I'm going to need the following theorem for a problem in here, and I was wondering if anyone could help me prove it?

Theorem:

For $\displaystyle p$ and $\displaystyle q$ polynomials, there exists a holomorphic extension $\displaystyle \text{Log}_h(p/q)=\ln|p/q|+i\arg(p/q)$ to the entire complex plane except along branch-cuts extending from each of the zeros and poles of the expression $\displaystyle p/q$ with $\displaystyle \arg(p/q)$ a continuous and analytic function in this domain. Additionally, the difference across each branch-cut is determined by the order of the branch point such that traveling in the positive sense around the branch point, this difference is $\displaystyle 2n\pi i$ for a zero of order n and $\displaystyle -2m\pi i$ for a pole of order m.

I believe I have a proof but not sure it's correct.