Asymptotic properties of the hyperbolic metric on the sphere with three conical singularities

The explicit formula for the hyperbolic metric ${\lambda }_{\alpha ,\beta ,\gamma }(z)\mathrm{\mid }dz\mathrm{\mid }$ on the thrice-punctured sphere $\mathbb{P}\mathrm{\backslash }\{z_1,z_2,z_3\}$ with singularities of order $\alpha ,\beta ,\gamma \le 1$ with $\alpha +\beta +\gamma >2$ at $z_1,z_2,z_3$ was given by Kraus, Roth and Sugawa in \cite{Rothhyper}. In this paper we investigate the asymptotic properties of the higher order derivatives of ${\lambda }_{\alpha ,\beta ,\gamma }(z)$ near the singularity and give some more precise description for the asymptotic behavior.