From uniformization theorem, it is known that every conformal class of metrics on a Riemann surface contains a unique hyperbolic metric. For a genus-$g$ Riemann surface with $n$ punctures, the punctures correspond to fixed points of the parabolic elements of the associated Fuchsian group. The question is that: what is the explicit local expression of this unique hyperbolic metric for such a surface around a puncture, associated with the fixed point $x$ of a parabolic element? A good reference is highly appreciated.

From uniformization theorem, it is known that every conformal class of metrics on a genus-$g$ Riemann surface with $n$ punctures such that $2g+n\ge 3$ contains a unique hyperbolic metric. The punctures correspond to the fixed points of the parabolic elements of the associated Fuchsian group. The question is that: what is the explicit local expression of this unique hyperbolic metric for such a surface around a puncture, associated with the fixed point $x$ of a parabolic element? A good reference is highly appreciated.