Question: > 
> Show that the mapping $w=z+\frac1z$ maps circles $|z|=p(p\ne 1)$ onto
>  ellipses $$\frac{u^2}{(p+\frac1p)^2}+\frac{v^2}{(p-\frac1p)^2}=1.$$
> 
> 
> 

I can parametrize the circle by $z(t)=pe^{it}, \ 0 \leq t\leq 2\pi$. Then, $w(t)=pe^{it}+\frac{1}{pe^{it}},$ but then how can I show that this does map circles onto ellipses?

Answer: 
**Hint**: Consider $g=\frac{1}{f''-3}$ and [Liouville's theorem](http://en.wikipedia.org/wiki/Liouville%27s_theorem_%28complex_analysis%29) .
Another way to see this is to note that $f''(z)$ is an entire analytic function that omits an uncountable number of values (a small interval around $3$). Then by [Picard's little theorem](http://en.wikipedia.org/wiki/Picard_theorem), the function must be constant.

The method in the other answer is more elementary, but this is the way I approached the problem. Knowledge of Picard's theorem allows you to do it without expending any effort.