Quotient of complex polynomials ( alhfors complex analysis) ## The Problem }{Q(z)}=\sum\limits_{k=1}^{n}\frac{1}{(z-\alpha_k)}$ Thus $\frac{P(z)}{Q(z)}=\sum\limits_{k=1}^{n}\frac{P(z)}{Q'(z)(z-\alpha_k)}$ This is about as far as I have traveled any ideas? ## Possible Ideas I get the feeling that I should use : ![enter image description here]( But I am not sure of it's relevancy.

We know that, for some $K\in\mathbb{C}\setminus\\{0\\}$, $Q(z)=K(z-\alpha_1)(z-\alpha_2)\cdots(z-\alpha_n)$. So, by partial fraction decomposition, $\frac{P(z)}{Q(z)}$ can be written as$$\sum_{k=1}^n\frac{\beta_k}{z-\alpha_k}.$$Besides,$$\beta_k=\lim_{z\to\alpha_k}(z-\alpha_k)\sum_{k=1}^n\frac{\beta_k}{z-\alpha_k}=\lim_{z\to\alpha_k}\frac{P(z)}{\frac{Q(z)}{z-\alpha_k}}=\frac{P(\alpha_k)}{Q'(\alpha_k)}.$$