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Mobius transform question

  1. Jan 13, 2010 #1
    [math][/math]
    [math]D^+=\left \{z:|z|<1,Im \left \{ z \right \}>0 \right \}[/math]
    so it represents the northen hemisphere of a circle with radius 1.
    [math]f(z)=\frac{2z-i}{2-iz}[/math]
    i need to find what is the picture of [math]f(d^+)=? [/math]

    i tried to solve it like this:
    my area is bounded by a line and a curve.
    i want to see what each one transforms to
    curve points:
    f(1)=1
    f(-1)=-1
    f(i)=i/3
    so it will look like this
    http://i46.tinypic.com/3129rg4.gif

    line points:
    f(1)=1
    f(-1)=-1
    f(0)=-i/2
    so it looks like this
    http://i49.tinypic.com/dnnh2u.jpg

    and when i try and see where the inside goes :
    f(i/2)=0
    so the answer represents their intersection area

    http://i49.tinypic.com/14bi2qr.gif

    but my prof thinks otherwise
    http://i48.tinypic.com/2jff52r.jpg

    who is worng here? why?
    so its in their intersections
     
  2. jcsd
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