(1). Use Fourier Transforms to find the complete solution x(t) for the displacement of the damped, driven harmonic oscillator for the case of critical damping a² = wo². As described in class, x(t) satisfies: d²x(t) dt² dx(t) dt 2 + 2α +w₁²x(t) = A(t) and we will take the driving term A(t) to be given by A(t) = [Alt≤T 0,lt > T Note: as in the underdamped case (a² < w¸²) that was solved in class, you will 0 need to separately consider three cases († < −t,-t≤t≤t,t

2. For the continuous time periodic signal x (t) = 2 + cos( t) + sin( t) determine the fundamental frequency wo and the Fourier coefficients Cn such that x (t) = En=-o Cn einw, t

Hello, can you help me to do the Fourier transform for this problem?

Evaluate the inverse Fourier transform f(t) (2π)-¹/2 F(w)e-iwt dw of the Lorentz distribution Y 1 F(w): π W² + y² 2 You will need to treat the cases t 0 differently when evaluating the integral by residues. =