$|z-1|=1$ is obviously a circle centered at $x=1$ and $y=0$ on z-plane. However, I am having a hard time visualizing it in the $w=z^2$. Is there an analytic way of getting the equation for $z$ in $w$-plane?
How should I do it?
Work done so far:
Nothing, i am not sure how to begin!
Thanks in advance!
If $z = 1 + e^{it}$, $w = z^2 = (1+e^{it})^2 = 1 + 2 e^{it} + e^{2it}$.
If you want an equation for the real and imaginary components of $z^2$, say $w = u + i v$, that turns out to be $$ {u}^{4}+2\,{v}^{2}{u}^{2}+{v}^{4}-4\,{u}^{3}-4\,{v}^{2}u-4\,{v}^{2}=0$$