Bulletin of the
Korean Mathematical Society BKMS

The explicit formula for the hyperbolic metric $\lambda_{\alpha,\,\beta,\,\gamma}(z)|dz|$ on the thrice-punctured sphere $\mathbb{P} \backslash \{0,\,1,\,\infty\}$ with singularities of order $0<\alpha,\,\beta<1$, $\gamma \leq 1$, $\alpha+\beta+\gamma>2$ at $0,\,1,\,\infty$ was given by Kraus, Roth and Sugawa in \cite{Rothhyper}. In this article we investigate the asymptotic properties of the higher order derivatives of $\lambda_{\alpha,\,\beta,\,\gamma}(z)$ near the origin and give more precise descriptions for the asymptotic behavior of $\lambda_{\alpha,\,\beta,\,\gamma}(z)$.

Keywords: conical singularities, hyperbolic metrics, special functions

MSC numbers: 30L10, 33D15