Also, a quick guess of the Tommy Riemann Mapping is:
\[
g(t) = f(1+t) - 1 - \frac{t}{2} = g_2t^2 + O(t^3)\\
\]
This is the only point on the boundary which has a second order expansion. Thereby, using a Bottcher coordinate \(G\), we have that:
\[
G(g(t)) = G(t)^2\\
\]
.......... Again, it's a quadratic Riemann surface......
\[
g(t) = f(1+t) - 1 - \frac{t}{2} = g_2t^2 + O(t^3)\\
\]
This is the only point on the boundary which has a second order expansion. Thereby, using a Bottcher coordinate \(G\), we have that:
\[
G(g(t)) = G(t)^2\\
\]
.......... Again, it's a quadratic Riemann surface......
