$\omega(z)=z+\frac 1 z$ is univalent function? I need to find if > $\omega(z)=z+\frac 1 z$ is **univalent** function **My attempt:** First, I know that: $\omega(z)=z+\frac 1 z$ $=(x+\frac{x}{x^2+y^2})+i(y-\frac{y}{x^2+y^2})$ > Now what?

If _univalent_ means _injective_ , then no: $\omega(\pm i)=0$.

More generally, it is clear that $\omega(1/z)=\omega(z)$, and it is easy to prove that this is the only possibility.

In general, no rational function of degree $\ge 2$ is injective, by the fundamental theorem of algebra.