The imaginary part of a response function describes how a system dissipates energy, since it is out of phase with the driving force. The Kramers–Kronig relations imply that observing the dissipative response of a system is sufficient to determine its in-phase (reactive) response, and vice versa.
But I thought that the real part (in phase) was dissipation, whereas the imaginary is just polarization or something. For example, with the complex conductivity, $j=\sigma E$. If $\sigma$ is purely imaginary, then it is like a capacitor: it doesn't actually take energy, right? You only get Joule Heating if $\sigma$ is real, I thought.
Is the article incorrect or am I not understanding something?