Fcos22πfct=F122cos22πfct-1+1…

Q: .3 \S

A: From the given problem:   L-125-1s32   We can write as:   2s-1s32=4s2-4s4+1s6   Now,…

Q: Find the Fourier transform of f (x) defined bv f x) = , a 10, > a and hence evaluate sin pa cos dp…

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Q: a) Find Fourier Transform f (x) = |x| > B and Evaluate Sin[2B]Cos[2x] da -8-

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Q: Find the Fourier transform of sinc (2t). (a) 2n A(w/2) 3. (b) п Д(u/4) A(w/8 (d) A(w/16) Moving to…

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Q: Find the Fourier Transform of Sa? – x², for |æ| a f(x) = and hence deduce that (sin s- s coss -) ds…

A: Fourier transform of gx=1 for x&lt;a0 for x&gt;a is ∫-aa1.e-isxdx. Calculate it:…

Q: Σ 4 NITX (пл)з (1- сos(пп)) sin. L f(x) = n=1 8

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Q: Find the exponential transform of the function {1, Pi/2<x<pi. and 0, otherwise} and write f(x) as a…

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Q: I find the Laplaa Transform ) f4): e" (3t*+ 1) - 5 sin at 5 sin 3t

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Q: 1.) Give the Fourier transform of the function f. Sx ha x = [0, 1] f(x) = 0 otherwise.

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Q: (7) Find the Fourier transform of the following functions: { {: cos x, if 0 1 (ii) f(x) = 0,

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Q: a Find the Fourier transform of the signal x(7) = 2 cos(m)[(1+3)-(1-3)], using the Fourier integral.

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Q: ) Find the first three Fourier-Legendre coefficients co, C1, C2 for the function f(x) = cos (号).…

A: Since you have asked multiple questions so as per guidelines we will solve the first question for…

Q: 3. Find the Fourier transform of f(x) = {1, |x| 1 Also evaluate .00 Sinx dx

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Q: Find the Fourier Transform of Seizt -1<t<1 f(t) = = 0 elsewhere

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Q: Find the la place transform of the following. Simplify your answer. f(+) = 2+² e²+ - sin²+

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Q: If the exponential Fourier transform of f (x) = e-alxl 2a %3D a2 +b2 What is the Fourier transform…

A: considering the function fx=e-ax=2aa2+p2   fx=e-ax=2aa2+p2 and fx=e-bx=2ba2+p2

Q: Find the Fourier sine transform of (sinx, 0a (2) %3D

A: Fourier transform

Q: O Show that the Fourier transform of | 3 -2<1<2 0 otherwise f() = {0 6 sin 20 is given by F (@) =

A: Since you have asked multiple question, we will solve the first question for you. If youwant any…