I'm trying to find the image of $|z-1|=1$ under the mapping $f(z)=z^2.$ I know that this is a circle of radius $1$ centered at $(1,0),$ given by $r=2\cos\theta.$ So I have $$z=2\cos\theta e^{i \theta}$$ and so $$f(z)=4\cos^2\theta e^{i 2 \theta.}$$
Is this correct and how do I interpret this geometrically?
Best Answer
It is correct. Geometrically, the image is a cardioid: $t\to (1+e^{it})^2$ with $t\in [0,2\pi)$.
P.S. Note that your expression $4\cos^2\theta e^{i 2 \theta.}$ implies that in polar coordinates we have $\rho(2\theta)=2(1+\cos(2\theta))$.