I'm trying to determine the image of the Joukowski map $J(z)=-\frac 1 2 (z+1/z)$ on the upper half unit disc. I have read this solution:
Proving $-\frac{1}{2}(z+\frac{1}{z})$ maps upper half disk onto upper half plane
The first answer listed there just establishes that the transformation maps values into the upper half plane, but doesn't quite tell us the image. The second answer finds an inverse, but it makes the solution process mysterious because it starts from already knowing the answer (that the image is $\mathbb C -[-1,1]$).
I read the Ahlfors reference, and after writing (note that he writes $z$ for the value and $w$ for the function's variable)
$$ z = \frac 1 2 (w+1/w) $$
$$ w = z-\sqrt{z^2-1} $$
he said:
The sign of the root is uniquely determined by the condition $|w|<1$, for $(z-\sqrt{z^2-1})(z+\sqrt{z^2-1})=1$.
I certainly see why that equation is true, but not what it tells me about the magnitude of $w$. It nearly looks like the $w\overline w$, but since $z$ is complex, that can't be so. It also looks like he's taking the product of two roots and observing that it's 1. But I don't know the significance of that if there is any.