In part 1, we had stopped before going to the frequency domain because we needed the Fourier transform of the sign function. This is where the technical difficulty appears, because we cannot simply write:

\begin{equation}\label{eq:sgnTF}

\operatorname{sgn}(\omega) = \int_{-\infty}^{\infty} \text{d} t \exp (-i \omega t) \operatorname{sgn}(t) \tag{5}

\end{equation} as the integral does not converge. One can however define\begin{align}

\label{eq:sgnvp}

&\operatorname{sgn}(\omega) = \lim_{\epsilon \to 0} \int_{-\infty}^{\infty} \text{d} t \exp (-i \omega t - \epsilon |t|) \operatorname{sgn}(t) = \nonumber \\

&- \lim_{\epsilon \to 0} \left [ \frac{1}{i \omega + \epsilon} + \frac{1}{i \omega - \epsilon} \right ]= \lim_{\epsilon \to 0} \frac{2i \omega}{\omega^2 + \epsilon ^2}= \mathcal{P} \left ( \frac{2i}{\omega}\right ) \tag{6}

\end{align}