Does there exist an analytic function whose domain is the complement of the closed unit disk and whose range is the open unit disk?
I am studying John B. Conway's book Functions of One Complex Variable for my own amusement. One of the exercises following Chapter III, Section 3, asks whether the open unit disk can be mapped conformally onto the punctured open unit disk. I solved that one (it's true), and then asked myself whether the converse were true as well. I immediately realized that that question is equivalent to whether the complement of the closed unit disk can be mapped conformally onto the open unit disk. Then I realized that I didn't even know the answer to the simpler question asked above.