what is the geometry of $W\subset \mathbb{C}$?

Question

Consider $$ W=\{\lambda\in \mathbb{C};\;\exists (x_1,x_2)\in \mathbb{C}^2;\, \lambda=x_1\overline{x_2}\;\hbox{and}\;|x_2|=1\}. $$

$W$ is unbounded subset of $\mathbb{C}$, what is exactly $W$?

Thank you!

Answer 1

Let $z \in \mathbb{C}$ be arbitrary. It can be written as $z = |z|e^{i\phi}$ for some $\phi \in \mathbb{R}$.

Hence, $z = x_1\overline{x_2}$ where $x_1 = |z|$ and $x_2 = e^{-i\phi}$. We have $x_1, x_2 \in \mathbb{C}$ and $|x_2| = 1$.

Therefore, $W = \mathbb{C}$.