Following the derivation of the massless free-boson two-point function given in Di Francesco, Mathieu and Sènèchal, I had an apparently stupid doubt. Look at the attached picture.
Where does the contribution $\lim_{\rho \rightarrow 0} \rho K'(\rho)$ to the integral in Eq. (2.100) go? Do they suppose it is zero? In this case Eq.(2.101) is not consistent with this requirement.
The only possible explanation I found is that the behaviour in $\rho\sim 0$ is not well-defined but I am not satified.
Any ideas??
It goes nowhere. The simplification being made between the first and second line is that $$-\int_0^r\frac{\rho}{\rho}\frac{\partial}{\partial \rho}(\rho K^\prime(\rho))~d\rho = -rK^\prime(r).$$