I am trying to understand the definition of the Eguchi-Hanson metric, and in particular the nature of the removable singularity at the origin when expressed as a metric on $\mathbb{C}^2$.
Background: On $\mathbb{C}^2$ with complex coordinates $z^1,z^2$, one looks for a rotationally symmetric Kahler metric of the form $\omega = i\partial\overline\partial \phi$, where $\phi = F(\rho)$ for $\rho = |z^1|^2+|z^2|^2$, with vanishing Ricci curvature. The only solutions are $ \omega_{Eucl} = i\partial\partial \rho$, which (up to scaling) is the Euclidean metric, and (up to some kind of scaling) the Eguchi-Hanson metric: $$ \omega_{EH} = i\partial\overline\partial\left(\sqrt{\rho^2 + 1}+\log(\rho)-\log(1+\sqrt{\rho^2+1})\right) $$ Now this metric has a singularity at the origin (as $\rho \rightarrow 0$, the $\log(\rho)$ term is unbounded), but many sources say that on passing to the quotient $\frac{\mathbb{C}^2}{\mathbb{Z}_2}$, where $\mathbb{Z}_2$ acts by multiplication by $-1$, the singularity disappears. Could anyone explain how to see this using the above formulation?