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I need to find a mapping for $z + \frac{1}{z}$ to the $x$-axis. I have that $f(i) = 0$, $f(-1)= -2$ and $f(1) = 2$. I am not sure as to what my professor seems to be asking us to think of in this example he used. What is the function that maps the line $z + \frac{1}{z}$ to the $x$-axis?

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    The preimage of the reals under this map is the real line (which is of course a circle on $\Bbb C^*$) in union with the unit circle. Is this relevant to your question? (To me, $z+\frac{1}{z}$ is a symbolic expression in the complex variable $z$. I don't know in what sense you understand it to be a line.)2012-04-30

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