Consider the Möbiustransformation $\displaystyle w=z^2.$ Find the image, by this transformation of:

a) First quadrant.

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

What I have:

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

Any help appreciated!!