# Mobius transform question

1. Jan 13, 2010

### nhrock3


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

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