Rational function with absolute value $1$ on unit circle 
What is the general form of a rational function which has absolute value $1$ on the circle $|z|=1$? In particular, how are the zeros and poles related to each other?

So, write $R(z)=\dfrac{P(z)}{Q(z)}$, where $P,Q$ are polynomials in $z$. The condition specifies that $|R(z)|=1$ for all $z$ such that $|z|=1$. In other words, $|P(z)|=|Q(z)|$ for all $z$ such that $|z|=1$. What can we say about $P$ and $Q$?

Define $M(z)$ the same as the high-voted answer. And we know (if you want to prove it, it's easy), if $f(z)$ is analytic and so is $\ \overline{f(\bar z)}\ $analytic. We choose $f(z) = R(1/z)$, so $\overline{R(1/\bar z )}$ analytic and thus $M(z)$ is analytic. 
