Show that the mapping $w=z+\frac1z$ maps the domain outside the circle $|z| = 1$ onto the rest of the $w$ plane.
I have already showed that circles $|z|=r_0 (r_0 \ne 1)$ are mapped onto ellipses $$\frac{u^2}{(r_0+\frac{1}{r_0})^2}+\frac{v^2}{(r-\frac{1}{r_0})^2}=1$$ with $r_0 \neq 1.$ In the course of proving this I had the equalities $u = (r_0 + \frac{1}{r_0}) \cos \theta$ and $v = (r_0 - \frac{1}{r_0}) \sin \theta$ with $0 \leq \theta \leq 2\pi$. For $|z| > 1$, my intuition tells me that values of $u$ and $v$ range over the $w$ plane, but I am not sure how to formally show this. I thought about letting the radii tend to infinity, but am not sure how the argument would go.