Relationship between the Real and Imaginary parts of a LTI causal system
I believe the motivation is to make the math easier. Without the logarithm, you have $G(\omega)=|G(\omega)|e^{j\angle G(\omega)}$. By taking the logarithm, you have $\ln G(\omega)=\ln |G(\omega)|+j\angle G(\omega)$; by removing the exponential function and turning it into a sum, the relationship between $|G(\omega)|$ and $\angle G(\omega)$ becomes much easier to handle.
KK only requires causality and linearity. Bode is more strict because it also requires no zeros in the upper half of the plane (but make sure to look at note 16 in the paper).

As far as I can see, the value of the real part of the zeros is irrelevant; only their existance matters. So, stability is not really involved in this analysis at all.

The implication (causal -&gt; analytic half plane) is true. I believe the &quot;vice versa&quot; (iff) is only true for stable systems, but I'm not 100% sure.


I don't know the answers to your questions 4 and 5.