How would conformal map from $\Bbb C\backslash\{z\in\Bbb C : Im(z)=0, Re(z)\le 0\}$ to the unit disc $\{|z|<1\}$ look like? Riemann's theorem guarantees the existence of a conformal mapping $f:\Omega\mapsto D$ where $\Omega=\Bbb C\backslash\{z\in\Bbb C : Im(z)=0, Re(z)\le 0\}$ is simply connecte...

Suppose I have a sector $D = \{0 < \arg z < \alpha\}$ where $\alpha \leq 2\pi$. If I apply the function $w = \frac{\zeta - i}{\zeta + i}$ from the upper half plane to the unit disc ($\zeta = z^{\frac{\pi}{\alpha}}$), I get that the vertex of the sector goes to -1 and $z = \infty$ goes to 1. I get...