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

The explicit formula for the hyperbolic metric $λ_{α,\,β,\,γ}(z)|dz|$ on the thrice-punctured sphere $\mathbb{P} \backslash \{z_1,\,z_2,\,z_3\}$ with singularities of order $α,\,β,\,γ\leq 1$ with $α+β+γ>2$ 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 $λ_{α,\,β,\,γ}(z)$ near the singularity and give some more precise description for the asymptotic behavior.

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