I've encountered the following integral while reading a paper: $$ F(u,v) = \int_0^\infty \frac{d\omega}{e^{\omega} - 1} \mathscr{P}\left\{ \ln\left( \frac{ \left( u - 2 \omega \right)^2 \left( v + 2 \omega \right)^2 }{ \left( u + 2 \omega \right)^2 \left( v - 2 \omega \right)^2 } \right) \right\} $$ Where the authors state that $\mathscr{P}$ denotes Cauchy's principal value.
I don't think this integral can be solved analytically, which is why the paper leaves it in this form. My question is what does the $\mathscr{P}$ mean exactly?
Naively, I can see that the integrand is going to be blowing up at the points $\omega = u$ and $\omega = v$, so I am guessing this is to deal with those singularities.
I've included a plot of the integrand for $F(2,4)$ for example here;
we see that there are singularities at $\omega \in \{1,2\}$. I'm guessing that the $\mathscr{P}$ is a way of evaluating the integral 'symmetrically' from each side of each singularity so that the result of the integral is finite, is this correct? And furthermore, is there a precise way to formulate how to do this integral in that case? (if that's even a correct understanding)
EDIT: In case it is helpful to anyone, here is the paper and I am referring to equation (III.44) (this is the function you get after setting $m_1 = m_2 = 0$, factoring the argument of the log, and I've set $u = |p_0| - |\mathbf{p}|$ and $v = |p_0| + |\mathbf{p}|$, and also thrown away the constants).