Solution
= Solution
Composing the maps gives
$$
\boxed{w\longmapsto\frac{e^w-1}{e^w+1}=\tanh\frac w2,}
$$
a conformal bijection from $S$ to the unit disc.
= Solution
Composing the maps gives
$$
\boxed{w\longmapsto\frac{e^w-1}{e^w+1}=\tanh\frac w2,}
$$
a conformal bijection from $S$ to the unit disc.