So say $z$ is on the unit circle $|z| = 1$. How many parts does the transformed region$$z \to e^z$$ have?
My work. Alright, so if something is on the unit circle, it's of the form$$z = a + bi,\text{ where }a^2 + b^2 = 1.$$So$$e^z = e^{a + bi} = e^a e^{bi} = e^a(\cos b + i\sin b),\text{ where }a^2 + b^2 = 1.$$So elements will have magnitude $e^a$ at angle $b$ subject to $a^2 + b^2 =1$, sure. But I am not sure where to go from here. Any help would be well-appreciated.
This can be proven through simple topological considerations. Since $\exp$ is a continuous function and the unit circle has only one connected component, its image under $\exp$ also has one connected component.