Given a complex valued polynomial $$P(z) = \sum_{n=0}^N c_n z^n, \ \ c_n \in \mathbb{C}$$
I'm trying to find some non-trivial series representation in terms of $z$ for $$f(z) = \frac{P^2(z)}{|P(z)|^2}.$$
However I'm at a loss for how to do this. It's not holomorphic, as it depends implicitly on the conjugate of $z$, so I don't think you can construct a power series or Laurent series, for instance, which would be the obvious thing to try.
Are there any other techniques that would work here? Maybe something like a Laurent series that works on arbitrary smooth functions?