# Complex Analysis: Conformal mapping, a strange question I encountered

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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How is $z+\frac{1}{z}$ a line? Is the question actually asking for the preimage of the real line under the map $z\mapsto z+1/z$? (This is just a guess.) –  anon Apr 30 '12 at 8:37
Circles are lines in the complex extended plane. A half circle is a line in the upper half plane. –  Low Scores Apr 30 '12 at 8:39
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.) –  anon Apr 30 '12 at 8:50