What shoul i compose to obtain this region? conformal map of weird region

Question

i need to transform $U=\{z\in \mathbb{C}: |z|>1,$ $R(z)<1$ onto the upper half plane $H = \{Im(z)>0\}$.

I serioulsy don't have any idei of how start this, i was trying to use $e^{iz}$ but i get to nowhere, anyone could help?

Answer 1

Make a picture of $U.$ Make a picture of $H$

$U$ is the area to the right of the imaginary axis excluding the half disk of radius 1.

$H$ is the plane above the real axis.

First thing we need is a way to rotate thing 90 degrees. Using complex numbers, this is easy, multiply by $i$

Next thing we need is a function that maps from $(1,\infty)$ to $(0,\infty)$. How about $\ln |z|$

$iz\ln|z|$