Result of a function (complex numbers)

I need to define the possible outcome of a function

$\displaystyle \displaystyle \[\omega \left( {\rm Z} \right) = \frac{{2i \cdot {\rm Z}}}{{{\rm Z} + 3}}\]$

whereas

$\displaystyle \displaystyle \[\omega = u + i \cdot v \in \mathbb{C}\]$

$\displaystyle \displaystyle \[{\rm Z} = x + i \cdot y \in \mathbb{C}\]$

and my $\displaystyle \displaystyle \[{\rm Z}\]$ obeys the equality

$\displaystyle \displaystyle \[\left| {{\rm Z} - 1} \right| = 2\]$

I find it fairly complicated, because while computer graphs it nicely as $\displaystyle \displaystyle \[\left| {u + i \cdot v} \right| = 1\]$ , my writings (fragment added below) and squared parts of equation were not going to be nearly as easily interpreted.

Maybe you know a smart and elegant way? It was not supposed to be a very complicated problem...

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