Suppose I have two polynomials $p(z)$ and $q(z)$ and a positive integer $n$. Suppose I wanted to define $r(z)=(\frac{p(z)}{q(z)})^{1/n}$ on $\Omega$ such that r(z) was analytic and single valued. On what kind of $\Omega$ could I do this and in what way would I define $r(z)$. Finally, would knowing that all the coeffecients of $p(z)$ and $q(z)$ were real help in any way?
Here is my thought process so far. I know in order for it to be analytic we will have to avoid the roots of $q(z)$. As for single valued, I need to worry about the fact that $\alpha^{1/n}$ is not single valued for complex values. Will I need to use log to define our function? I have recently learned about branches(cuts,points, etc.). However, I am really confused about some of it still as seen by my struggles here. There is some stuff on this website and others somewhat related to this topic but I need an in depth focused discussion. Thanks for the help!
If $\Omega$ is a domain and $f$ is analytic on $\Omega$ and not equal to zero, then you can define $\log f$ to be analytic (and single valued!) on $\Omega$. So in your case, if neither $p(z)$ nor $q(z)$ is zero on $\Omega$, then $p(z)/q(z)$ is analytic and nonzero on $\Omega$, so you can define $\log p(z)/q(z)$, and then you can take the $n^{th}$ root to be $e^{1/n \log p(z)/q(z)}$.
Whenever you are taking to take something to a nonintegral power, always think about using logarithms, then you just need to understand how to define logarithms!