Finding a holomorphic transformation

Question

I am trying to understand methods for finding holomorphic mappings between two regions of the complex plane. I am trying to find a holomorphic mapping between the region $$U_1=\{z\in \Bbb C:Im(z) \lt\frac{1}{2} \text{ and } z \neq0\}$$ and $$U_2 = \{z \in \Bbb C: |z+i|\gt1\}$$ I have prior to this shown that the region of $U_1$ is equal to the region that satisfies the inequality $|z|\lt |z-i|$. I don't know if I'm going to want to transform this inequality, to get the desired result? We know that, by this inequality $$|z+i|\lt |z+i-i|<|z|$$ for $z \in U_1$, so if we want $|z+i|\gt 1$ we want, $|z|\gt 1$? So how do we transform z to guarantee this?

Answer 1

I believe the easiest way would be to perhaps do it step by step.

As you have pointed out, $U_1$ can be expressed as $\{z \in \Bbb C : |z|\lt |z-i|\}$. So start by putting $w=\frac{z-i}{z}$. Under this transformation $U_1$ becomes $\{w \in \Bbb C: |w| > 1\}$.

This is almost what you'd like but not quite. Finally put $\zeta=w+i$, which should hopefully give you what's required.