Use conformal maps or combinations of conformal maps such as linear fractional transformations, powers, roots, sin z, log z, etc., to find a one-to-one analytic function mapping the given region D onto the upper half-plane U.
I'll try to do (1)
If anyone could tell me if I'm on the right track and provide me with some help on the other problems, I would really appreciate it, thanks!
Again visualize this set. This is the horizontal infinite strip between and . Let us raise this stip up to the positive side by the mapping . Now we will compress this strip to the reason for this will soon be apparent. We do this by a dilation mapping in this case . Note the following mapping and look what happens with the points. The point , becomes . And what is that? That is the upper half plane!
Therefore, the required mapping is .
Can you the last one?
You can find a detail discussion on conformal mapping here. I hope it helps you.