Finding a Mobius transformation that maps the upper half-plane to the inside of a unit-disk, such that point $i$ is mapped to $0$ and $\infty$ to $-1$ Finding a Mobius transformation that maps the upper half-plane $\{Im(z)>0\};z \in \mathbb{C}$ to the inside of a unit-disk, such  that point $i$ is mapped to $0$ and $\infty$ to $-1$.
Okay, to be clear, I know that a Mobius transformation $w$ is of the form:
$$w=\frac{az+b}{cz+d};ad-bc\neq0.$$
I am very aware of what a unit disk is. I have done assignments like finding an mobius transformation that maps some points ($\mathbb{C}$) $a,b,c$ to $d,e,f$. -Where these points were not infinity.
But nothing like this problem that I have here? How is this done? I think I just need one more point and know it's picture to be able to figure is out. But how?

One very useful Mobius transformation to remember is the Cayley transform defined by
$$ f(z)=\frac{z-i}{z+i}$$
This maps the upper half plane to the open unit disk because the upper half plane is precisely the set of points in $\mathbb{C}$ which are closer to $i$ than to $-i$. Note also that $f(i)=0$.
This map doesn't quite fit your requirements because $f(\infty)=1$, but $-f(z)$ works instead.
