The question is to find a conformal map from $\mathbb{D}$ to $\{z\in\mathbb{C}:Rez>0\}$ such that $z=1$ goes to $z=0$.
My thoughts:
We know that map $f(z)=\frac{1+z}{1-z}$ maps $\mathbb{D}$ to the right half plane. However, $f(1)=\infty\neq1=f(0)$. So is there a way that an algebraic way for me to manipulate this map so that $f(1)=f(0)$ while still preserving that it is conformal?