Show that the given function maps the open unit disk into the upper half plane

[Raziel2701]

Homework Statement

4. Let w = f(z) = i(\frac{1-z}{1+z}). Show that f maps the open unit disk {z \in C | z < 1} into the upper half-plane {w\in C|Im(w) >0}, and maps the unit circle {z\in C||z|=1} to the real line.

Homework Equations

I was given this hint:

"set w=i(\frac{1-z}{1+z}) and use the formula Im(w)= \frac{1}{2i}(w -\bar{w})"

The Attempt at a Solution

This is cliche but, what does this mean in English? I've been trying to decipher some of this stuff, in order for me to know what to do, I must first understand what I'm being asked to do, so that would be my first request.

The second thought I have of this is that the hint given also doesn't mean much to me. So what exactly would be the topic I could read on to help me get more information on this concept I'm being tested on? I'm at a library right now, so if I were to pick up a book on complex variables, what topic more or less is this problem covering?

I need to be pushed on the right direction to solve this problem, right now I'm just more or less in the dark. The set notation is a bit cryptic for me. I get that z is an element of the set of complex numbers, but what exactly is it being said after the "|"?

Thanks!
And pardon the rough formatting.

 

[Raziel2701]

Is it sufficient to show that since for the open disk z < 1, then if I plug a value less than one into w I get a result that is less than one and imaginary, and thus, it maps any point of the open disk into the upper half plane?

The same for the unit circle where |z| = 1. If I plug one I get zero for w, and thus I get the point (0,0) which is in the real line and therefore w maps any point on the unit circle to the real line?

 

[hunt_mat]

So you have shown that the map w, maps boundaries to boundaries, now choose a z with |z|<1 and show that Im(w)>0, so perhaps write z=re^{i\theta} with r<1 and see if Im(w)>0.