Let $H =\{ z = x +iy \in \mathbb C : y>0 \}$ and $D = \{ z \in \mathbb C : |z| <1\}$ be the open unit disc . Suppose $f$ is a mobius transformation which maps conformaly onto $D$. Suppose that $f(2i) = 0$. Pick each correct statement from below.
$1.$ $\ f$ has a simple pole at $z = -2i$.
$2.$ $\ f$ satisfies $f(i)\overline{f(-i)} = 1$
$3.$ $\ f$ has essential singularity at $z = -2i$
$4.$ $\ |f(2+2i)| = \frac{1}{\sqrt5}$
I know that any analytic function from $H$ onto $D$ is of the form $f(z) = e^{i\theta}\frac{z-z_o}{z-\bar z_o}$, where $z_o$ is the zeroes of $f$, so $f(z) = e^{i\theta}\frac{z-2i}{z-\overline 2i}$,, thus $f(z) = e^{i\theta}\frac{z-2i}{z+2i}$. Thus $f$ has a simple pole at $z = -2i$. So 1) is true and 3) is false.
Now $f(i) = \frac{-e^{i\theta}}{3}$ and $f(-i) = -3e^{i\theta}$
Therefore $f$ satisfies $f(i)\overline{f(-i)} = 1$, so 2) is also true.
for 4) $|f(2 +2i)| = |\frac{2}{2 +4i}| = \frac{1}{\sqrt 5} $
I would be thankful if someone check my solution.