# Möbius transformation

• Dec 3rd 2011, 01:12 PM
Homing
Möbius transformation
Consider the Möbiustransformation $w=z^2.$ Find the image, by this transformation of:

b) The region bounded by $x=1,$ $y=1$ and $x+y=1.$

What I have:

$w=z^2=(x+iy)^2=x^2-y^2+2ixy,$ so let $f(z)=f(x+iy)=x^2-y^2+2ixy,$ but now I don't know what's next. At least for part a) I have $x,y\ge0$ and for part b) I have a triangle with vertices $(0,0), (0,1)$ and $(1,0),$ but I don't know how to solve the problem though.

Any help appreciated!!
• Dec 4th 2011, 04:37 AM
Homing
Re: Möbius transformation
• Dec 4th 2011, 05:09 AM
FernandoRevilla
Re: Möbius transformation
Hint Write the transformation in the form $\begin{Bmatrix} u=x^2-y^2\\v=2xy\end{matrix}$ and find previously the image of the boundaries.
• Dec 4th 2011, 11:59 AM
Homing
Re: Möbius transformation
You mean writing $w(x+iy)=u+vi$ ? For part a) I have $w(0)=0,$ but for part b) I have a triangle, it's just like putting the coordinates on $w$ ? I still don't get how to solve the problem. :(