I was recently exploring domain colouring, when the transformation $z \mapsto z^z$ changed the complex plane thus:
This made me wonder how we could mathematically describe the coloured 'boomerang' region, which represents the range of the function $z^z$ (the white areas have no values at their corresponding points). Of course, one way to describe the region would be,
$S = \{ \left(x, y \right) : \left(x, y \right) \in \mathbb{R}^2, x+iy = z^z, z \in \mathbb{C} \}$
Which corresponds to the range $\{ z : z \in \mathbb{C}, z = w^w, w \in \mathbb{C} \}$
But is there any way of finding the range in another form?
If $z=\exp W(\ln w)$ with $W$ a branch of the Lambert-$W$ function,$$z^z=\exp(z\ln z)=\exp(\ln w)=w.$$This works for all $w\in\Bbb C\setminus\{0\}$.