I'm also self studying the Ahlfors Complex Analysis book.
A question asks:
Suppose $R(z)$ is some rational function which is real on the circle |z|=1 in the complex plane. The question asks, how are the zeros and poles situated?
I have consulted and understood the argument made here:
If a rational function is real on the unit circle, what does that say about its roots and poles?
But, I can't understand why conclusions can be drawn for zeros/poles outside the unit circle: the equation $R(z) = \overline{R(1/\overline{z})}$ should hold only on the unit circle $|z|=1$, so I'm not sure why any conclusion can be made for zeros/poles $\alpha_i$ that are not on the unit circle.
Thanks for your help.
Best Answer
Two rational functions that agree on the unit circle are equal on their common domains, by the identity theorem for holomorphic functions. So the equation in your question holds for all complex numbers. In particular $a$ is a zero of $R$ if and only if $1/\overline{a}$ is a zero of $R$, and similarly $a$ is a pole of $R$ if and only if if $1/\overline{a}$ is a pole of $R$. In both cases the order is preserved.