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

  1. Sep 23, 2010 #1
    1. The problem statement, all variables and given/known data
    4. Let w = f(z) = [tex]i(\frac{1-z}{1+z})[/tex]. Show that f maps the open unit disk {z [tex]\in[/tex] C | z < 1} into the upper half-plane {w[tex]\in[/tex] C|Im(w) >0}, and maps the unit circle {z[tex]\in[/tex] C||z|=1} to the real line.

    2. Relevant equations
    I was given this hint:

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

    3. 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 "|"?

    And pardon the rough formatting.
  2. jcsd
  3. Sep 23, 2010 #2
    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?
  4. Sep 23, 2010 #3


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    Homework Helper

    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 [tex]z=re^{i\theta}[/tex] with r<1 and see if Im(w)>0.
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