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 and polynomials, there exists a holomorphic extension to the entire complex plane except along branch-cuts extending from each of the zeros and poles of the expression with 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 for a zero of order n and for a pole of order m.

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