How does Galitskii's integral converge? In V M Galitskii's 1958 paper "$\textit{Energy spectrum of a non-ideal Fermi Gas}$," he builds the following integral as part of a longer expression for the real part of the self-energy (eqn 26'). It is:
$$ \int d^{3}p'\int d^{3}k P\frac{n_{p'}}{q^{2}-k^{2}} $$
where $P$ indicates principal value, $n$ is the occupation number, $q$, $p'$ and $k$ are vectors, and 
$$ q = \frac{1}{2}(p - p') $$
Without some extra condition, I can't see how this integral converges, since $|k|$ can run from 0 to $\infty$. Nevertheless, it is well-known to converge (Galitskii's derivation appears in  and Walecka's many-body text).
However, it seems Galitskii indicates the extra condition in a way I cannot understand. So, the heart of my question, I think, lies here. If I read correcty, Galitskii indicates that the integral can converge when $k^{2}$ is of the order of $(p')^{2}$. But, since I must integrate over $p'$ and $k$, I don't see how to apply this condition.
Any help is much appreciated.
 A: Terminology: XY+Z has two terms (XY and Z) and XY has two factors (X and Y).
In equation (23) he had a separate integral (with only one term inside the integral) with just one principal value, $P\frac{1}{k^2-q^2}$, as a factor but that was with the amplitudes inside the integral too as additional factors.
Later, in equations (26) and (26'), there is no  claim that $n_{p'}P\frac{1}{k^2-q^2}$ all by itself is a convergent integrand. All the talk about $k$ being close to $q$ is about pulling other factors out by saying that sure they change, but only their values near $k\approx q$ affect the integral and that the factors are roughly constant there so you can approximate (26) with (26') by pulling those factors outside the integral.
So you have one term of an expression inside an integral. If an author writes $\int A-B$ this doesn't mean that it equals $\int A-\int B$ and it doesn't mean that $\int A$ or $\int B$ even exist. So I can't figure out why you are asking about an integral of just one term of an integral of a sum of two terms.
Back in (23) there were separate integrals for the two principle values, now there isn't. So asking about the integral of just one term doesn't make sense.
I understand this isn't really an answer, but I literally just can't find support for any of your claims. Either that the integral you mention comes up as its own separate integral, or the talk about it converging (its hard to discuss it converging if it never comes up).
So I'm basically disbelieving your question.
