Let $r,\sigma,\phi$ be positive real variables in:
$$
\lim_{\phi\to 0} \left[ \sqrt{r^2+\sigma^2-2 r\sigma\cos(\phi)}
+ r\cos(\phi)\ln\left(-r\cos(\phi)+\sigma+\sqrt{r^2+\sigma^2-2 r\sigma\cos(\phi)}\right)
-r-r\cos(\phi)\ln(r)-r\cos(\phi)\ln(1-cos(\phi))\right]
$$
Note. Arising from this question:
Could this be called Renormalization? .
Can't proceed without knowing the outcome of this limit. Please help.
Bonus. It would be even nicer if someone can calculate the accompanying integral:
$$
\int_0^{2\pi}\left[ \sqrt{r^2+\sigma^2-2 r\sigma\cos(\phi)}
+ r\cos(\phi)\ln\left(-r\cos(\phi)+\sigma+\sqrt{r^2+\sigma^2-2 r\sigma\cos(\phi)}\right)
-r-r\cos(\phi)\ln(r)-r\cos(\phi)\ln(1-cos(\phi))\right]d\phi
$$
Otherwise I have to do it numerically, which is feasible anyway.