The Fourier transform $\tilde{f}(\omega)$ of a suitable function $f(t)$ is defined as $\tilde{f}(\omega)=$ $\int_{-\infty}^{\infty} f(t) e^{-i \omega t} d t$. Consider the function $h(t)=e^{\alpha t}$ for $t>0$, and zero otherwise. Show that

$$\tilde{h}(\omega)=\frac{1}{i \omega-\alpha},$$

provided $\Re(\alpha)<0$.

The angle $\theta(t)$ of a forced, damped pendulum satisfies

$$\ddot{\theta}+2 \dot{\theta}+5 \theta=e^{-4 t},$$

with initial conditions $\theta(0)=\dot{\theta}(0)=0$. Show that the transfer function for this system is

$$\tilde{R}(\omega)=\frac{1}{4 i}\left[\frac{1}{(i \omega+1-2 i)}-\frac{1}{(i \omega+1+2 i)}\right]$$