The complex permittivity of a material is typically written as epsilon=epsilons+i*sigma/omega.(epsilons: the real part, sigma: conductivity, omega: angular frequency). As we know, any material has a finite conductivity(even for insulator, conductivity is very small but not zero). Therefore, the complex permittivity would be singular for a DC field (omega=0, infinite imaginary part). How to resolve this singularity? As we know, In reality, for a DC field, only the real permittivity is used for a dielectric material.
DC and AC properties are intimately coupled to each other by Fourier transformation. What you observe in the frequency domain as a function of log frequency, you observe the same in the DC step response experiment as a function log log time. Thus if you observe a dispersion in he frequency domain, you will observe a time dependent permittivity if you apply a DC step.
DC and AC properties are intimately coupled to each other by Fourier transformation. What you observe in the frequency domain as a function of log frequency, you observe the same in the DC step response experiment as a function log log time. Thus if you observe a dispersion in he frequency domain, you will observe a time dependent permittivity if you apply a DC step.
In the experiments dealing with complex permittivity results are presented approximately from frequency of 1 mikroherz (using alternating current methods). This corresponds to one period with a duration of 11 days and 13 hours and 46 minutes. The response to unit step is also a partial solution. However, you need a lot of time without EMI.
If you apply a static electric field to a material with conductivity you will get a conduction current. It's a resistor. However taking your starting point of complex permittivity and following through the analysis to get the complex wave propagation constant you will eventually arrive at the idea of skin depth for frequencies where the imaginary part of the permittivity is much larger than the real part.
In deriving the characteristic impedance and propagation constant of a material typically the results will be written: Z0=sqrt((j.w.mu)/(j.w.eps + sigma)) and gamma=sqrt((j.w.eps + sigma).(j.w.mu)) .
if we divide the numerator and denominator of the characteristic impedance by the angular frequency we can write Z0=sqrt((j.mu)/(j.(eps -j.sigma)/w)) and if we bring the angular frequency outside the square root in the propagation constant we get gamma=w.sqrt((j.(eps -j.sigma/w).(j.mu)) . Here would could consider the term: (eps -j.sigma/w) as the complex permittivity as a convenience but mathematically this term goes to infinity as w goes to zero. However if you consider the full expression there is always an w multiplier so the overall expression is finite.
The other reason for considering complex permitttivity is that there is a loss mechanism in the polarisability of most dielectrics which is frequency dependent and this may be better considered as a complex permittivity than a real conductance as there is no dc conductance here. See: https://en.wikipedia.org/wiki/Permittivity#Complex_permittivity
PS I used e^jwt assumption I think you may be using e^-iwt so flip the signs of the complex parts if you wish.
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