I have uploaded the final set of lecture notes, covering L'Hopital's rule and the remainder of the material on applications of analytic maps to Laplace's equation, here. (The material on analytic continuation is in the supplement uploaded previously.) In particular I have put in a fairly detailed derivation of Re arcsin z.

On the other hand, since the resulting expression is not all that illuminating, and rather long to work with, if you wish you may express your answer to the homework in terms of Re arcsin z if that is useful. Note that it must be Re arcsin z -- you are not to just leave it in terms of Re arccos z. But the two are very closely related, so this should not be much of a problem.

If anyone is a bit rusty on arclength integrals (as used on question 1 of the final homework), they are defined here (see section 3). Also, the hint means that the proof of Liouville's Theorem as given in the lecture notes (see section 36) may have some useful ideas which can be adapted.

The promised additional practice problems, etc., will be posted tomorrow.

Please let me know if you have any questions!

Nathan