Answer to Question #198659 in Complex Analysis for smi

Question

If z + 1/z=2 cos theta , theta belongs to R.  Show that |z|=1 and for any n belongs to Z, z^n + 1/z^n  =2 cos n theta.

Answer 1

We note $|z|=r$. With this notation the imaginary part of $z+1/z$ is $(r-1/r) \sin\theta$. Now as $2\cos \theta$ is a real number, its imaginary part is zero. Therefore there are two cases :

1. $r-1/r=0 \rightarrow r=|z|=1$
2. $\sin \theta = 0 \rightarrow z\in \mathbb{R}$ but in this case we have $z=\pm 1, |z|=1$

Therefore, in any case we have $|z|=1$ and so $z$ can be written as $e^{i\theta}$. Using this expression we find that for any $n\in \mathbb{Z}$ we have

$z^n + 1/z^n = e^{in\theta} + e^{-in\theta}=2\cos n\theta$