Isolated singularities of conformal hyperbolic metrics

J. Nitsche proved that an isolated singularity of a conformal hyperbolic metric is either a conical singularity or a cusp one. We prove by developing map that there exists a complex coordinate $z$ centered at the singularity where the metric has the expression of either $\displaystyle{\frac{4\alpha^2\vert z \vert^{2\alpha-2}}{(1-\vert z \vert ^{2\alpha})^2}\vert $\mathrm{d}$ z \vert^2}$with$\alpha>0$or$\displaystyle{\vert z \vert ^{-2}\big(\ln|z|\big)^{-2}|dz|^{2}}$.