I'm reading through T. Needham's "Visual Differential Geometry and Forms" and am a bit stumped by exercise 27 (chapter 7).
The exercise is about the metric of hemispheres in 3D hyperbolic space $\mathbb{H}^3$ and illustrates that these hemispheres (which end on the horizon) are hyperbolic planes $\mathbb{H}^2$.
The first few parts of the problem pose no issue. We start from the metric of the hemisphere in polar coordinates ($\theta,\phi$) and with coordinate transformations $u=\tan\theta$ and $u=1/\sinh \xi$ a conformal metric is obtained
$$ds^2 = \frac{d\phi^2 + d\xi^2}{\sinh^2 \xi}.$$
The last step is where I'm stuck. It asks to find a conformal mapping $(\phi,\xi \to x,y)$ such that the standard $\mathbb{H}^2$ metric $ds^2 = \frac{dx^2 + dy^2}{y^2}$ is recovered. It has a hint
Set $dy/y = d\xi /\sinh \xi$,
but that does not seem to help me. I can solve that equation to $y(\xi)$, but it seems to me that having a map $y(\xi)$ then implies having $x(\phi)$ in order to keep a conformal metric, which then cannot obey the Cauchy-Riemann equations for this conformal map.
Am I misreading the hint or just on the wrong track? Any general hints to how to approach the problem?