Can you map the open unit disk conformally onto $\{ z: 0 < |z| < 1 \}$?

Question

I only find that is not possible from the punctured disk to the unit disk, but in the other direction is possible or not ? If not, counter example, if yes please provide the mapping.

This is problem 15 from section 3 chapter 3, functions of complex variables Conway book.

Answer 1

$\textbf{Solution:}$

Let $D = \{z \in \mathbb{C}: |z| < 1 \}$ and $D^{*} = \{z \in \mathbb{C}: 0 < |z| < 1 \}$.

Consider the Mobius transformation that takes $D$ to the right half plane

$ z_1 = \displaystyle\frac{1 + z}{1 - z} $

then by a rotation, we can obtain the left half plane

$ z_2 = -z_1 $

Finally, the left half plane can be mapped conformally to $D^{*}$ by the exponential transformation

$ z_3 = e^{z_2} $

Hence, the desired conformal map is $f: D \to D^{*}$

$ \boxed{ f(z) = \exp\{-\displaystyle\frac{1+z}{1 - z}\} } $