Formula for conformal mappings.

I have this problem

Find a formula to send $A=\{ z\colon 0<\Im(z)<1 \}$ to the unit disc.

My main idea was to send $A$ to the upper half plane considering $f(z)=e^{z\pi}$ and then send it to the unit disc, but this could not be the only way to do this. How can I obtain a formula to obtain all the possible conformal mappings?

• – Nosrati Jul 23 '17 at 1:39

The awesome thing about conformal maps from the disk $\mathbb{D}$ to itself is that there aren't really too many, and they can be described quite succinctly: they are all of the form

$$f_{a,w}(z) = e^{ia}\frac{z -w}{1 - \bar{w}z}$$

for $a \in \mathbb{R}, w \in \mathbb{D}$.

Thus, to get a large number of conformal maps from $A$ to $\mathbb{D}$, start with just one conformal map $g: A \to \mathbb{D}$, and consider the collection of maps $f_{a,w} \circ g$. These are still conformal maps from $A$ to $\mathbb{D}$. Can you see why these are all the possibilities, using the assertion that the $f_{a,w}$ are all the conformal maps from $\mathbb{D}$ to itself?

• I had that idea, considering that $f_{a,w}$ and composing with $e^{z\pi}$. My doubt was if it could be another function different than that composition that could satisfy the condition. – MonsieurGalois Jul 22 '17 at 21:54