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I came across the following problem:
Let $f(z)=z+1/z$ for $z \in \mathbb C$ with $z \neq 0$.
Then how can I prove that "$f$ maps the unit circle to a subset of the real axis" ? Can someone point me in the right direction? Thanks in advance for your time.

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If $z$ is on the unit circle, then $z=e^{i\theta}$ for some $\theta\in[0,2\pi)$. Now $$f(e^{i\theta})=e^{i\theta}+e^{i(-\theta)}=\cos(\theta)+i\sin(\theta)+\cos(-\theta)+i\sin(-\theta)=2\cos(\theta).$$Here we've used the fact that $\cos(\theta)$ is an even function and $\sin(\theta)$ is odd.

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Hint If $z$ is on the unit circle, then

$$\frac{1}{z}=\frac{\bar z}{|z|^2}=\bar{z}$$

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  • $\begingroup$ Thanks a lot sir.I have got it. +1 from me. $\endgroup$ – user52976 Feb 8 '13 at 4:05

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