Find a Mobius transformation mapping the unit disk {|z| < 1} into the right half-plane and taking z = −i to the origin.
My workings:
$\phi(t) = \frac{az+b}{cz+d}$
We map -i to the origin (0) by taking the numerator and equating it to 0, knowing b = +i (since we started at -i).
That is:
$az+i=0 \implies i = b, a = 1$, which is correct.
However, how do we map $|z|$ to the upper right half plane (i.e. Re(w) = 0)?
Thanks.
Best Answer
Watch that first step: $-ai+b=0\implies b=ai$. Let's have $-1\to i$, so that $-a+ai=i(-c+d)$. And $0\to1$, giving $b=d$. So $c=-a$.
So, we get $f(z)=\frac{az+ai}{-az+ai}$ or $\boxed{f(z)=\frac{z+i}{i-z}}$.
By specifying the values at $3$ points, the Möbius transformation is determined.