I have a some trouble with solving the following problem:
Map area conformally to upper half-plane
$S$ = {$z \in \mathbb{C}: Im(z) > 0$} $\cup $ {$z\in \mathbb{C}: |z+i| > \sqrt2 $}.
I tried to solve the problem as follows: first I tried to find a conformal mapping that takes my region to a certain angle (in fact, by the properties of a conformal mapping, we even know what the corner is, because conformal mappings are characterized by "conservatism of angles". Therefore, it is enough just to find the angle between the two generalized circles presented).
I did the following: $-1, 0, 1$ $\in S$. Then, for a conformal mapping $w_1$ of this set into an angle, it is sufficient just to use a mapping of the form:
$w_1(z) = \frac{z-1}{z+1}\cdot a$
$1 = w_1(0) = \frac{0-1}{0+1}\cdot a \Rightarrow a = -1 \Rightarrow w_1(z) =\frac{1-z}{1+z}$.
It's easy to understand that the angle to which this mapping will be translated will be equal to $\frac{3\pi}{4}$. However, upon direct verification, it turned out that this is not the case. This follows at least from the fact that $w_1(i) = -i$. This value, while obviously belonging to the area, will not belong to the desired display (ie, corner).
I hope I was able to convey my message clearly.