What do you have so far?
for (i)(ii)(iii)(iv) respectively.
how about the behaviour
I'll try (i) for :
You can let and then then . Now consider (i) at various points around the circle . At 1/2, and so . Now start going around that circle and show how the term changes smoothly between the values of and except when you cross the line segment between zero and one and therefore the function has a single-valued analytic component over this circle except for the line segment (branch-cut) between zero and one.