Sketch the image of $|z-1| = 1$ in the $z$-plane, and sketch its image under the mapping $\omega = z^2$.

Sketch the image of $|z-1| = 1$ in the $z$-plane, and sketch its image under the mapping $\omega = z^2$.

I'm not exactly sure what this is asking me to do. I understand that the image in the $z$-plane is a circle of radius $1$ centered at $(1,0)$, but how exactly am I supposed to map this? Is it just the square of each point?

• Right, the image is just the set containing $w = z^2$ for each $z$ on the circle. Hint: It is a cardioid. – arkeet Oct 13 '16 at 20:15
• It might help to think about it in polar coordinates, where the equation of the circle $|z-1|=1$ is $r = 2 \cos \theta$. – arkeet Oct 13 '16 at 20:18
• How would I find $\omega = z^2$ analytically? I can see how it looks like a cardioid by plotting a few points, but is there a particular expression for $z$ that can be evaluated here? – Oliver G Oct 14 '16 at 1:05