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{\bf Question}

 Let $X$ be the open wedge in ${\bf C}$ bounded by the positive
real axis and the Euclidean ray from the origin making angle
$\theta = \frac{2\pi}{3}$ with the positive real axis.  Write down
a bijective analytic map $\varphi: X\rightarrow {\bf H}$ with
analytic inverse. Use this map $\varphi$ to pull back the
hyperbolic element of arc-length from ${\bf H}$ to $X$.  (Give the
hyperbolic element of arc-length on $X$ explicitly as $\mu(z) \:
|{\rm d}z|$.)

\medskip
\noindent Write down the equations determining the hyperbolic
lines in $X$.

\medskip

{\bf Answer}

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$\phi:X \longrightarrow {\bf{H}},\ \ \phi(z)=z^{\frac{3}{2}}$

$\phi$ is bijective and has analytic inverse (and is analytic)

The element of arc-length on $X$ is

$$ds_X=\frac{1}{\rm{Im}(\phi(z))}|\phi'(z)||\,dz|=
\frac{1}{\rm{Im}(z^{\frac{3}{2}})}\frac{3}{2}|z^{\frac{1}{2}}||dz|$$

[Write $z=|z|e^{i {\rm{arg}}(z)}$ where
$0<\rm{arg}(z)<\frac{2}{3}\pi$]

$$ds_X=\frac{3}{|z|\sin(\frac{3\rm{arg}(z)}{2})2}|\,dz|$$

\un{$\omega=z^{\frac{3}{2}}$}\ \ \un{$z=pe^{i\theta}$}

\un{$\rm{Re}(\omega)=c$}\ \
$p^{\frac{3}{2}}\cos\left(\ds\frac{3\theta}{2}\right)=c$

\un{$|\omega-a|^2=r^2$}\ \ $\begin{array} {l}
|z^{\frac{3}{2}}-a|^2=r^2\\z^3-az^{\frac{3}{2}}-a\bar{z}^{\frac{3}{2}}+a^2=r^2\\
\un{z^3-a^2\rm{Re}\left(z^{\frac{3}{2}}\right)=r^2-a^2}
\end{array}$
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