Prove in detail the following properties of the Fourier Sine and Cosine Transforms. These transforms are given by C[f](w) = Fc(w): S[f](w) = Fs(W) = 2 πT √ f(x) cos(wx) dx, 2 = f(x) sin(wx) dx. a) For any a > 0, we have the following. C[f(ax)](w) = — Fc (~) α 1 S[f(ax)](w) = ±F, (~2) α b) ō d C[xf](w) == dw F,(w) d S[xf](w) = -Fc(w) dw C[e](w) S[e](w) 2 1. = π 1+w2 2 w π 1+w2

Could you explain to me how to find the inverse Fourier transform of the function a. f(w) = 9e b. f(w) = -(w+4)² 32 1+iw 2 6-w +5iw

2. Determine the Laplace transform of the following functions. Here you may use common identities and known transforms of basic functions, but clearly show each step and briefly state which identities/properties you use in each step (i.e. Shift Theorems, Derivative Theorem etc.). f(t) = te-2t cos(4t) f(t) = t² sin2(t) + t cos² (t)

Apply the theorem of differentiation of transforms to find the Laplace transform of the given function. f(t) = 9t sin 3t Click the icon to view the table of Laplace transforms. L{9t sin 3t} = S> C. (Type an expression using s as the variable.)

Compute the Fourier transform A(w) of the following function: 12 /2a² cos wot = -e S(t) a You can assume that a > 0 and wo > 0. Hint: You can write the cosine as a sum of exponentials. Is there any value of w for which the Fourier transform is equal to zero?

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