I don’t understand what is the meaning of the omega prime (meaning the thing that looks like w’) in the Kramers-Kronig relations, or why it is there.
See en.wikipedia.org/wiki/Kramers-Kronig_relations for an explanation of the Kramers-Kronig relations. The gist of this is that if some linear physical process in the time domain causes some effect, and you look at the real and imaginary parts of the Fourier transform of that effect, you find the real and imaginary parts have to be related by certain integral equations. I need to find functions that satisfy these integral equations - hence my interest.
In my understanding (which is obviously incomplete), the real and imaginary parts are functions of frequency and are traditionally expressed as f1(w) and f2(w) where w (actually lowercase omega) is the frequency. However, in the integral equations that relate them, there also appears w’ in several places, such as w^2 - w’^2, which make it clear w’ is different from w. What is w’?
I’ve found a few references that explain it more or less as wiki does, none of which explain w’.
Can any of the SDMBers help straighten me out with this admittedly very obscure mystery?
That is correct, w’ is a dummy intergration variable. The K-K relation essentially states that if a function has a certain form, we can use Cauchy’s theorem, and complex contour integration to show that the real and imaginary components have this relation.
Let’s see how this looks: (perhaps there is a more general relationship, but I only know the one for EM theory)
If we have a function
f(w) = f_o[1+ int_0^infinity {g(t)exp(iwt)dt}]
int_0^infinity means we are integrating from zero to infinity (integration variable t). g(t) is a real finite valued function. Then K-K states that the real and imaginary parts of f(w) are related as you have read. (you can also show that the real part is an even function and the imaginary part is an odd function)
From this, you should be able to see how to build a function that satisfies the KK relations. This occurs in EM with g(t) being the time-dependent susceptibility (often represented as chi(t)) and f(w) the frequency dependent permitivitty (often represented as epsilon(w)).
Oh, wow, of course! Now that you tell me this, I see I should have read it that way from the start, obviously! It’s just an integration variable, and I should have seen that because it appears as dw’ in both cases. I suppose it’s written w’ to emphasize it’s got the dimensions w has and suggest sweeping across all frequencies w might take on.
I think what threw me was the idea w was a function of something else unsaid and w’ was its derivative. I’m too clumsy with math to recover from such a bizarre tangent.