Well I really need your help here because I need to sketch the curve $|z-1|=1$ in the z-plane and then its image under the $w=z^{2}$ but the thing is that I dont know how to sketch that function.
In class we only chose a line and then we saw that $w$ only open its argument, we covered too that when we have a circle it bend it like if we have a donut, but I dont know how this may helps.
Well, the question is, how can I sketch this function and its image?, I am lost in this 🙂 thanks in advance for your help I really need it.
Best Answer
Your circle is parametrized by $\gamma(t) = 1 + \exp(it)$, so its image under squaring is parametrized by $$ \gamma(t)^{2} = \bigl(1 + \exp(it)\bigr)^{2} = \dots. $$ Alternatively, if you prefer to work with real vectors instead of complex numbers, you can write $\gamma(t) = (1 + \cos t, \sin t)$, and note/recall that squaring sends $(x, y)$ to $(x^{2} - y^{2}, 2xy)$.