analytic function from the unit disk onto $\{z: \vert z \vert >1\}$ which is not bijection

Question

Examples of analytic function from the unit disk onto $\{z: \vert z \vert >1\}$ which is not bijection?

Can anyone tell me what should be the line of thought. I think of the function $1/z$, need help.

Answer 1

There exists a bijection from $\{z:|z|<1\}$ to $\{z:\Re z > 0\}$, that is, $\phi(z) = \frac{1+z}{1-z}$.

Then the function $f(z) = e^{\phi(z)}$ is your desired function.