I am slightly confused as to the meaning of 'the principle part' of a limit, specifically in relation to the Kramers-Kronig relations as derived here on page 61. The source uses the 'Principal part' in reference to an integral, which I understand as the standard Cauchy Principal part defined as
$P \int _{-\infty}^{\infty} dx f(x) = lim_{\epsilon \rightarrow 0} [\int _{-\infty}^{a-\epsilon} dx f(x) + \int _{a+\epsilon}^{\infty} dx f(x)]$
for $f(x)$ singular at a.
Now what is causing me confusion is that the source uses the same symbol in reference to a limit
$\frac{i\omega}{\omega ^2 + \epsilon ^2} |_{\epsilon \rightarrow 0} = P \frac{i}{\omega}$
What does this mean? I don't know how to interpret this. The same symbol is used, but it wouldn't have a definition in terms of the integral... I haven't found a formal definition of 'principal value' for a limit such as this one.
The specific part of the text is here:
EDIT: Another thing I just noticed that I am slightly uneasy about: between equations (7.20) and (7.21) in the image, it seems that
$\int d\omega ' [P(\frac{1}{\omega - \omega '})] v(\omega ' )$
(whatever the thing in the square brackets formally means- I still haven't found an answer...) has been turned into:
$P[\int d\omega ' (\frac{1}{\omega - \omega '}) v(\omega ' )]$
Of course, the formal meaning of $P(\frac{1}{\omega - \omega '})$ in terms of limits may make absolutely obvious why this is a valid thing to do, but I raise the issue now in case it is not and requires further explanation; hopefully whosoever is able to answer the primary question will be able to add a note on this too.