More chores = less sex!

[from the NY Times] It's official: when men do more work around the house, the couple has less sex! Now, where is my newspaper ?!

The Kramers-Kronig relations - part 2

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}

The Kramers-Kronig relations - part 1

[See part 2 for some technical aspects]
Very nice derivation of the Kramers-Kronig relations (on Wikipedia, of all places), exploiting the relation between the even and odd components of a function \(\chi (t)\) and the real and imaginary parts of its Fourier transform \(\chi (\omega) = \chi ' (\omega) + i \, \chi '' (\omega)\).

One usually invokes the analyticity of \(\chi (\omega)\) in the upper half-plane, which must first be derived from the causality: \(\chi (t) = 0\) for \(t < 0\). Complex integration along a well-chosen contour then yields the Kramers-Kronig relations in their standard form [1]:

\begin{align}
\label{eq:KK}  
\chi (\omega) &= \frac{1}{i \, \pi} \mathcal{P} \int_{-\infty}^{\infty} \text{d} \omega ' \frac{\chi (\omega ')}{\omega ' - \omega} \quad \text{or, for the components:} \nonumber \\
\chi '(\omega) &= \frac{1}{\pi} \mathcal{P} \int_{-\infty}^{\infty} \text{d} \omega ' \frac{\chi ''(\omega ')}{\omega ' - \omega} \\
\chi ''(\omega) &= -\frac{1}{\pi} \mathcal{P} \int_{-\infty}^{\infty} \text{d} \omega ' \frac{\chi '(\omega ')}{\omega ' - \omega} \nonumber
\end{align}
where \(\mathcal{P}\) denotes Cauchy's principal value.