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

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.)

7

A function f(x) is given by (in the picture). Find the Fourier transform of f(x) with hint (in the picture to)

Please see the attached below

6

Help me solve this problem

HW: Find the z transform for sin (ak):