The Green's function captures the response of an oscillator to an instantaneous impulse. In class, we derived
the Green's function by considering a tophat force and then taking the limit of the duration of the force to
zero). However, there are other ways of obtaining an instantaneous impulse.
Consider an undamped oscillator initially at rest at the origin. At t = 0, a force of the form
$$F[t] = F_0 e^{-t/\Delta t}$$
is applied (where $\Delta t$ is a constant). Show that in the limit $\Delta t \to 0$, the leading order non-zero response of the
oscillator is
$$x[t] = (F_0 \Delta t) \left(\frac{1}{m \omega_0} Sin[\omega_0 t]\right)$$
where we can identify the second term with the Green's function, with an appropriate shift in the time
coordinate. (Feel free to use Mathematica to simplify any algebra).
