2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/223960We describe, in terms of generalized elliptic integrals, the hyperbolic metric of the twice-punctured sphere with one conical singularity of prescribed order. We also give several monotonicity properties of the metric and a couple of applications.13 pagesComplex VariablesClassical Analysis and ODEs30F45Twice-punctured hyperbolic sphere with a conical singularity and generalized elliptic integraltext